handbook of metric fixed point theory
TRANSCRIPT
HANDBOOK OF METRIC FIXED POINT THEORY
Handbook of Metric Fixed Point Theory
Edited by
William A. Kirk Department of Mathematics, The University of Iowa, Iowa City, lA, U.S.A.
and
Brailey Sims School of Mathematical and Physical Sciences, The University of Newcastle, Newcastle, Australia
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-5733-4 ISBN 978-94-017-1748-9 (eBook) DOI 10.1007/978-94-017-1748-9
or
and
Printed on acid-free paper
All Rights Reserved © 2001 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2001 Softcover reprint of the hardcover 1 st edition 200 1 No part of the material protected by this copyright notice may be reproduced utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage retrieval system, without written permission from the copyright owner.
Contents
Preface
Contraction mappings and extensions
W. A. Kirk
1.1 Introduction
1.2 The contraction mapping principle
1.3 Further extensions of Banach's principle
1.4 Caristi's theorem
1.5 Set-valued contractions
1.6 Generalized contractions
1. 7 Probabilistic metrics and fuzzy sets
1.8 Converses to the contraction principle
1.9 Notes and remarks
2
Examples of fixed point free mappings
B. Sims
2.1 Introduction
2.2 Examples on closed bounded convex sets
2.3 Examples on weak' compact convex sets
2.4 Examples on weak compact convex sets
2.5 Notes and remarks
3
Classical theory of nonexpansive mappings
K. Goebel and W. A. Kirk
3.1 Introduction
3.2
3.3
3.4
3.5
Classical existence results
Properties of the fixed point set
Approximation
Set-valued nonexpansive mappings
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xi
1
3
7
14
15
18
20
23
25
35
35
36
40
43
47
49
49
50
64
69
78
vi
3.6 Abstract theory
4
Geometrical background of metric fixed point theory
S. Pros
5
4.1 Introduction
4.2 Strict convexity and smoothness
4.3 Finite dimensional uniform convexity and smoothness
4.4
4.5
4.6
Infinite dimensional geometrical properties
Normal structure
Bibliographic notes
Some moduli and constants related to metric fixed point theory
E. L. Fuster
6
5.1 Introduction
5.2
5.3
Moduli and related properties
List of coefficients
Ultra-methods in metric fixed point theory
M. A. Khamsi and B. Sims
7
6.1 Introduction
6.2
6.3
6.4
6.5
6.6
Ultrapowers of Banach spaces
Fixed point theory
Maurey's fundamental theorems
Lin's results
Notes and remarks
Stability of the fixed point property for nonexpansive mappings
J. Garcia-Falset, A. Jimenez-Me/ado and E. Llorens-Fuster
7.1 Introduction
7.2 Stability of normal structure
7.3 Stability for weakly orthogonal Banach lattices
7.4 Stability of the property M(X) > 1
7.5 Stability for Hilbert spaces. Lin's theorem
7.6 Stability for the T-FPP
7.7
7.8
Further remarks
Summary
79
93
93
93
98
108
118
127
133
133
134
157
177
177
177
186
193
195
197
201
201
204
212
217
223
228
231
236
Contents
8
Metric fixed point results concerning measures of noncompactness
T. Dominguez, M. A. Japon and G. Lopez
9
8.1 Preface
8.2
8.3
8.4
8.5
8.6
8.7
8.8
Kuratowski and Hausdorff measures of noncompactness
¢-minimal sets and the separation measure of noncompactness
Moduli of noncompact convexity
Fixed point theorems derived from normal structure
Fixed points in NUS spaces
Asymptotically regular mappings
Comments and further results in this chapter
Renormings of £1 and Co and fixed point properties
P. N. Dowling, C. J. Lennard and B. Turett
9.1 Preliminaries
9.2 Renormings of £1 and Co and fixed point properties
9.3 Notes and remarks
10
Nonexpansive mappings: boundary/inwardness conditions and local theory
w. A. Kirk and C. H. Morales
10.1 Inwardness conditions
10.2 Boundary conditions
11
10.3 Locally nonexpansive mappings
10.4 Locally pseudocontractive mappings
10.5 Remarks
Rotative mappings and mappings with constant displacement
W. Kaczor and M. Koter-Morgowska
11.1 Introduction
12
11.2 Rotative mappings
11.3 Firmly Iipschitzian mappings
11.4 Mappings with constant displacement
11.5 Notes and remarks
Geometric properties related to fixed point theory in some Banach function lattices
S. Chen, Y. Cui, H. Hudzik and B. Sims
12.1 Introduction
VII
239
239
240
244
248
252
257
260
264
269
269
271
294
299
299
301
308
310
320
323
323
323
330
333
336
339
339
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12.2 Normal structure, weak normal structure, weak sum property, sum property and uniform normal structure 343
12.3 Uniform rotundity in every direction 356
12.4 B-convexity and uniform monotonicity 358
12.5 Nearly uniform convexity and nearly uniform smoothness 362
12.6 WORTH and uniform nonsquareness 367
12.7 Opial property and uniform opial property in modular sequence spaces 368
12.8 Garcia-Falset coefficient 377
12.9 Cesaro sequence spaces 378
12.10 WCSC, uniform opial property, k-NUC and UNS for cesp 380
13 391
Introduction to hyperconvex spaces
R. Espinola and M. A. Khamsi
14
13.1
13.2
13.3
13.4
13.5
13.6
13.7
13.8
Preface
Introduction and basic definitions
Some basic properties of hyperconvex spaces
Hyperconvexity, injectivity and retraction
More on hyperconvex spaces
Fixed point property and hyperconvexity
Topological fixed point theorems and hyper convexity
Isbell's hyperconvex hull
13.9 Set-valued mappings in hyperconvex spaces
13.10 The KKM theory in hyperconvex spaces
13.11 Lambda-hyperconvexity
Fixed points of holomorphic mappings: a metric approach
T. Kuczumow, S. Reich and D. Shoikhet
14.1
14.2
14.3
14.4
14.5
14.6
14.7
Introduction
Preliminaries
The Kobayashi distance on bounded convex domains
The Kobayashi distance on the Hilbert ball
Fixed points in Banach spaces
Fixed points in the Hilbert ball
Fixed points in finite powers of the Hilbert ball
14.8 Isometries on the Hilbert ball and its finite powers
14.9 The extension problem
14.10 Approximating sequences in the Hilbert ball
14.11 Fixed points in infinite powers of the Hilbert ball
14.12 The Denjoy-Wolff theorem in the Hilbert ball and its powers
14.13 The Denjoy-Wolff theorem in Banach spaces
391
393
394
399
405
411
415
418
422
428
431
437
437
438
440
447
450
454
460
465
469
472
481
483
490
Contents
15
14.14 Retractions onto fixed point sets
14.15 Fixed points of continuous semigroups
14.16 Final notes and remarks
Fixed point and non-linear ergodic theorems for semigroups of non-linear mappings
A. Lau and W. Takahashi
15.1 Introduction
16
15.2 Some preliminaries
15.3 Submean and reversibility
15.4 Submean and normal structure
15.5 Fixed point theorem
15.6 Fixed point sets and left ideal orbits
15.7 Ergodic theorems
15.8 Related results
Generic aspects of metric fixed point theory
S. Reich and A. J. Zaslavski
16.1 Introduction
17
16.2 Hyperbolic spaces
16.3 Successive approximations
16.4 Contractive mappings
16.5 Infinite products
16.6 (F}-attracting mappings
16.7 Contractive set-valued mappings
16.8 Nonexpansive set-valued mappings
16.9 Porosity
Metric environment of the topological fixed point theorems
K. Goebel
18
17.1 Introduction
17.2 Schauder's theorem
17.3 Minimal displacement problem
17.4 Optimal retraction problem
17.5 The case of Hilbert space
17.6 Notes and remarks
Order-theoretic aspects of metric fixed point theory
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496
502
507
517
517
518
519
523
527
532
538
545
557
557
557
558
561
564
567
568
569
570
577
577
579
586
597
604
608
613
x
J. Jachymski
19
18.1 Introduction
18.2 The Knaster-Tarski theorem
18.3 Zermelo's fixed point theorem
18.4 The Tarski-Kantorovitch theorem
Fixed point and related theorems for set-valued mappings
G. Yuan
19.1
19.2
19.3
19.4
19.5
19.6
19.7
Index
Introduction
Knaster-Kuratowski-Mazurkiewicz principle
Ky Fan minimax principle
Ky Fan minimax inequality-I
Ky Fan minimax inequality-II
Fan-Glicksberg fixed points in G-convex spaces
Nonlinear analysis of hyperconvex metric spaces
613
614
623
630
643
643
644
651
653
659
662
666
691
Preface
The presence or absence of a fixed point is an intrinsic property of a map. However, many necessary or sufficient conditions for the existence of such points involve a mixture of algebraic, order theoretic, or topological properties of the mapping or its domain. Metric fixed point theory is a rather loose knit branch of fixed point theory concerning methods and results that involve properties of an essentially isometric nature. That is, the class of mappings and domains satisfying the properties need not be preserved under the move to an equivalent metric. It is this fragility that singles metric fixed point theory out from the more general topological theory, although, as many of the entries in this Handbook serve to illustrate, the divide between the two is often a vague one.
The origins of the theory, which date to the latter part of the nineteenth century, rest in the use of successive approximations to establish the existence and uniqueness of solutions, particularly to differential equations. This method is associated with the names of such celebrated mathematicians as Cauchy, Liouville, Lipschitz, Peano, Fredholm and, especially, Picard. In fact the precursors of a fixed point theoretic approach are explicit in the work of Picard. However, it is the Polish mathematician Stefan Banach who is credited with placing the underlying ideas into an abstract framework suitable for broad applications well beyond the scope of elementary differential and integral equations. Around 1922, Banach recognized the fundamental role of 'metric completeness'; a property shared by all of the spaces commonly exploited in analysis. For many years, activity in metric fixed point theory was limited to minor extensions of Banach's contraction mapping principal and its manifold applications. The theory gained new impetus largely as a result of the pioneering work of Felix Browder in the mid-nineteen sixties and the development of nonlinear functional analysis as an active and vital branch of mathematics. Pivotal in this development were the 1965 existence theorems of Browder, Gohde, and Kirk and the early metric results of Edelstein. By the end of the decade, a rich fixed point theory for nonexpansive mappings was clearly emerging and it was equally clear that such mappings played a fundamental role in many aspects of nonlinear functional analysis with links to variational inequalities and the theory of monotone and accretive operators.
Nonexpansive mappings represent the limiting case in the theory of contractions, where the Lipschitz constant is allowed to become one, and it was clear from the outset that the study of such mappings required techniques going far beyond purely metric arguments. The theory of nonexpansive mappings has involved an intertwining of geometrical and topological arguments. The original theorems of Browder and Giihde exploited special convexity properties of the norm in certain Banach spaces, while Kirk identified the underlying property of 'normal structure' and the role played by weak compactness. The early phases of the development centred around the identification of spaces whose bounded convex sets possessed normal structure, and it was soon
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xii
discovered that certain weakenings and variants of normal structure also sufficed. By the mid-nineteen seventies it was apparent that normal structure was a substantially stronger condition than necessary. And, armed with the then newly discovered GoebelKarlovitz lemma the quest turned toward classifying those Banach spaces in which all nonexpansive self-mappings of a nonempty weakly compact convex subset have a fixed point. This has yielded many elegant results and led to numerous discoveries in Banach space geometry, although the question itself remains open. Asymptotic regularity of the averaged map was an important contribution of the late seventies, that has been exploited in many subsequent arguments. A turning was the discovery by Alspach in 1980 of a nonempty weakly compact convex subset of LIla, 1] which admitted a fixed point free nonexpansive mapping. This was quickly followed by a number of surprising results by Maurey. Maurey's ideas were original and set the stage for many of the more recent advances. The asymptotic embedding techniques recently initiated by Lennard and Dowling represent a novel development. For example, combined with one of Maurey's results their work shows that a subspace of L 1 [0, 1] is reflexive if and only if all of its nonempty closed bounded convex subsets have the fixed point property for nonexpansive mappings.
All of these developments are reflected in the contents of this Handbook. It is designed to provide an up-to-date primary resource for anyone interested in fixed point theory with a metric flavor. It should be of value for those wishing to find results that might apply to their own work and for those wishing to obtain a deeper understanding of the theory. The book should be of interest to a wide range of researchers in mathematical analysis as well as to those whose primary interest is the study of fixed point theory and the underlying spaces. The level of exposition is directed toward a wide audience, embracing students and established researchers.
The focus of the book is on the major developments of the theory. Some information is put forth in detail, making clear the underlying ideas and the threads that link them. At the same time many peripheral results and extensions leading to the current state of knowledge often appear without proof and the reader is directed to primary sources elsewhere for further information. Because so many authors have been involved in this project, no effort has been made to attain a wholly uniform style or to completely avoid duplication. At the same time an attempt has been made to keep duplication to a minimum, especially where major topics are concerned.
The book begins with an overview of metric contraction principles. Then, to help delineate the theory of nonexpansive mappings and alert the reader to its subtleties, this is followed by a short chapter devoted to examples of fixed point free mappings. A survey of the classical theory of nonexpansive mappings is taken up next. After this, various topics are discussed more or less randomly, including the underlying geometric foundations of the theory in Banach spaces, ultrapower methods, stability of the fixed point property, asymptotic renorming techniques, hyperconvex spaces, holomorphic mappings, generic properties of the theory, nonlinear ergodic theory, rotative mappings, pseudocontractive mappings and local theory, the non expansive theory in Banach function lattices, the topological theory in a metric environment, order-theoretic aspects of the theory and set-valued mappings.
We wish to thank all the contributors for enthusiastically supporting the project and for giving generously of their time and expertise. Without their contributions this
PREFACE xiii
Handbook would not have been possible. Special thanks are due to Mark Smith, who did almost all of the type-setting and formatting - an invaluable contribution - and to Tim Dalby. Both devoted many hours to proofreading the original manuscripts, and thanks to their efforts many oversights (including some mathematical ones) have been corrected. Lastly we must thank Liesbeth Mol, whose enthusiasm, understanding and gentle encouragement made our dealings with the publishers both a pleasant and an effective one.
ART KIRK AND BRAILEY SIMS, IOWA CITY AND NEWCASTLE, APRIL 2001
Chapter 1
CONTRACTION MAPPINGS AND EXTENSIONS
W. A. Kirk
Department of Mathematics
The University of Iowa
Iowa City, IA 52242-1419 USA
kirklllmath.uiowa.edu
1. Introduction
A complete survey of all that has been written about contraction mappings would appear to be nearly impossible, and perhaps not really useful. In particular the wealth of applications of Banach's contraction mapping principle is astonishingly diverse. We only attempt to touch on some of the high points of this profound and seminal development in metric fixed point theory.
The origins of metric contraction principles and, ergo, metric fixed point theory itself, rest in the method of successive approximations for proving existence and uniqueness of solutions of differential equations. This method is associated with the names of such celebrated nineteenth century mathematicians as Cauchy, Liouville, Lipschitz, Peano, and, especially, Picard. In fact the precursors of the fixed point theoretic approach are explicit in the work of Picard. However it is the Polish mathematician Stefan Banach who is credited with placing the ideas underlying the method into an abstract framework suitable for broad applications well beyond the scope of elementary differential and integral equations. Accordingly we take Banach's formulation as our point of departure in Section 2. It is remarkable in its simplicity, yet it is perhaps the most widely applied fixed point theorem in all of analysis. This is because the contractive condition on the mapping is -easy to test and because it requires only the structure of a complete metric space for its setting.
The key ingredients of the Contraction Mapping Principle as it first appeared in Banach's 1922 thesis [3] are these. (M,d) is a complete metric space and T: M -t M is a contraction mapping. Thus there exists a constant k < 1 such that
d (T (x), T (y)) :'S kd (x,y)
for each x,y E M. From this one draws three conclusions:
(i) T has a unique fixed point, say Xo.
(ii) For each x E M the Picard sequence {rn (x)} converges to Xo.
(iii) The convergence is uniform if M is bounded.
W.A. Kirk and B. Sims (etis.), Handbook of Metric Fixed Point Theory, 1-34. © 2001 Kluwer Academic Publishers.
2
In fact condition (iii) can be put in much more explicit form in terms of error estimates.
(iiih d(Tn(x),xo)::::: 1 ~ kd(x,T(x)) for each x E M and n ~ 1;
(iiih d (rn+1 (x) ,xo) ::::: 1 ~ kd (rn+1 (x) ,Tn (x)) for each x E M and n ~ 1.
In particular, there is an explicit rate of convergence:
(iv) d (rn+l (x) ,xo) ::::: kd(rn(x),xo).
A primary early example of an extension of Banach's principle is a theorem of Caccioppoli [16] which asserts that the Picard iterates of a mapping T converge in a complete metric space M provided for each n ~ 1, there exists a constant en such that
d (rn (x) , rn (y)) ::::: end (x, y)
for all x, y E M, where L~l en < 00.
The Contraction Mapping Principle has seen many other extensions particularly to mappings for which conclusions (i) and (ii) hold. In many of these instances (especially ones which reduce to Banach's principle under an appropriate renorming) it is possible to obtain (iii) as well. We give an overview of these facts below. We begin with an explicit proof of Banach's theorem (one of many) along with one of its canonical applications. We then take up many of the extensions. We conclude with a brief discussion of converses of Banach's theorem. Many other parts of this volume are devoted to the limiting case k = 1, where in general it is possible to conclude at most that T has a (not necessarily unique) fixed point.
Before proceeding we turn to a simple example to illustrate the usefulness of the contraction mapping principle. Consider the Volterra integral equation
u(x)=f(x)+ foxF(X,Y)U(Y)dY, (1.1)
where f and the kernel F are defined and continuous on, respectively, [0, a] and [0, a] x [0, a]. By employing the standard method of successive approximations it is possible to show that 1.1 has a unique continuous solution for any F. On the other hand, if the operator T : C [0, a] --> C [0, a] is defined by setting
T(u(x)) = f(x) + foX F(x,y)u(y)dy, (1.2)
then it is easy to see that for u, vEe [0, a]
liT (u) - T (v) II ::::: aK Ilu - vII
where K = SUPO~x,y~a IF (x,y)1 and 11·11 is the usual supremum norm on C [0, a]. Banach's contraction principle thus immediately yields a unique solution (with convergence of successive approximations) on any interval for which aK < 1. The problem is that in order to obtain a solution one must either restrict the size of the interval [0, a] or the magnitude of the kernel F. This is not serious since in the first instance standard continuation arguments can then be applied to extend the solution.
Contraction mappings 3
On the other hand, A. Bielecki [6J discovered another way to remedy this 'problem'. By assigning a new norm 1I-II.~; >. > 0, to C [0, aJ as follows:
Ilull)' = sup [exp(->.x) lu(x)lJ, O:S;x:S;a
it is possible to show that for all u, v E C [0, aJ , K IITea) - T (v)ll>. ::; ~ lIu - vII)',
(1.3)
where K is defined as above. (For details, see e.g., [39J.) It is then clear that for>. sufficiently large T is indeed a contraction mapping on the Banach space (C [0, aJ , 11·11),). A direct application of the contraction mapping principle now yields the desired solution.
2. The contraction mapping principle
Let (M, d) be a metric space. A mapping T : M -> M is said to be lipschitzian if there is a constant k 2: ° such that for all x, y E M
d(T(x),T(y))::; kd(x,y). (2.1)
The smallest number k for which 2.1 holds is called the Lipschitz constant of T.
Definition 2.1 A lipschitzian mapping T : M -> M with Lipschitz constant k < 1 is said to be a contmction mapping.
Theorem 2.2 (Banach's Contmction Mapping Principle). Let (M,d) be a complete metric space and let T : M -> M be a contmction. Then T has a unique fixed point Xo. Moreover, for each x E M,
lim Tn (x) = Xo n-HX)
and in fact for each x E M,
n = 1,2,···.
Proof. Since T is a contraction mapping we know that for each x E M,
d(T(x),T2(X)) ::; kd(x,T(x)).
Adding d(x, T(x)) to both sides of the above yields
d(x,T(x)) + d(T(x),T2(x)) ::; d(x,T(x)) + kd(x,T(x))
which can be rewritten
d(x,T(x)) - kd(x,T(x))::; d(x,T(x)) - d(T(x),T2(x)).
This in turn is equivalent to
d(x, T(x)) ::; (1 - k)-l [d(x, T(x)) - d(T(x), T 2(x))J.
Now define the function cp : M -> ~+ by setting cp(x) = (1- k)-ld(x, T(x)), for x E M. This gives us the basic inequality
d(x,T(x)) ::; cp(x) - cp(T(x)), xEM.
4
Therefore if we fix x E M and take m, n E N with n < m, we obtain
m
d(m(x),Tm+1(x)):S Ld(T(x),T+1(x)) :S rp(Tn(x)) - rp(rm+1(x)). i=n
(Notice that the last inequality comes from cancelation in the telescoping sum.) In particular by taking n = 1 and letting m ..... 00 we conclude that
00
Ld(T(x),Ti+1(x)):s rp(T(x)) < 00.
;=1
This implies that {m(x)} is a Cauchy sequence. Since M is complete there exists Xo E M such that
lim m(x) = Xo n~oo
and since T is continuous
Xo = lim m(x) = lim m+1(x) = T(xo). n--+oo n--+oo
Thus Xo is a fixed point of T.
In order to see that Xo is the only fixed point of T, suppose T (y) = y. Then by what we have just shown
Xo = lim m(y) = y. n~oo
Returning to the inequality
d(m(x),rm+1(x)) :S rp(m(x)) - rp(Tm+1(x)),
upon letting m -+ 00 we see that
d(m(x),xo):S rp(m(x)) = (1- k)-ld(Tn(x),m+1(x)).
Since (1 - k)-ld(Tn(x), m+1(x)) :S 1": k d(x, T(x)) we obtain
kn d(m(x),xo) :S 1- kd(x,T(x)).
This provides an estimate on the rate of convergence for the sequence {m(x)} which depends only on d (x, T (x)). •
While the above estimate is sharp, in practical situations it might be necessary to approximate the values of m(x). Here a natural question arises. If one replaces the sequence {m(x)} with {Yn} where Yo = x and Yn+1 is 'approximately' T(Yn), then under what conditions will it still be the case that limn~oo Yn = xo? The following positive answer to this question was given by Ostrowski in [87]. (For a recent generalization of this result as well as related literature, see [52].)
Theorem 2.3 Let (M, d) be a complete metric space, let T : M ..... M be a contraction mapping with Lipschitz constant k E (0,1), and suppose Xo E M is the fixed point ofT. Let {en} be a sequence of positive numbers for which limn~oo en = 0, let Yo E M, and suppose {Yn} C M satisfies
n= 1,2,···.
Contraction mappings
Then limn -+oo Yn = Xo·
Proof. Let Yo = x and observe that
Thus
d(Tm+1(x), Ym+l) :S d(T(rm(x» , T(Ym» + d(T(Ym), Ym+l)
:S kd(Tm(x),Ym) +Em m
:S Lkm-iEi. i=O
m
:S Lkm-iEi+d(Tm+1(x),xo). i=O
5
Now let 10 > O. Since limn -+oo En =-0 there exists N E N such that for m ;::: N, Em :S E. Then .
m N m
L km-iEi = L km-iEi + L km-iEi i=O i=O i=N + 1
N m
:Skm-NLkN-iEi+E L km- i. i=O i=N+l
Hence
m . (kN +1 ) lim ~ km-'Ei:S 10 -- • m--->oo~ 1-k
i=O
Since 10 > 0 is arbitrary, and since limm-+ood(rrn+1(x),xo) limm--->oo Ym+ 1 = XO·
o we conclude that
• The following theorem is noteworthy in that the mapping T is not even assumed to be continuous!
Theorem 2.4 Suppose (M, d) is a complete metric space and suppose T : M -+ Mis. a mapping for which TN is a contraction mapping for some positive integer N. Then T has a unique fixed point.
Proof. By Banach's Theorem TN has a unique fixed point x. However,
so T(x) is also a fixed point of TN. Since the fixed point of TN is unique, it must be the case that T(x) = x. Also, if T(y) = Y then TN(y) = y proving (again by uniqueness) that y = x. •
In considering lipschitzian mappings an obvious question that arises immediately is whether it is possible to weaken the contraction assumption even a little bit and still obtain the existence of fixed points. In a broad sense the answer is no and here is an example. Begin with the complete metric space e[O,l] and consider the closed
6
subspace M of C[O, 1] consisting of those mappings f E C[O,I] for which f(l) = 1. Since M is a closed subspace of a complete metric space, M is itself complete. Now define T : M -> M by taking T(f) to be the function in M obtained by setting
T(f)(t) = tf(t) t E [0,1].
If f,g E M then IT(f) - T(g)1 E C[O, 1], so by the Maximum Value Theorem IT(f)T(g)1 attains its maximum value at some point to E [0,1]. We then have
d(T(f), T(g)) = sup IT(f)(t) - T(g)(t)1 = to If(to) - g(to)1 ::::: d(f, g). tEIO,l]
But if f f= 9 it must be the cast that f(t) f= g(t) for some t E [0,1] and since f(l) = g(l) = 1, this in turn implies to < 1. Therefore, if f,g E M and f f= g,
d(T(f),T(g)) < d(f,g).
Now suppose T(f) = f for f E M. This implies that for each t E [0,1], f(t) = tf(t). This implies that f(t) = 0 for all t E [0,1). On the other hand, f(l) = 1. This contradicts the assumption that f is continuous, so T can have no fixed point in M. Therefore the Banach Contraction Mapping principle does not even extend to the following slightly more general class of mappings.
Definition 2.5 A mapping T : M -> M is said to be contractive if
d(T(x),T(y)) < d(x,y)
for each x, y E M with x f= y.
The next obvious question is whether there is a meaningful fixed point result for the contractive mappings. The answer is yes, but the class of spaces to which it applies is much more restrictive.
Theorem 2.6 Let (M, d) be a compact metric space and let T : M -> M be a contractive mapping. Then T has a unique fixed point Xo, and moreover, for each x E M, limn~ooTn(x) = Xo.
Proof. The existence of a fixed point for T is easy. Introduce the mapping 1jJ : M ->
~+ by setting
1jJ(x) = d(x,T(x)), xEM.
Then 1jJ is continuous and bounded below, so 1jJ assumes its minimum value at some point Xo E M. Since Xo f= T(xo) implies
1jJ(T(xo)) = d(T(xo),T2(xo)) < d(xo,T(xo)) = 1jJ(xo)
it must be the case that Xo = T(xo).
Now let x E M and consider the sequence {d(Tn(x), xon. If m(x) f= xo,
d(Tn+1(x),xo) = d(m+1(x),T(xo)) < d(m(x),xo),
so {d(Tn(x), xon is strictly decreasing (until perhaps it reaches xo). Consequently the limit
r = lim d(Tn(x),xo) n~oo
Contraction mappings 7
exists and r 2: o. Also, since M is compact, the sequence {T'(x)} has a convergent subsequence {Tnk(x)}, say limk-+ooTnk(x) = z E M. Since {Tn(x)} is decreasing,
r = d(z,xo) = lim d(T'k(X),XO) = lim d(Tnk+L(X),xo) = d(T(z),xo). k--+oo k--+oo
But if z i- Xo then d(T(z),xo) = d(T(z),T(xo» < d(z,xo). This proves that any convergent subsequence of {T'(x)} converges to Xo, so it must be the case that
lim Tn(x) = Xo. n .... oo
• 3. Further extensions of Banach's principle
The strength of the Contraction Mapping Principle lies in the fact that the underlying space is quite general (complete metric) while the conclusion is very strong, including even error estimates. There have been numerous extensions of a milder form of Banach's principle which asks only that the fixed point be unique and that the Picard iterates of the mapping always converge to this fixed point .. We discuss some of the more well known of these, especially from a historical perspective, in this section.
The first such generalization to receive significant attention is the following result of Rakotch [98].
Theorem 3.1 Let M be a complete metric space and suppose f : M --> M satisfies
d (J (x),J (y» ~ a (d (x,y» d (x,y) for each x,y E M,
where a : R+ --> [0,1) is monotonically decreasing. Then f has a unique fixed point, X, and {r (x)} converges to x for each x E M.
Subsequently Boyd and Wong [8] obtained a more general result. In this theorem it is assumed that 'IjJ : R+ --> R+ is upper semicontinuous from the right (that is, rn ! r 2: O:=}- limsup'IjJ(rn) ~ 'IjJ(r».
n .... oo
Theorem 3.2 Let M be a complete metric space and suppose f : M --> M satisfies
d (J (x), f (y» ~ 'IjJ (d (x,y» for each x,y EM,
where 'IjJ : R+ --> [0,(0) is upper semi-continuous from the right and satisfies 0 ~ 'IjJ (t) < t for t > O. Then f has a unique fixed point, x, and {r (x)} converges to x for each XEM.
Proof. Fix x E M and let Xn = fn(x), n = 1,2,· ... We break the argument into two steps.
Step 1. liIDn .... ood(xn'Xn+l) = o. Proof Since f is contractive the sequence {d(xn , X n+1)} is monotone decreasing and bounded below so limn .... "" d(xn, X n+1) = r 2: o. Assume r > O. Then
8
Step 2. {xn} is a Cauchy sequence.
Proof Suppose not. Then there exists I': > 0 such that for any kEN, there exist mk > nk 2:: k such that
(3.1)
Furthermore, assume that for each k, mk is the smallest number greater than nk for which (3.1) holds. In view of Step 1 there exists ko such that k 2:: ko ~ d (Xk, Xk+l) ::; 1':. For such k we have
1':::; d(xmk,xnk )::; d(xmk,xmk-d +d(Xmk-l,Xnk )
::; d(xmk,xmk-d +1':::; d(Xk,Xk-l) +1':
This proves limk~oo d (xmk , x nk ) = 1':. On the other hand,
d (xm., Xnk) ::; d (Xmk , xmk+d + d (Xmk+l, Xnk+l) + d (Xnk+l, Xnk)
::; 2d (Xk, Xk-l) + 'Ij; (d (Xmk,Xnk ))·
It follows that I': ::::: 'Ij; (I':) - a contradiction.
The proof is completed by observing that since {fn(xn is a Cauchy sequence and M is complete, limn~oo fn(x) = Z E M. Since f is continuous, f(z) = z. Uniqueness of z follows from the contractive condition. •
REMARK. Boyd and Wong also show in [8] that if the space M is metrically convex, then the upper semi continuity assumption on 'Ij; can be dropped. Matkowski has extended this fact even further in [77] by showing that it suffices to assume that 'Ij; is continuous at 0 and that there exists a sequence tn 10 for which 'Ij; (tn ) < tn.
Since it is the explicit control over the error term that contributes so much to the wide-spread usefulness of Banach's principle, the following variant of the Boyd-Wong theorem due to Browder [11] is also of interest.
Theorem 3.3 Let X be a complete metric space and M a bounded subset of X. Suppose f : M -t M satisfies
d (j (x), f (y)) ::; 'Ij; (d (x,y)) for each x, y E M,
where 'Ij; : [0,00) -t [0,00) is monotone nondecreasing and continuous from the right, such that 'Ij; (t) < t for all t > O. Then there is a unique element x E M such that {fn (xn converges to x for each x EM. Moreover, if do is the diameter of M, then
and 'lj;n (do) -t 0 as n -t 00.
Another variant is due to Matkowski [76]. In this result the continuity condition on 'Ij; is replaced with another condition.
Theorem 3.4 Let M be a complete metric space and suppose f : M -t M satisfies
d (J (x), f (y)) ::::: 'Ij; (d (x,y)) for each x, y E M,
Contraction mappings 9
where 'If; : (0,00) ---+ (0,00) is monotone nondecreasing and satisfies limn--+oo 'If;n (t) = 0 for t > O. Then f has a unique fixed point x, and liIDn->oo d (r (x) , x) = 040r every xEM.
Proof. Fix x E M and let Xn =In(x), n = 1,2,···. As before we break the argument into two steps.
Step 1. limn->oo d(xn, Xn+l) = O.
Proof 0 ~ limsupd(xn ,xn+1) ~ limsup'lf;n(d(x,Xl)) = O. n--+oo n--+oo
Step 2. {xn} is a Cauchy sequence.
Proof Since 'If;n (J) ---+ 0 for f > 0, 'If; (c) < c for anye: > O. In view of Step 1 given any e: > 0 it is possible to choose n so that
Now let
K (xn,c) = {x EM: d(x,xn) ~ e:}.
Then if z E K (xn,c),
d(J(z) ,xn) ~ d(J (z) ,f(xn)) +d(J (xn) ,xn) ~ 'If; (d(z,xn)) +d(xn+l,xn) ~ 'If; (e:) + (c - 'If; (e:)) = e:.
Therefore I: K(xn,c) ---+ K(xn,c) and it follows that d(xm,xn) ~ e: for all m::::: n. This completes Step 2.
The conclusion of the proof follows as in Theorem 3.2. • A somewhat different approach which has also received substantial attention is the following formulation due to Meir and Keeler [78].
Theorem 3.5 Let (M, d) be a complete metric space and suppose I : M ---+ M satisfies the condition: Given e: > 0 there exists 8 > 0 such that
e: ~ d(x,y) ~ c +8 ~ d(J(x) ,/(y)) < e:.
Then I has a unique fixed point x, and limn->oo r (x) = x for each x E M.
Clearly the Meir-Keeler condition implies that I is contractive
(X#y~d(J(x),/(y))<d(x,y)).
Thus I is continuous, and if I has a fixed point it must be unique. Also it is easy to see that the Meir-Keeler condition implies d (r (x), r+1 (x)) i 0 as n -+ 00. It only remains to show that in fact {r (x)} is a Cauchy sequence and this is easily accomplished by assuming the contrary and obtaining a contradiction.
For a recent variant of the Meir-Keeler theorem in ordered Banach spaces see [22]. Also, following in the spirit of the Boyd-Wong approach, T. C. Lim [70] has recently characterized the Meir-Keeler condition in terms of a function 'IiJ.
10
Finally we take up a variant of Rakotch's approach due to Geraghty [38J. In this case we also present the details.
Let S denote the class of those functions a : jR+ -> [0,1) which satisfy the simple condition a(tn) -> 1 =} tn -> O.
Theorem 3.6 Let (M, d) be a complete metric space, let J : M -> M, and suppose there exists a E S such that Jar each x, y E M,
d(f(x),J(y)) :::: a(d(x,y))d(x,y).
Then J has a unique fixed point Z E M, and {r(x)} converges to z, Jar each x E M.
Proof. Fix x E M and let Xn = In(x), n = 1,2,···. Yet again we break the argument into two steps.
Step 1. limn->ood(xn,Xn+l) = o. Proof Since J is contractive the sequence {d(xn, Xn+l)} is monotone decreasing and bounded below, so limn->oo d(xn, xn+d = r 2': o. Assume r > O. Then by the contractive condition
n = 1,2,···.
Letting n -> 00 we see that 1 :::: limn->ooa(d(xn,xn+d), and since a E S this in turn implies r = O. This contradiction establishes Step 1.
Step 2. {xn} is a Cauchy sequence.
Proof Assume limsupd(xn,xm) > O. By the triangle inequality m,n_oo
so by the contractive condition
Under the assumption limsupd(xn,xm) > 0, Step 1 now implies mIn-co
limsup(l- a(d(Xn,xm))-l = +00 m,n--+oo
from which
limsupa(d(xn,xm)) = 1. m,n_oo
But since a E S this implies limsupd(xn,xm) = 0 - again a contradiction. m,n----+co
The proof is completed as in the previous results. • The key step in proving the existence of a fixed point in each of the proofs just given involved showing that given x E M, {r(x)} is a Cauchy sequence (and then invoking continuity of I). It is possible to carry this idea much further. First we need some
Contraction mappings 11
notation. For any mapping F : M -t M we use keF) to denote the Lipschitz constant of Fj thus
keF) = sup {d(F~(;:~(Y)) : x,y E M, x =I Y}.
Note in particular that if F, G : M -t M are two lipschitzian mappings, then
keF 0 G) ~ k(F)k(G).
Now let (M, d) be a complete metric space and suppose T : M -t M is lipschitzian. Fix x E M, and let Xn = Tn(x), n = 1,2,···. By the triangle inequality, if m > n, then
m
d(xn, xm) ~ L d(Xi, Xi+1). i=n
Consequently {xn} is a Cauchy sequence if
00
L d(Xi, XiH) < 00,
i=l
and since
it follows that 00 00
Lk(T) < 00 => Ld(x;,Xi+1) < 00.
;=1 i=l
Also
k(Tm+n) ~ k(rn)k(Tm).
Specifically [k(Tn)]l/n ~ k(T) and so
koo(T) := lim sup [k(rn)]l/n n-+oo
exists. This is all we really need at this point. However it is possible to say more. Replacing T with TP in [k(Tn)]l/n ~ k(T) and taking the pth root of both sides we obtain
p= 1,2,· ...
Also, a simple calculation shows that
. [k(rn)]l/n hm = l.
n-+oo [k(TnH )]l/nH
These two facts lead to the conclusion:
koo(T) = lim [k(rn)]l/n = inf {[k(Tn)]l/n : n = 1,2,· .. }. n--+oo
Now, by the Root Test for convergence of series, if koo(T) < 1 then E~l k(Ti) < 00.
This leads to the following theorem.
12
Theorem 3.7 Let M be a complete metric space and let T : M ---t M be a lipschitzian mapping for which koo(T) < 1. Then T has a unique fixed point z E M, and for each x E M the sequence {Tn(x)} converges to z.
Proof. In view of what was shown just prior to the statement of the theorem, all that remains is to show that the fixed point of T is unique. However this follows from the fact that if koo(T) < 1 then for n sufficiently large, k(Tn) < 1. •
There is a point to the previous development. One might ask whether qualitatively, Theorem 3.7 result is stronger than Banach's contraction principle, and indeed the answer is no. To see this we introduce the concept of equivalent metrics.
Two metrics, p and d on a space M are said to be equivalent if there exist positive numbers a and b such that for each x, y E M
ad(x,y) ::; p(x,y) ::; bd(x,y).
From this it follows that if T : M ---t M then
Moreover,
lIb d(T(x),T(y)) ::; -p(T(x),T(y)) ::; -kp(T)p(x,y) ::; -kp(T)d(x,y).
a a a
b This implies kd(T) ::; -kp(T). Similarly,
a
b p(T(x),T(y)) ::; bd(T(x),T(y))::; bkd(T)d(x,y)::; -kd(T)p(x,y);
a
b hence kp(T) ::; -kd(T). Consequently
a
( b) lin lin and since lim - = lim (~) = 1 we conclude
n~oo a n-H)O b
Hence (kd)oo (T) = (kp)oo(T), that is, koo(T) is the same for all equivalent metrics on M.
Now suppose (M, d) is a complete metric space and let T : M ---t M be a mapping for which koo(T) < 1. Then if ,\ E [O,l/koo (T)),
,\nd(Tn(x) , Tn(y)) ::; ,\nk(Tn)d(x, y)
with
lim [,\nk(rn)]l/n = '\koo(T) < 1. n~oo
Contmction mappings
Therefore
00
r)..(x,y):= LAnd(Tn(x),r(y)) < 00,
n=O
and moreover
This proves that 1').. and d are equivalent metrics on M.
Finally, for x, y E M
00
r)..(T(x), T(y)) = L And(Tn+1(x), Tn+1(y)) n=O
00
n=O
= (I/A) h(x,y) - d(x,y)]
::; (I/A)r)..(x,y).
13
Thus kr>. (T) ::; t < 1. This proves that T is a contraction mapping on the metric space (M,r)..). Also, since r A and d are equivalent metrics, (M,rA) is a complete space. It is now possible to invoke Banach's original theorem to conclude that T has a unique fixed point Xo (with limn->ood(Tn(x),xo) = 0 for each x EM).
The preceding ideas lead to the following.
Theorem 3.8 Suppose (M, d) is a metric space and suppose T : M --t M is a mapping for which (kd)oo(T) < 1. Suppose also that T is continuous relative to a metric p on M for which (M,p) is complete, and suppose p(x,y) ::; d(x,y) for each X,y E M. Then T has a unique fixed point Xo, and moreover limn->oo r(x) = Xo for each x E M.
Proof. There exists a metric d' on M which is equivalent to d and such that T : (M, d') -+ (M, d') is a contraction mapping. Therefore, for each x E M, {Tn (x)} is Cauchy sequence in (M,d')j hence {Tn(x)} is a Cauchy sequence in (M,d) and, since p ::; d, {Tn(x)} is in fact a Cauchy sequence in (M,p) as well. Therefore, relative to p, limn->oo Tn(x) = Xo for some Xo E M and the conclusion follows because T is continuous on (M,p). •
We now turn to a principle of a different kind. In this result the contractive condition is imposed only at the first step. This paves the way for the result discussed in the next section.
Theorem 3.9 Suppose M is a complete metric space and suppose T : M -+ M is a continuous mapping which satisfies for some 'P : M -+ jR+,
d(x,T(x))::; cp(x) - cp(T(x)), xEM. (*)
Then {Tn(x)} converges to a fixed point ofT for each x E M.
14
Proof. This is a piece of the argument used in the proof of Banach's contraction principle. The condition (*) implies that {<p(T'(x))} is monotone decreasing and hence limn~oo <p(T'(x)) = r 2 o. By the triangle inequality, if m, n E Nand m > n then
m-l
d(Tn(x), T""(x)) ~ L d(yi(x), yi+1(x)) ~ <p(rn(x)) - <p(T""(x)) i=n
so limm,n~oo d(T'(x), rm(x)) = O. Since M is complete there exists z E M such that limn~oo Tn(x) = z and by continuity of T, z = T(z). •
REMARK. In the above result, one can obtain an estimate on the rate of convergence of {Tn(x)} by referring back to the inequality
m-l
L d(Ti(x),Ti+1(x)) ~ <p(Tn(x)) - <p(Tm(x)). i=n
This yields
d(rn(x),T""(x)) ~ <p(rn(x)) - <p(Tm(x)),
and if T(z) = z, upon letting m -+ 00 one has
d(rn(x), z) ~ <p(rn(x)) - <p (z).
FURTHER REMARKS. In [53] Jachymski gives a detailed comparison of the relationship between Theorems 3.2 and 3.4, as well as several related contractive conditions. In particular it is shown that a number of mutually equivalent fixed point principles are methodologically reducible to Matkowski 's condition of Theorem 3.4, and that Theorem 3.2 of Boyd and Wong essentially improves on Theorem 3.3 of Browder.
4. Caristi's theorem
We only mention Caristi's theorem in passing here because the relationship between this theorem and order-theoretic principles is taken up in detail in the article by J. Jachymski elsewhere in this volume. On the other hand it also can be viewed as an extension of the contraction mapping principle.
In view of the fact that continuity of T was essential to the proof of Theorem 3.9 it is remarkable to find that the result remains true without such an assumption if a mild continuity assumption is placed on the mapping <po This observation is an outgrowth of Caristi's study [18] of mappings satisfying inwardness conditions. In its original and traditional formulation, Caristi's theorem stated as follows. (A function <p : M -+ R is said to be lower semicontinuous at x if for any sequence {xn } C M,
lim Xn = x EM=> <p (x) ~ lim inf<p(xn).) n--+oo n--+oo
Theorem 4.1 Let M be a complete metric space, suppose <p : M -+ R is lower semicontinuous and bounded below, and suppose 9 : M -+ M satisfies:
d(x,g(x)) ~ <p(x) - <p(g(x)), x EM.
Then 9 has a fixed point.
Contraction mappings 15
Shortly after the appearance of Caristi's Theorem, which actually follows from a variational principle formulated earlier by Ekeland [34], numerous papers were published devoted to various proofs. Caristi's original proof involved an intricate transfinite induction argument, an argument Wong [123] was able to considerably simplify. One of the shortest proofs is a straightforward Zorn's Lemma argument (e.g., [10], [61]) based on Brondsted's partial order. Proofs which invoke weaker set-theoretic premises were discovered by Browder [12], Brezis-Browder [9], Penot [95], Siegel [1l0], among others. Numerous papers have been published which are devoted to related and equivalent formulations of the Caristi-Ekeland principle, for example Kasahara [60], Park [90], [91], Takahashi [1l4]. A proof based solely on the basic axioms of Zermelo and Fraenkel has been given by Manka [74].
An elegant and direct proof of Caristi's theorem is found in Deimling [29], § 11. Another elementary and straightforward approach is due to Brezis and Browder [9] via a general principle on ordered sets which itself has a simple mathematical induction proof. A recent paper of Oettli and Thera [85] provides yet another order-theoretic approach to Caristi's theorem.
We remark that in Caristi's theorem, the mapping 'P above need only be assumed to be lower semicontinuous relative to sequences Xn -'> X E M for which 'P (xn +1) :S 'P (xn ) ,
n2:1.
5. Set-valued contractions
Banach's Contraction Mapping Principle extends nicely to set-valued mappings, a fact first noticed by S. Nadler [84J (also see [75]). The key idea is the following. If A and B are nonempty closed bounded subsets of a metric space and if x E A, then given E > 0 there must exist a point y E B such that
d(x,y) :S H(A,B) + E,
where H(A,B) denotes the Hausdorff distance between A and B. This is because the definition of Hausdorff distance assures that for any I-' > 0
A ~ Np+JL(B)
= {x: dist(x,B) < P+I-'}
where p = H(A, B).
Theorem 5.1 Let (M, d) be a complete metric space, and let 9J1: be the collection of all nonempty bounded closed subsets of M endowed with the Hausdorff metric H. Suppose T : M -'> 9J1: is a contraction mapping in the sense that for some k < 1 :
H(T(x),T(y)) :S kd(x,y), x,yEM.
Then there exists a point x E M such that x E T(x).
Proof. Select Xo E M and Xl E T(xo). By the observation immediately preceding the statement of the theorem there must exist X2 E T(Xl) such that
Similarly, there exists X3 E T(X2) such that
d(X2,X3) :S H(T(Xl),T(X2)) + k2.
16
An induction argument yields a sequence {xn} in M with the property that Xi+! E T(Xi) for each i E ]\\I and for which
Therefore
d(Xi, Xi+l) :'S H(T(Xi_I), T(Xi)) + ki
:'S kd(Xi_l, Xi) + ki
:'S k[H(T(Xi-2),T(Xi-d) + ki- l ] + ki
:'S k2 d(Xi-2, Xi-d + 2ki
:'S" .
:'S ki d(xo, Xl) + iki.
~d(Xi'Xi+I):'S d(xo, Xl) (~ki) + ~iki < 00.
This proves that {xn} is a Cauchy sequence, so since M is complete there exists X E M such that limn-->oo Xn = x. Also, since T is continuous,
lim H(T(xn),T(x)) = O. n-->oo
Since Xn E T(Xn-I),
lim dist(xn,T(x)) = lim inf{d(xn, y) : y E T(x)} = O. n~oo n~oo
This implies that
inf{d(x,y) : y E T(x)} = 0,
and since T(x) is closed it must be the case that X E T(x). • In contrast to Banach's theorem, the preceding theorem does not assert that the fixed point x is unique. Indeed, it need not be. In [106] an example in ]R2 is given of a multi valued contraction mapping whose values are compact and connected yet the mapping has a disconnected fixed point set. It is also shown in [106] that if X is a fixed point of a multivalued contraction mapping T defined on a closed convex subset of a Banach space and if T (x) is not a singleton, then T always has at least one additional fixed point distinct from x. On the other hand, Ricceri [103] has shown that the fixed point set is an absolute retract if M is a closed and convex subset of a Banach space and T has closed convex values.
There is an interesting stability result (due to T. C. Lim [69]) that holds for set-valued contractions (hence ordinary contractions as well). Such results find applications, for example, in the study of iterated function systems ([19], [35]). We begin with a technical lemma.
Lemma 5.2 Suppose M and!m are as in the preceding theorem, and let T;, : M -+!m, i = 1,2, be two contraction mappings each having Lipschitz constant k < 1. Then, if F(TI) and F(T2) denote the respective fixed point sets ofTI and T2,
1 H(F(TI)' F(T2)) :'S --k sup H(TI (x), T2(X)).
1- xEM
Contraction mappings
Proof. Let E> 0, choose c > 0 so that c 2:::'.:"=1 nkn < 1, and set CE
El = (1 - k)·
Select Xo E F(TIl, and then select Xl E T2(XO) so that
d(xo, Xl) ::; H(Tl(XO), T2(XO)) + El·
Since H(T2(xIl, T2(XO)) ::; k d(Xl, xo) there exists X2 E T2(XI) such that
d(X2,Xl) ::; kd(Xl,XO) + kq.
Now define {xn} inductively so that Xn+1 E T2(Xn) and
Then
n = 1,2,···.
d(xn+1,xn) ::; kd(xn, Xn-l) + knEl
::; k(k d(xn-l, Xn-2) + kn-lEl) + knEl
= k2 d(Xn-l, Xn-2) + 2knEl.
Continuing in this fashion we obtain
d(Xn+l,Xn)::; knd(Xl,XO) +nknEl·
Therefore 00 00
n=m n=m
17
Since the right side of the above tends to 0 as m -+ 00, this proves that {xn} is a Cauchy sequence with limit, say z. Since T2 is continuous, limn-+oo H(T2(xn), T2(z)) = O. Also, since xn+1 E T2(Xn) it must be the case that z E T2(Z), that is, z E F(T2). Furthermore
00
n=O 00
::; (1- k)-ld(Xl,XO) + LnknEl n=l
< (1- k)-l(d(Xl,XO) + E)
::; (1- k)-l(H(Tl (xo),T2(XO)) + 2E).
Reversing the roles of Tl and T2 and repeating the argument just given leads to the conclusion that for each Yo E F(T2) there exist Yl E Tl (yo) and W E F(Tl) such that
d(yo,w)::; (1- k)-1(H(Tl (Yo),T2(YO)) + 2E).
Since E > 0 is arbitrary, the conclusion follows. • Theorem 5.3 Suppose M and rrn are as in the preceding theorem, and let Ti : M -+ rrn, i = 1,2,··· be a sequence of contraction mappings each having Lipschitz constant k < 1-Iflimn-+oo H(Tn(x), To(x)) = 0 uniformly for X E M, then
lim H(F(Tn),F(To)) = O. n-+oo
Proof. Let E > O. Since limn-+ooH(Tn(x),To(x)) = 0 uniformly it is possible to choose N E N so that for n:2: N, sUPxEMH(Tn(x),To(x)) < (1- k)E. By the lemma, H(F(Tn),F(To)) < E for all such n. •
18
6. Generalized Contractions
There is a vast amount of literature dealing with technical extensions and generalizations of Banach's theorem. Most of these results involve a common underlying strategy. One assumes that a self-mapping f of a complete metric space M satisfies some general (and frequently quite complex) contractive type condition (C) which implies that (1) the sequence of Picard iterates of the mapping, or some related sequence is Cauchy, and (2) the limit of such a sequence is always a fixed point of the mapping. The condition (C) usually involves a relationship between the six distances {d(x,y),d(f(x),f(y)),d(x,f(y)),d(f(x),y),d(x,f(x)), d(y,f(y))} for each pair x, y E M, and continuity of the mapping mayor may not be assumed. People who want to fully acquaint themselves with this literature are directed to the survey of Rhoades [102] which covers the period up through the mid-seventies, a paper by Hegedus [42], a subsequent survey by Park and Rhoades [92], an analysis of [102] by Collac;;o and Silva [25], as well as references found in these sources. Further escalations in the level of complexity can be found in a paper by Park [89] and in Liu's recent observations [72] involving Park's conditions.
Here we describe an approach which is reasonably elegant, yet sufficiently general to include many of the interesting cases in the work alluded to above. As before we assume (M, d) is a metric space with T : M --> M. For x E M let
O(x) = {x,T(x),T2(x), .. . },
and let O(x, y) denote O(x) U O(y) for x, y E M. Let ¢ be a contractive gauge function on jR+. This means ¢ : jR+ --> jR+ is continuous, nondecreasing, and satisfies ¢( s) < s for s > o.
Clearly if ¢(s) = ks for s E jR+ and fixed k E [0,1), then a mapping T : M --> M satisfying
d(T(x), T(y)) :; ¢(d(x, y))
for all x, y E M is a contraction mapping in the sense of Banach and thus if M is complete T has a unique fixed point z E M. It is a far reaching extension of this approach inspired by Felix Browder that we take up here. We begin with the following fact due to Walter [122].
Theorem 6.1 Let M be a complete metric space and suppose T : M --> M has bounded orbits and satisfies the following condition. For each x E M there exists n(x) EN such that for all n ~ n(x) and y E M,
d(T"(x),T"(y)):; ¢(diam(O(x,y))). (6.1 )
Then there exists z E M such that limlc-->oo Tic (x) = z for each x EM.
Proof. This is essentially the proof of Browder [13]. There are four steps. For x E M we use the notation xk = Tk(x), k = 0,1,2,· ...
Step 1. If m = max{n(x),n(y)}, then diam(O(xm,ym)):; ¢(diam(O(x,y))).
Proof. Suppose n ~ m and r ~ O. Then any two elements (u,v) of O(xm,ym) are of one of the forms: (xn,yn+T), (xn+T,yn), (xn,xn+T), or (yn,yn+T). In the first case we have
d(u,v) = d(T"(X),T"(yT)):; ¢(diam(O(x,yT))):; ¢(diam(O(x,y))).
Contmction mappings 19
The other three cases follow by similar inequalities.
Next we define sequences {k(i)} c N and Ai eM by
k(O) = 0; k(i + 1) = k(i) + max {n(xk(i)), n(yk(i))}
and A- = 0 (xk(i) yk(i)) i = 0 1 2 ...
t '" " .
Step 2. diam(Ai+l) :s; ¢>(diam(Ai)) for i = 0,1,2,· ...
Proof For i = 0 this is just a restatement of (1). Now let i be arbitrary and let € = xk(i), TJ = yk(i), and J-L = max{n(xk(i)),n(yk(i))}. Applying Case (1) we have
(6.2)
However €" = xk(i)+" = xk(i)+max{n(xk(il),n(yk(il )} = xk(iH). Similarly, TJ" = yk(i)+" =
yk(iH). Thus diam(O(€",TJ")) = diam(Ai+l) and (6.2) coincides with (2).
Now let ai = diam(Ai).
Step 3. limi .... oo ai = o. Proof From (2) and the fact that ¢> is decreasing we have aiH :s; ¢>(ai) ::; ai. Thus {ai} is decreasing so there exists a 2: 0 such that limi .... oo ai = a :s; ¢>(a). Since ¢>(a) < a if a> 0, it must be the case that a = O.
Step 4. We have shown that limi-+oo diam(Ai) = limi .... oo diam(O(xk(i),yk(i))) = o. This
clearly implies limk-+oo diam( O( xk, yk)) = O. Thus both {xk} and {yk} are Cauchy sequences and have the same limit, say z E M. Since y E M is arbitrary, we conclude that in fact limk .... oo xk = z for each x E M. •
Corollary 6.2 If in addition to the assumptions of Theorem 6.1 we assume that T is continuous, then T(z) = z.
By strengthening the assumption (6.1) to require that n(x) = 1 for all x EMit is possible to conclude that T has a fixed point without assuming continuity.
Theorem 6.3 Let M be a complete metric space and suppose T : M -> M has bounded orbits and satisfies the following condition. For each x, y E M,
d(T(x),T(y)):S; ¢>(diam(O(x,y))). (6.3)
Then T has a unique fixed point z E M and limk-+oo Tk (x) = z for each x EM.
Proof. By Theorem 6.1 there exists z E M such that limk-+oo Tk(x) = z for each x E M. Assume z =1= T(z). Then diam(O(z)) = a > O. From this it is possible to select two sequences {p(k)} and {q(k)} such that
o :s; p(k) < q(k) and for which lim d (zP(k) , zq(k)) = a. k .... oo
Since limk .... oo zk = z there exists ko EN such that for k, I 2: ko, d(zk, zl) :s; a/2. Hence for some p with 0 :s; p < ko, it must be the case that p(k) == p for infinitely many k.
20
Therefore there is a subsequence {r(k)} of {q(k)} such that limk-->oo d(zP, zr(k)) = Q. If r( k) = q infinitely often, then d( zP, zq) = Q. Otherwise there exists a subsequence of {s(k)} of {r(k)} with s(k) --> 00 as k --> 00 and this implies d(zP,z) = Q. In any case, there exist p, q 2': ° such that d(zP, zq) = Q. If p, q 2': 1 then (6.3) implies
d(zP,zq) = d(T(zP-l),T(zq-l))::; ¢>(diam(O(zp-l,zq-l)))::; ¢>(diam(O(z)).
Since Q = d(zP, zq) = diam(O(z)) this gives 0< ::; ¢>(o<).
On the other hand, if d(z, zq) = 0<, then since limk-->oo zk = z
In either case we have a contradiction since ¢>(o<) < 0< if 0< > 0. Hence O(z) = {z}. •
In an attempt to extend the contraction principle in another direction Jachymski and Stein [56] have formulated the following conjecture.
Conjecture 6.4 Let (M, d) be a complete metric space, k E (0,1), and T : M --> M. Let J be a given finite subset of N, and assume T satisfies the condition
min {d (Ti (x) ,T (y)) : i E J} ::; kd(x,y).
Then T has a fixed point.
The above obviously reduces to Banach's principle if J = {I}. The conjecture has also been confirmed ([55]) for the sets J = {p, 2p}, J = {p, 3p}, J = {2p, 3p}, p E No See [113] for further discussion.
7. Probabilistic metrics and fuzzy sets
In an attempt to respond to classical concerns about imprecision in the natural world, K. Menger [79), [80) introduced the concept of a probabilistic metric space. These are spaces in which the distance between points is a probability distribution on R+ rather than a real number. A short time later L. A. Zadeh [127] introduced the notion of a fuzzy set in a similar attempt to deal with situations in which the imprecision is not of obvious probabilistic nature. In both settings it is possible to introduce 'metric' concepts and ask about the corresponding fixed point theory for 'contraction' mappings (along with fixed point theory in a broader sense), as well as applications.
There is no analogue of Banach's contraction mapping theorem for complete probabilistic metric spaces in general, but a number of positive results are known. In order to describe these we need to introduce precise definitions. As usual, let R+ = [0,00). A mapping F : R+ --> [0,1) is called a (distance) distribution function if it is nondecreasing and left-continuous, with F (0) = ° and sUPxEIR F (x) = 1. We denote the set of all such functions D+ and we use eo to denote the specific distribution function defined by
{ ° if x = 0; eo (x) = 1 if x> 0.
The definition also entails the notion of a t-norm. A mapping T : [0,1] x [0,1) --> [0,1] is called a t-norm if for any a, b, c, d E [0, 1] :
Contraction mappings
(i) T (a, 1) = a;
(ii) T(a,b) =T(b,a);
(iii) T(c,d) 2:T(a,b) ifc2:aandd2:b;
(iv) T(T(a,b) ,c) = T(a,T(b,c)).
21
Let S be a nonempty set. A probabilistic metric space (also often called a PM-space or Menger space) is an ordered triple (S,F,T) which satisfies the following conditions.
(I) F: S x S -t D+ is a symmetric function which satisfies (denoting F (p,q) = Fp,q for (p, q) E S x S) :
(1) Fp,q = eo if and only if p = q;
(2) Fp,r (x) = 1 and Fr,q (y) = 1 => Fp,q (x + y) = 1 for all p,q, rES and all x,yEjR+.
(II) T is a t-norm on (S, F) which satisfies
Fp,q (x + y) 2: T (Fp,r (x) ,Fr,q (y))
for every p, q, rES, x, y E jR+.
A Hausdorff topology on a probabilistic metric space (S, F, T) is given by the neighborhood system 11 = {Uq (e,.\)} , q E S, .0,.\ > 0, where
Uq (.0,.\) = {p E S : Fp,q (e) > 1 - .\}.
If SUPxE(O,l) T (x, x) = 1, this topology is metrizable. A sequence {Pn} in S is said to be T-convergent to pES if given .0,.\ > ° there exists N = N (p,.\) E N such that Fpn,p (e) > 1 - .\ whenever n 2: N; {Pn} is T-Cauchy if given .0,.\ > ° there exists N = N (p,.\) EN such that Fpn,p= (e) > 1-,\ whenever n, m 2: N; and (S,F, T) is said to be complete if every T-Cauchy sequence has aT-limit.
The first 'contraction type' fixed point theorem of note in the setting described above is due to Sehgal and Bharucha-Reid [111]. This theorem applies to probabilistic metric spaces for which the t-norm T is the function min. These spaces are metrizable according to the above definition.
Theorem 7.1 Let (S,F,T) be a complete probabilistic metric space for which the tnorm T is min. Suppose f : S -t S is a continuous mapping for which there exists k E (0,1) such that for all p, q E Sand a.ll u > 0,
Ffp,(q (ku) 2: Fp,q (u).
Then f ha.s a unique fixed point pES, and r (q) -> P for each q E S.
It is also noted in [111] that every complete metric space easily gives rise to an induced complete probabilistic metric space; thus it is actually possible to derive Banach's contraction mapping principle from the above result.
Since the appearance of Theorem 7.1 several approaches to the fixed point problem in probabilistic metric spaces have been undertaken. One approach seeks to identify those t-norms which are strong enough to assure that the sequence of Picard iterates of the mapping at a point is a Cauchy sequence (e.g., [40]). Another approach, initiated by
22
Hicks in [44], has been to modify the contractive definition. However it was noted in [97] (also [107]) that a contraction mapping in the sense of [44] is in fact a standard contraction mapping on a related metric space to which Banach's theorem applies. A recent generalization of the approach of Hicks is given in [88]. Another quite recent approach (Tardiff [118]) imposes growth conditions on the distribution functions and yields the following.
Theorem 7.2 Let (S, F, T) be a complete probabilistic metric space and T a continuous t-norm which is stronger than T (x, y) = max {x + y - 1, O}, x, y E [0,1]. If for all p,q E S
100 InudFp,q(u):S 00,
then any contraction in the sense of Theorem 7.1 has a unique fixed point.
In concluding our very abbreviated review of literature on this topic, we mention that a comparison of various contractive conditions in probabilistic spaces can be found in Tan [115]. Also a rather technical generalized contraction principle is given in Chang, et al [20], which is then applied to a class of differential equations in probabilistic metric spaces.
We now turn to fuzzy sets. Let (M,d) be a metric space. A fuzzy set in M is a mapping A : M ----> [0,1]. For x E M, A (x) denotes the 'grade of membership' of x in A. We denote the collection of all fuzzy sets in M by \l (M) .
For A E \l (M) and a E [0,1] the a-level set of A, denoted A"" is defined by
Aa={XEM:A(x)2a} ifaE(O,l]; and
Ao = {x EM: A (x) > O}.
A number of metrics are used on subspaces of fuzzy sets (e.g., see [30]), and in [59] Kaleva shows how Banach's contraction mapping theorem can be applied to obtain existence of solutions of fuzzy differential equations for fuzzy set-valued mappings of a real variable whose values are normal, convex, upper semicontinuous compactly supported fuzzy sets in ]Rn.
We now describe the fundamental contraction principle for fuzzy metrics. This result is due to Heilpern [43].
Definition 7.3 A fuzzy subset A of a metric linear space M is an approximate quantity if sUPxEM A (x) = 1 and if its a-level sets are compact convex subsets of X for each a E [0,1]. The collection of approximate quantities of \l (M) is denoted 2!1 (M) .
For A,B E 2!1(M) and a E [0,1] define
p", (A,B) = inf {d(x,y) : x E A""y E Bo,};
D (A,B) = supH (A""Ba) ,
'" where H denotes the usual Hausdorff distance between closed subsets of M.
Definition 7.4 Let S be an arbitrary set and M any metric linear space. A mapping F : S ----> 2!1 (M) is called a fuzzy mapping.
Contmction mappings 23
Theorem 7.5 Let M be a complete metric linear space and let F : M -> 2lJ (M) be a fuzzy mapping. Suppose there exists k E (0,1) such that lor each x, y E M
D(F(x),F(y)):::; kd(x,y).
Then there exists x· E M such that {x·} c F (x·) .
Using the above theorem as a point of departure there have been many extension of the standard metric fixed point theorems to a fuzzy setting. For the most part the methods used to obtain these results mimic their non-fuzzy counterparts. The topological theory is another matter. For example in Diamond, et al [30J a direct proof is given which shows that the space of fuzzy sets on a compact metric space with the so-called 'sendographic' metric has the fixed point property for continuous maps.
8. Converses to the contraction principle
There are several versions of the 'converse' of Banach's contraction mapping principle, each focusing on different underlying assumptions. The first result of this type, and in some sense the most elegant, is due to Bessaga [5J. One of the most accessible of the several proofs of Bessaga's result is the adaptation of Wong's proof of [124J which is found in Deimling's book [28], pp. 191-192. An application of Bessaga's converse in which the uniqueness assumption is dropped in exchange for a weaker conclusion is given in [105J.
We devote this section to Bessaga's original proof. Related questions, especially those pertaining to foundational aspects of the converse of Banach's theorem, are found in [52J and elsewhere in this volume.
Theorem 8.1 Suppose S is an arbitrary nonempty set and suppose I : S -> S has the property that I and each of its iterates fn has a unique fixed point. Then for each >. E (0,1) there is a metric P)' on S such that the space (S, p),) is a complete metric space and for which P)' (f (x),J (y)) :::; >.p), (x,y) for each X,y E S.
Proof. Define two equivalence relations in S as follows. (1) x '" y ¢} either x = y or r-l (x) =f r (x) = r (y) =f r-l (y) for some n E N; (2) x ~ y ¢} fP (x) = f q (y) for some p, q E N. Clearly x '" y '* x ~ y. For XES let
[xJ = {y E S : X '" y} and [SJ = {[xJ : XES}.
Next, for arbitrary [xJ E [SJ let
([xlJ = {[yJ E [SJ : 3 Xl E [xJ, YI E [yJ such that Xl ~ YI}·
Similar! y, set
[[SJJ = {[[xlJ : [xJ E [S]} .
Observe that for each u, v E Seither [[uJJ = [[v]] or [uJ rt. [[vlJ . Finally define
[xJ :::; [yJ ¢} 3 Xl E [xJ, YI E [yJ such that fk (Xl) = YI for some k 2: o. Then:::; assigns an order type to each of the sets [[xJJ similar to one of the following: some natural number n 2: 1, the set w of all natural numbers, the set w· of all negative integers, or the set w· +w of all integers. Consequently it is possible to 'label' each set
24
[[Xll according to its order type in a natural way. Observe that the case w' +w requires the Axiom of Choice1 . For [yJ E [[xll , select Yo E [yJ, set ~ (yo) = 0, and define
{I if u E [YoJ+ ;
~ (u) = 0 if u E [yo] ; -1 if Yo E [u]+,
where [yJ+ denotes the successor of [yJ relative to ~.
The convention just described defines a function ~ : S -> Z with the properties
x E [yJ =* ~(y) = ~(x)
x E [yJ+ =* ~ (y) = ~ (x) + 1,
For every xES set foo (x) = x, where x is the unique fixed point of f. Now define a metric p on S x S by setting
i=l i=l
where p and q are the smallest nonnegative numbers for which fP (x) = r (x'). (Apply the usual convention I:?=1 .xe(x)+i = 0.)
It is obvious that p(T(x),T(x'» ~ .xp(X,X') for each X,X' E S. Also if xES and x of- x, then ~ un (x» -> 00 and
00
p (r (x), x) = L .xeW(x»)+i -> o. i=l
Next observe that if x of- x' then either p 2: 1 or q 2: 1 and consequently
p (x, x') 2: min { .xe(x)+!, .xe(X')+!} .
Now suppose {xn} is a Cauchy sequence in (S, p) and suppose ~ (xn) ...... 00. Then (by passing to a subsequence) we may suppose that ~ (xn) ~ N < 00 for all n 2: 1. Since .x E (0,1) ,this implies
.xe(xn )+! 2: .xN+!;
hence p(xn,xm ) 2: .xN+! for all Xn of- Xm which contradicts the assumption that {xn} is Cauchy. Therefore if {xn} is a Cauchy sequence in (S, p) then ~ (xn) -> 00, and it follows that {xn} converges to x. This proves that (S,p) is complete. •
REMARK. For the conclusion of Theorem 8.1 it suffices only to assume that fn has at least one fixed point for n 2: 1 and that some fn has a unique fixed point. (This is the formulation given in [28J.) Under this assumption it is possible to apply Theorem 8.1 to fn and obtain a complete metric relative to which fn is a contraction, apply Theorem 2.4 to conclude that f has a unique fixed point, then apply Theorem 8.1 to f and obtain another complete metric relative to which f is a contraction.
Another question arises. Instead of starting with an abstract set, suppose one has a mapping of a metric space into itself which has a unique fixed point. Then when does an
1 In fact Bessaga notes in [5] that Theorem 8.1 implies this form of the Axiom of Choice.
Contraction mappings 25
equivalent metric exists relative to which the mapping is a contraction? The following theorem is due to Janos [48].
Theorem 8.2 Suppose (M, p) is a compact metric space, and suppose f : M -+ M is continuous and satisfies
Then for each A E (0,1) there exists a metric PA on M such that the space (M, PAl is a compact metric space and for which PA (j (x) , f (y)) ::; >'P>. (x, y) for each x, y E M.
In [81] Meyers modified the ideas in Janos's proof to obtain another version of the converse to Banach's theorem by considering continuous self-mappings of a metrizable topological space X. Such a map f is said to be contractifiable if f is a contraction mapping relative to an appropriate metrization of X. If f is a contraction mapping on X then there is a point e E X and an open neighborhood U of e such that
(8.1)
r (x) -+ e for each x EX; (8.2)
r(U)-+{e}. (8.3)
The explicit meaning of the last assertion is that for each neighborhood V of e there is an integer n (V) > 0 such that r (U) c V for all n :co: n (V) . In [81] (also see [82]) Meyers proved that if f is continuous and satisfies (8.1) - (8.3) for some e E X and some open neighborhood U of e, then f is contractifiable. Completeness of X is not assumed. This result, in essence, was subsequently rediscovered by Leader in [67]. The following is an explicit statement of Leader's result.
Theorem 8.3 Suppose T maps a metric space (M,p) into itself. Then there exists a metric 2: which is topologically equivalent to P and for which T : (M,2:) -+ (M,2:) is a contraction mapping with fixed point p if and only if:
(a) lim Tn (x) = p for each x E M; and n->oo
(b) the limit in (a) holds uniformly in some neighborhood V ofp.
9. Notes and remarks
1. It would be difficult to overestimate the number of variations on the contraction mapping theme which appear in the literature. Some are attempts to unify underlying principles; others represent specific adjustmcnts designed for particular applications.
A recent example is an article of Chen [23]. The point of departure is the following result found in [66]. Let M be a complete metric space, let T : M -+ M, and
(C) suppose for each 0 < a < b there exists L (a, b) E (0,1) such that
d(T(x) ,T(y))::; L(a,b)d(x,y)
for all x,y E M.
26
Then T has a unique fixed point z, and limn Tn (x) = z for each x E M.
Since T satisfies C it is possible to set,
L (a) = sup {d(T (x), T (y)) /d (x,y) : d (x,y) = a},
and observe that if a' < a then L (a) ::; L (a', a) E (0,1) . In particular
d(T(x) ,T(y))::; L(d(x,y))d(x,y)
and it is possible to reach the desired conclusion, for example, via Theorem 3.6. (It suffices to select a number b > 0 for which d (u, T (u)) < b for some u E M and then replace M with the space
M' = {x EM: d(x,T(x))::; b}.
Under these circumstances, if L (an) -> 1 and if an -+> 0 then by passing to a subsequence we may suppose there exists 0 < a < b such that a ::; an ::; b and reach a contradiction.)
2. Often in the study of contraction mappings assumptions which are not fully needed are made out of habit. For example many theorems which invoke continuity actually hold under the weaker assumption that the mapping f : (M, d) -> (M, d) in question has a close graph. (This means that for {x;} c M the conditions limi Xi = x and limi f (Xi) = y imply f (x) = y.). This observation applies to the contractive condition as well. For example the assumption d(J(x),f2(X)) ::; kd(x,f(x)) for x E M, k E
(0,1) suffices to assure {r (x)} is a Cauchy sequence. (See [45J for an early example of this line of thinking.)
3. (Set valued contractions) Extensions of set-valued contractions in the spirit of some of the early extension of Banach's theorem have also been undertaken. Let (M, d) be a complete metric space, and let CB (M) and K (M) denote, respectively, the space of all nonempty closed bounded subsets and all nonempty compact subsets of M endowed with the Hausdorff metric. It is known that if T : M -> K (M) satisfies
H (T(x) ,T(y)) ::; k (d (x,y)) d(x,y) for x,y E M,x '" y,
where k : (0,00) -> [0,1) satisfies lim sup k (8) < 1 for each t > 0, then T has a fixed s~oo
point. S. Reich ([100], [101]) has asked whether K (M) can be replaced with CB (M) in this result. A partial affirmative answer has recently been given in [51J; also see [26J. For a variation on this approach, see [21J.
4. (Topological extensions) In 1971, Cain and Nashed [17J extended the contraction mapping theorem to Hausdorff locally convex topological vector spaces (lEJ, {1'laJ"'EA) (where {1·la} aEA is a family of seminorms generating the topology of lEJ) as follows: For X c lEJ, a mapping f : X -> X is said to be a contraction mapping if for every a E A, there exists ka < 1 such that
If (x) - f (y)l", ::; k", Ix - Yla for every x, y E X.
They showed that the Banach contraction principle is still valid for such mappings defined on sequentially complete subspaces of X.
There have been a number of extensions of the contraction mapping theorem and related results from a metric setting to a uniform space setting. Among the first were extensions given by Knill [62J and Tarafdar [116J. Those interested in pursuing this
Contraction mappings 27
aspect of the theory further might want to check the survey by Lee [68], the papers [1], [41], [83]' and certainly the book by Hadzic [41] and citations therein. A fixed point theorem for generalized contractions defined in gauge spaces has recently been given by Frigon [37]. This result is obtained via an application of the set-valued theorem of Nadler [84].
5. (Structure of fixed point sets) As noted above, a multivalued contraction mapping whose values are compact and connected may have a disconnected fixed point set. With sufficiently strong additional assumptions this is not the case. A nonempty closed subset P of a Banach space X is called a-paraconvex, a E [0,1] , if for every T > ° and every open ball D ofradius T such that Dnp =I- 0, it is the case that dist (q, P) :::; aT for every q E conv {D n P}. It is shown in [109] that if F : X ....... 2x is a contraction mapping with Lipschitz constant /, where a + / < 1, and if F takes values in the collection of a-paraconvex subsets of X, then Fix (F) is a (continuous single-valued) retract of X.
6. (Ultrametric spaces) Ultrametric spaces are metric spaces for which a much stronger version of the triangle inequality holds.
Definition 9.1 A metric space (M, d) is an ultrametric space if, in addition to the usual metric axioms, the following property holds for each x, y, z EM:
d(x,z):::; max{d(x,y),d(y,z)}.
It is immediate from the definition that if d(x, y) =I- dey, z) then in fact
d(x,z) = max{d(x,y),d(y,z)}.
Therefore each three points of an ultrametric spaces represent vertices of an isosceles triangle. This leads to an anomaly.
(1) If B(aj Tl) and B(bj TZ) are two closed balls in an ultrametric space, with Tl :::; T2, then either B(ajTl) n B(bjTZ) = 0 or B(ajTl) <:;; B(bjTZ). In particular if a E B(b; TZ), then B(a; Tl) <:;; B(b; TZ).
An ultrametric space M is said to be spherically complete if every descending sequence of closed balls in M has nonempty intersection. Thus a spherically complete ultrametric space is always complete (where only sequences of balls whose radii tend to ° are considered) .
The following is now an immediate consequence of (1).
(2) If ~ is a family of closed balls in a spherically complete ultrametric space and if each two members of ~ intersect, then n~ =I- 0.
Another consequence of (1).
(3) If B(a; Tl), B(b; TZ) are closed balls in an ultrametric space, if B(a; Tl) <:;; B(b; TZ), and if b 1:- B(a; Tl), then deb, a) = deb, z) for each z E B(a; Tl).
Proof. Since b 1:- B(a;Tl) and z E B(a;Tl), d(b,a) > d(z,a). Hence d(b, a) = d(b,z) .
•
28
The following fixed point result is due to Priess-Crampe [96].
Theorem 9.2 An ultrametric space M is spherically complete if and only if every strictly contractive mapping T : M -t M has a (unique) fixed point.
7. (Applications in logic and programming languages) For an excellent non-technical account of how fixed-point theorems, and Banach's theorem in particular, come in to play in the study of denotational semantics in programming languages, see de Bakker and de Vink [27]. It is interesting to note that the metric involved in this case is also an ultrametric. Another variation on the metric theme is found in Seda [108] where the author investigates the role of quasi-metrics (non-symmetric metrics) and uses the Banach contraction principle as a substitute for the Knaster-Tarski theorem in the semantics of logic programs.
8. (Remetrization) Banach's theorem obviously holds in any metric space which is completely metrizable, but the following fact is somewhat surprising. In [36] it is shown that any separable metric space (M, d) has an equivalent metric p which has the property that every contraction mapping of (M, p) -t (M, p) has unique fixed point. The approach in [36] has a deep topological flavor.
9. (Hybrid extensions) A view to applications has motivated numerous extensions of Banach's theorem. An early example is a result of Krasnoselskii [63] which combines the metric theory with the topological.
Theorem 9.3 Let K be a closed convex nonempty subset of a Banach space X, and suppose that U and V are mappings of K into X which satisfy:
(i) ifx,YEK,thenU(x)+V(Y)EK;
(ii) U is continuous and U (K) is compact;
(iii) V is a contraction mapping.
Then there exists x E K such that x = U (x) + V (x).
(See [15] for a very recent extension of this approach.)
Krasnoselskii's approach can be thought of as a very early precursor to the study of k-set contractions (k < 1) and condensing mappings taken up elsewhere in this volume.
10. (Local contractions) A mapping f : (M, d) -t (M, d) is said to be a local contraction (with constant k E (0,1)) if for each x E M there exists a neighborhood N (x) of x such that for each u, v E N (x) , d (J (u) ,j (v)) ~ kd (u, v) ; if it is assumed only that d (J (u), f (x)) ~ kd (u, x) for each u E N (x), then f is called a local radial contraction. These are two of the standard localizations of the contraction mapping condition (d., Rakotch [99] and Holmes [46] respectively).
In [99] it is shown that if (M,d) is complete and if f: (M,d) -t (M,d) is a local radial contraction with the property that Xo and f (xo) are joined by a path of finite length for some Xo E M, then f has a fixed point. In [47] it is shown that this fact actually is a direct consequence of Banach's theorem. In [46] Holmes proves that if (M, d) is
REFERENCES 29
connected and locally connected, and if f : (M, d) ---+ (M, d) is a homeomorphism of M onto M which is also a local radial contraction on M, then in fact there exists a metric 8 on M which is equivalent to d and for which f : (M, ti) ---+ (M, ti) is a global contraction. However, as noted in [47], the assertion in [46] that (M,ti) is complete is false. (It is noted in [58] that because of another incorrect assertion in [46] the proof of [47] requires minor adjustments.)
Additional comments on the local theory appear elsewhere in this volume.
11. (Continuous mappings). There is even a contraction mapping link to the fixed point problem for continuous mappings. In [57] it is shown that a continuous mapping f of a complete metric space (M, d) into itself has a fixed point if and only if there exists a E (0,1) and a mapping 9 : M --t M which commutes with f and satisfies the conditions: 9 (M) ~ f (M) and d (g (x), 9 (y)) :s: ad (f (x), f (y)) for all x, y E M. This of course reduces to the Banach contraction principle if f is the identity map.
12. (The converse problem) Some other variations and extensions of Janos's result are given in Opoitsev [86]. The following topological variant is proved in [117]. A mapping f : X ---+ 2x (X a topological space) is said to be a topological contraction if f is upper semicontinuous with closed values having the property that f (A) = A for a nonempty closed subset A of X implies A is a singleton.
Theorem 9.4 Lei X be a compact Hausdorff topological space and let f: X ---+ 2x
be a topological contraction with nonempty closed values. Then there is a unique point Xo E X such that {xo} = f (xo) = n':=or (X) .
13. (Random theory). Contraction mappings also enter into the study ofrandom fixed point theory. This theory has its roots in a paper by Bharucha-Reid [4]. Let (M,d) be a complete separable metric space and let (n,~) be a measurable space (i.e., ~ is a sigma-algebra of subsets of 12.) A mapping T : 12 x M ---+ M is said to be a random operator if, for any x E M, Tx := T (., x) : 12 --t M is measurable (Tx- 1 (E) E ~ for each open subset E of M.) The fixed point set of T (w,·) is the set
{xEM:x=T(w,x)}.
A random fixed point of T is a measurable function x : 12 --t M such that x (w) = T (w, x (w)) for all wEn. If T is a set-valued mapping then a random fixed point of T is a measurable selection x of the mapping F defined by
F (w) = {x EM: x E T (w, x)}.
In [126] it is shown that if T is a suitably defined random set-valued contraction then the function F is measurable (and hence T has a random fixed point). This result is also an application of Nadler's fixed point theorem [84] for set-valued contraction mappings. For additional results see the citations in [125].
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Chapter 2
EXAMPLES OF FIXED POINT FREE MAPPINGS
Brailey Sims
Mathematics, School of Mathematical and Physical Sciences
The University of Newcastle
NSW, 2308, Australia
1. Introduction
In this short chapter we collect together examples of fixed point free nonexpansive mappings in a variety of Banach spaces. These examples help delineate the class of spaces enjoying the Epp, the w-Epp, or the w* -Epp. We begin by recalling the relevant definitions.
Let X be a Banach space. A mapping T : C <:;; X --7 X is nonexpansive if IITx -Tyll :S Ilx - yll, for all x, y E C. The fixed point set of T is Fix(T) := {x E C : Tx = x}.
We say that the space X has the fixed point property (Epp) if for every nonempty closed bounded convex subset C of X and every nonexpansive mapping T : C --7 C we have Fix(T) =f. 0.
Similarly, X is said to have the weak fixed point property (w-Epp) iffor every nonempty weakly compact convex subset C of X and every nonexpansive mapping T : C --7 C we have Fix(T) =f. 0.
If X is the dual space of a given Banach space E, X = E*, we say that X has the weak* fixed point property (w* -Epp) if for every nonempty weak* compact (that is, !T(X,E)-compact) convex subset C of X and every nonexpansive mapping T: C -t C we have Fix(T) =f. 0. Which subsets of X are weak* compact depends on the choice of pre-dual. Thus, when discussing the w* -Epp it is important that we have a specific pre-dual E in mind.
Clearly, we have Epp ===} w-Epp, with the two properties coinciding if X is reflexive, and when X = E* we have Epp ===} w* -Epp ===} w-Epp. Finding characterizations of those spaces enjoying the Epp, the w-Epp, or the w* -Epp are perhaps the three most fundamental questions of metric fixed point theory. All three questions remain open.
Much of the effort expended on metric fixed point theory has gone into identifying widely applicable and easily verifiable sufficient conditions for either the Epp, the wEpp, or the w* -Epp. The results of these efforts occupy a considerable portion of this handbook. This chapter approaches the questions from the opposite direction by identifying spaces which fail one or more of these properties.
35
W:A. Kirk and B. Sims (eds.), Handbook of Metric Fixed Point Theory, 35-48. © 2001 Kluwer Academic Publishers.
36
Unfortunately, known examples of fixed point free nonexpansive mappings are rather sparse. With the exception of Alspach's example (or modifications of it, see section 4), the mappings concerned are adaptations of affine maps (indeed, modified shifts), or minor variants thereof. This dearth of examples is a major impediment to a fuller understanding of metric fixed point theory and the discovery of informative new examples would be an important step forward.
In the following section we will document examples that demonstrate failure of the fpp. Subsequent sections will deal with more specialized examples that demonstrate failures of the w*-fpp in duals of certain Banach spaces and finally Alspach's famous demonstration that the w-fpp fails in LdO, 1].
2. Examples on closed bounded convex sets
Example 2.1 CO fails the fpp.
Let C = B;};, := {(xn) E Co : 0 S Xn S 1, all n} and define two affine maps by
T1(xn ) := (1, Xl, X2, ... )
and T2(Xn ) := (1 - Xl, Xl, X2, ... ).
Then for i = 1, 2 and any X,y E CO we easily see that IITiX - T;YII = Ilx - YII. SO, both T1 and T2 are nonexpansive, indeed metric isometries, and map C into C. On the other hand, the only possible fixed point for T1 is (1,1,1, ... ) while the only possible fixed point for T2 is (!,!,!, ... ) neither of which is in co.
It is possible to generalize the above examples in the way illustrated by the next example.
Example 2.2 CO fails the fpp with a contraction; that is a mapping T for which IITxTyll < IIx - yll whenever X t y.
As an alternative to the presentation in example 2.1, we will describe the current example using the standard Schauder basis; e1, e2, e3,'" of co, where en := (On,i) with On,n = 1 and On,i = 0 for i t n.
Let (An) be a decreasing sequence of real numbers converging 1. Define,
C := {~tnAnen : (tn) E Co with 0 S tn S I}
and an affine map T on C by,
T (~tnAnen) := A1e1 + ~ tnAn+1en+1'
Straight forward calculations show that T is a mapping of C into C that is always nonexpansive and a contraction, provided the sequence (An) is strictly decreasing, whose only possible fixed point is (A1, A2, A3,"') ¢ co.
We would like to have examples of fixed point free non-affine nonexpansive maps on nonempty closed bounded convex subsets of CO. Here is a simple example of such a map due to C. Lennard [private communication, 1995].
Examples 37
Example 2.3 CO fails the fpp with a non-affine contraction.
Let C be defined as in example 2.1 and let (Pn) be any real sequence that strictly decreases to 1. Define T by
Tx := (1, xl! , ... , x{;k , ... ) . P1 Pk
Then T is readily seen to be a non-affine contraction mapping C into C. Furthermore, if P := I1~=1 Pn is finite, then a simple calculation shows that T is fixed point free.
To put the next example into context it is important to recall that CO enjoys the w-fpp.
Example 2.4 Co fails the fpp for a contraction and on a set which is compact in a topology only slightly coarser than the weak topology.
The ideas underlying this somewhat interesting example should be clear to anyone familiar with properties of the summing basis for CO. However, some of the details are both tedious and technical and will only be sketched. The interested reader is referred to [6] for a fuller account.
Let a = (a(n)) be a strictly decreasing sequence of 'weights' in 100 satisfying a::; a(n) ::; j3, for some 0 < a ::; j3 < 00. Define elements of CO by: ao := 0 and
an := (a(l), ... ,a(n),O,O, . .. ) for n = 1,2,3, ....
and let K be the closed convex hull of {an}::"=o. Thus, K consists of all vectors of the form
00
I>nan = (a(l)(l - AO), a(2) (1 - (AO + A1))' a(3) (1 - (AO + A1 + A2))"" ), n=0
where An ::::: 0, for all n, and E~=o An = 1.
If Ta denotes the affine map defined on K by,
Ta ( a(l)(l- AO), a(2)(1- (AO+A1)), ... ) := (a(l), a(2)(1- AO), a(3) (1- (AO+A1)), ... ),
then we have the following.
Lemma 2.5 (i) Ta maps K into K,
(ii) Ta is a contraction,
(iii) Ta is fixed point free in K.
Proof. To establish (i) it suffices to note that for An ::::: 0 and E~=o An = 1, we have
00
T( I>nan) = (a(l), a(2)(1 - AO), a(3)(1 - (AO + A1)), ... ) n=O
00
= LAn-Ian E K. n=l
38
To verify (ii) note that for x = (a(1)(1-Ao),a(2)(1-(AO+AI»), ... ) and y = (a(1)(1-
J.L0), a(2)(1 - (J.L0 + J.LI»), ... ) we have
IIx - yll = sup{ a(1)1J.L0 - Aol,a(2)1J.L0 - AO + J.LI - All,···},
while IITax - TaYIl = sup{ a(2)1J.L0 - Aol, a(3)1J.L0 - AO + J.LI - All,···}·
Since a = (a( n») is a strictly decreasing sequence, we now readily see that Ta is a contraction. Note: if the weights (a(n») were only required to be decreasing then Ta would be nonexpansive, but not necessarily contractive.
Finally, suppose that x = (a(1)(1 - AO), a(2) (1 - (Ao + AI»), ... ) were a fixed point of
Ta; that is,
x = Tax = (a(1),a(2)(1- Ao),a(3)(1- (Ao + AI»)' ... ).
Then, we would have AO = 0, Al = 0, ... contradicting the requirement that 2::::'=0 An = 1, and so we have (iii). •
We now introduce a topology Ea into Co which is only slightly coarser than the weak topology , but with respect to which K is compact.
To define this topology, we regard a = (a(n») as an element of £i and define
Ea:= ker(a) = {(y(n» E h : Ly(n)a(n) = o}.
Thus, Ea is a norm closed, but not weak* closed (as a ¢ CO), co-dimension one subspace of £1 = Co. So, Ea is a weak*-dense, and hence, norming subspace for co. Indeed simple calculations show that for x E CO,
a: ,6l1xll :<::: sup{J(x) : I E Ea, 11/11 :<::: 1} :<::: 114
We define Ea:= a(co,Ea). That is, E is the smallest locally convex linear topology on Co for which all the elements of Ea are continuous as linear functionals on co.
The topology Ea may be seen as only 'slightly' coarser than the weak topology, a(co'£l), on CO, being induced by a norming codimension one subspace of £1. None-the-less it displays some unusual, though not too pathological, properties. Here are some examples. A sequence (xn) in CO is Ea convergent to x E Co if and only if for every I E Ea, we have I(xn) --> f(x). Closures are sequentially determined in the Ea topology. However, the norm is not Ea-Iower semi-continuous and Mazur's theorem is not valid for the Ea topology. The sequence an does not have any weakly convergent subsequences, but
an ~ ao = O. This will be used to show that K is Ea-compact. However, first we need the following lemma.
Lemma 2.6 K is Ea-closed.
Proof. For n = 1,2, ... let
00
Xn = L A~n)dk = (a(1)(1 - A&n», a(2) (1 - (A&n) + A~n»), ... ), k=O
Examples
where A~n) :::: ° and ~%':o Ain) = 1, be such that Xn ~ x = (fLl a(l), J.L2a(2) , ... ).
Choosing f:= (1/a(1),-1/a(2),0,0, ... ) E Ea we have
f(xn - x) = (1 - A6n) - J.Ld - (1 - A6n ) - Ain) - J.L2) -t 0.
That is \ (n) A1 -t J.L1 - J.L2·
Similarly, choosing f := (0, 1/a(2), -1/a(3), 0, 0, ... ) we obtain,
A~n) -t P'2 - J.L3,
and in general, \ (n) Ak -t J.Lk - J.Lk+1·
Thus, for k = 1,2, ... Ak := J.Lk - J.Lk+1 = lim Ak :::: °
n
and x = (J.L1a(1) , (J.L1 - Ada(2), (J.L1 - A1 - A2)a(3), ... ) E co·
So we must have
and then, provided J.L1 ::; 1,
00
00
x = LAkdk E K. k=l
But, given E > ° there exists N so that
00 N
fLl = L Ak < L Ak + E/2, k=l k=l
and there exists n for which
IAk - Ain ) 1 ::; E/2N, for k = 1,2, ... ,N.
Thus, N
J.L1 ::; L A~n) + E ::; 1 + E,
k=l and so J.L1 ::; 1, as required.
00
as L A~n) = 1, k=O
39
• Since an ~ ao, we have that {an};;:"=o is [a-compact. The [-compactness of K then follows from Lemma 2.6, the definition of [a, and the following general result from Banach space theory (see, for example, [6] for a proof).
Lemma 2.7 Let X be a separable Banach space and let M be a closed norming subspace of X*. If Dc X is a(X,M)-compact then co(D) is a(X,M)-precompact.
This example suggests the following open question: Does a nonempty closed bounded convex subset of Co have the fpp if and only if it is weakly compact? See [6J for more evidence in support of this.
40
3. Examples on weak" compact convex sets
Example 3.1 h = Co with the equivalent dual norm 11111' := IIrll V III-II fails the w*-fpp.
This example is due to T. C. Lim [4) and provides us with a nonexpansive map T on a domain C that is a w* -compact minimal invariant set for T of diameter 2.
We first show that II . II' is indeed an equivalent dual norm for h. To this end, for x E Co define
IIxll':= IIx+1I + IIx-1i
Then II . II' is an equivalent norm on CO satisfying IIxll :s: IIxll' :s: 211xll and so it suffices to show that for I E h we have
11111' = sup{J(x) : x E Co, IIxll':S: I}.
Now for x E Co with IIxll' :s: 1 let
Yi = { ~i Then lIyll' :s: IIxll' :s: 1 and
00
if Ii x; > 0 otherwise.
I(x) := LJ;x; i=1 00
i=1
:s: lIy+llllrll + lIy-1iIIrII
= ("Y+""r" + lIy-lIllrll)lIyll' lIyll' lIyll'
:s: (IIrll V IIrll)lIyll'
:s: 11/11'·
To see the reverse inequality note that 11/+11 (or IIrll) can be approximated arbitrarily well by I(x) where the Xi are a suitable choice of 0 or 1 (0 or -1) and so IIxll':S: 1.
Now let C = {I E l1 : J; ~ O,II/II':S: I} and define T by
00
TI:= (1- LJ;, iI, h, ... ). ;=1
C is closed and bounded with respect to II . II and since the unit ball centred at 0 in the same norm is w*-compact we have C is a weak*-compact convex subset of h. It is readily verified that T is a fixed point free affine mapping of C into C. Furthermore C is a minimal invariant set for T. To see this note that for any I = (1m) E C the successive iterates are:
00
TI=(I- Llm,iI,h, ... ) 1
00
T2j = (0,1- Llm,fI,h, ... ) 1
Examples 41
00
T 3 f = (0,0,1- 2.:.fm,h,h, ... ) 1
So, Tn f ~w' 0. Thus ° belongs to any nonempty T-invariant w*-compact convex subset K of C. Hence the n'th basis vector, P-n = Tn(O), is in K. It follows that C = co{ en} ~ K ~ C, so K = C.
We conclude by showing that T is a metric isometry (hence certainly a nonexpansive mapping) on C.
Given f,g E C let P:= {i : fi - gi 2: O} and N := {i : I; - gi < O}. In the case that L (fi - gil 2: L (gi - fi) we have iEP iEN
00 ,
IITf - Tgll' = II (2.:.(9i - fi), h - gl, h - 92, ... ) II i=l
= II (2.:.(9i - fi) - 2.:.(fi - gil, h - gl, h - g2, ... )) II' iEN iEP
v __ -----
negative
= Max{2.:.(fi - gil, 2.:.(gi - fi)} iEP iEN
= Ilf-gll·
The equality follows similarly in the case when Ilf - gil' = L (gi - 1;). iEN
Example 3.2 h = c' with its natural norm fails the w' -Hp for an affine contraction.
It will be convenient to take the dual action of h on c to be
where (fn) E hand (xn) E c. In particular then, regarding x = (-1,1,1, ... ) E c as a weak' continuous linear functional over h. we see that,
{ f E 11 : h = f I;} = ker x ,=2
is a w' -closed hyperplane and consequently the set
C = {f : fi 2: 0, h = f fi :; I,} '=2
being the intersection of ker x and weak' closed halfspaces is itself convex and weak' closed. Obviously, C c 2Bt, so C is weak' compact.
Now, let 0 E (0,1] and let (Ek) C [0,1) be a sequence such that L%':l Ek < 00 and so TI%':1 (1 - Ek) > 0. Define a mapping by
42
then T is clearly an affine mapping. We claim that T is a fixed point free nonexpansive mapping of C into C and further, T is a contraction if all the Ek are strictly positive.
To prove the T-invariance of K we need to show that (Tf)n ~ 0, (Tfl! = "L.':=1(Tf)k and (T 1)1 :S l. The first two are obvious, for the third observe that,
00
(Tfh = 8(1- fll + L(1- Ek)fk+1 :S 8(1- h) + h :S 8 + (1- 8)h :S 8 + (1- 8) = l. k=1
We next show that T is always nonexpansive:
IITf -Tgil 00 00
= 18(g1 - h) + L(l- Ek)(fk+1 - gk+1)1 + 18(g1 - fill + L(l- Ek)lik+1 - gk+11 k=1 k=1
00 00
= 1(8 - 1)(g1 - h) + L Ek(fk+1 - gk+1)1 + 18(g1 - h)1 + L(1- Ek)lfk+1 - gk+11 k=1 k=1
00 00
:S (1 - 8)lg1 - hi + L Eklfk+1 - gk+11 + 81(g1 - fIll + L(l - Ek)lfk+1 - gk+11 k=1 k=1
00
k=1
= Ilf-gll· Now suppose that Ek > 0, for all k, and that liT f - Tgil = Ilf - gil, then the above contains only equalities. Hence
and
00
k=1 00
= 1(1- 8)(g1 - fill + I L Ek(gk+1 - ik+1)1 k=1
I L Ek(fk+1 - gk+1) I = L Eklfk+1 - gk+1l· k=1
To satisfy (1) we must either have
(1 - 8)(g1 - h) ~ ° and
00
L Ek(gk+1 - fk+1) :S ° k=1
(1)
(2)
or the reverse. Both cases follow a similar proof so we will prove the first case only. From (2) we see that the elements of the sum "L.~1 Ek(fk+1 - gk+1) are either all negative or all positive, so we must have
ik ~ gk, for k ~ 2.
But also, g1 ~ h, and hence 00 00
h = L fk 2: L gk = g1 ~ ft. k=2 k=2
Examples 43
Thus !k = gk for k :::: 1; that is, I = 9 and so T is a contraction.
Lastly we show that T is indeed fixed point free. Suppose there were an I E C with TI = I. Then, for n:::: 3 we would have,
(f)n = (1 - En-2)ln-l = (1 - En-2)(1 - En-3)···(1 - El)h
Thus, if h = a, then In = a for n :::: 3 and also 6(1 - !I) = a, whence !I = 1 and we have the contradiction:
00
!I = (Tlh = 6(1 -!I) + 2)1 - Ek)lk+l = a i= fI. k=1
Consequently we must have h of a and, since In -+ a as n -+ 00, we have
r In a n':'~ h = .
This means I1~1 (1 - Ed = a which contradicts L:~1 Ek < 00.
When T is a contraction there can be only one minimal invariant set for T, but we do not know if Cis itselfthat minimal invariant set. However, when Ek = a, for k = 1,2, ... , this is not the case [2]. There is a smaller weak' closed convex T-invariant set; namely,
C' = {f E C:!I = I}
and a slightly more subtle variant of the argument used in example 3.1 shows that in this case C' is in fact the unique minimal invariant set for the nonexpansive map T. Indeed, simple calculations show that in these cases the orbit of any point of C under T converges weak' to 10 := (1,1, a, a,· .. ). So, 10 is in any set which is T-invariant and it suffices to note that the closed convex hull of the orbit of 10 is C'. Computer experiments show that when the Ek are not all zero C' need not be T-invariant.
Example 3.3 A non-affine example in 11 = c'.
In the same spirit as example 2.3 C. Lennard [private communication, 1995] has given a non-affine variant of example 3.2 in the case when 6 = 1 and Ek = a, for all k.
Let C be defined as in example 3.2, and let (Pn) be any sequence of real numbers that strictly decreases to 1. Define T by
TI:= 1,1- L 1+1,_2_, ... , k+l, .... (
00 I Pj [Pi rk ) j=1 Pj PI Pk
Then one may verify that T is a non-affine contraction of C into C. Furthermore, if P := I1~=1 Pn is finite, then T is readily seen to be fixed point free.
4. Examples on weak compact convex sets
Although the question had been raised more than twenty years earlier it was not until 1981 that Dale Alspach gave an example, drawn from ergodic theory, showing that not all Banach spaces enjoy the w-FPP.
Example 4.1 Alspach's example [1]
Here we take C to be the set
C:= {f E Ll[a, 1] : a::; I::; 1, [ I dx = n
44
As the intersection of an order interval with a hyperplane in an order continuous Banach lattice, C is weak compact.
The mapping T is essentially the baker transform of ergodic theory. Formally, for f E C
Tf(t) .= { (2f(2t)) 1\ 1 for a:; t:s; ~j . (2 f (2t - 1) - 1) V a for ~ < t :s; 1.
It is clear from the above description that T is an isometry on C.
We now show that T is fixed point free and hence LIla,l], and any space containing an isometric copy of it, fails to have the w-fpp.
Intuitively the idea is simple. First observe that the successive iterates of any point in C under T assume values closer to a or 1. Hence any fixed point for T must be a function which assumes only the values a or 1. By the 'ergodic' nature of T it then follows that such a function must be either constantly a or constantly 1, and neither of these functions lie in C.
The details follow.
For any f E C we have T f (t) = 1 if and only if either
a :s; t :s; ~ and ~:s; f(2t) :s; 1
or ~<t:S;l and f(2t-l)=1.
Furthermore if ~ :s; t:s; 1 and Tf(t) = 1, then Tf(t - ~) = 1.
Now, suppose f is a fixed point for T then
A:={t:f(t)=l}
={t:Tf(t)=l}
= {t : a :s; t :s; ~ and ~ :s; f (2t) :s; I} U {t : ~ < t :s; 1 and f (2t - 1) = I}
= Ht: ~ :s; f(t):s; I} U H + ~t: f(t) = I}
= Ht: ~ :s; f(t) < I} u ~A u (~+ ~A)
Since the three sets in the above union are mutually disjoint and each of the last two sets has measure one half that of A it follows that:
is a null set. But, then
B1 := {t: ~:s; f(t) < I}
Bl={t:~:S;Tf(t)<l}
:::JH:~:S;f(t)<~}
and so B 2 := {t: ~ :s; f(t) < ~} is also a null set. Continuing in this way we have
1 1 Bn := {t : 2n :s; f(t) < 2n - 1 }
is a null set for n = 1, 2, ... , hence
00
{t : a < f ( t) < I} = U Bn n=l
Examples
is null and
f == XA (where meas(A) = 11 XA = ~). From the definition of T we have
so, up to sets of measure zero,
A = !A U (~+ ~A).
Continuing to iterate under T yields
A = ±A U (± + ±A) U (! + ±A) U (i + ±A)
A = ~A U (~+ ~A) U (± + ~A) U ...
et hoc genus omne.
45
Thus, the intersection of A with any dyadic interval (and hence any interval) has measure one half that of the interval, an impossibility for a set which is not of full measure.
Notice that, unlike the previous example, the domain C of the baker transform T is not a minimal invariant set. This follows since
diam(C) = 1,
as 1::: diam(C) ::: IIx[o,~] - x[~,1]111 = 1,
while for any f E C we have -~ ::; f - ~ ::; ! hence
1 t 1 1 IIf - 2x[o,1]lh = Jo If - 21::; 2'
Thus, C is not diametral and therefore not a minimal invariant set.
Indeed there seems to be no known explicit example of a non-trivial minimal invariant set for a nonexpansive map on a weak compact convex set.
Example 4.2 Sine's modification of the Alspach example
Robert Sine [9] gave the following modification to example 4.1 which allows us to take as the domain C of our fixed point free nonexpansive mapping the whole order interval of 0 ::; f ::; l.
For f E C:= {g : 0 ::; g::; 1} let Sf := X[O,l] - f, then 5 defines a mapping of Canto C with IISf - Sgll = Ilf - gil for all f,g E C.
An argument similar to that for Alspach's example shows that the composition ST, where T is the baker transform of 4.1, is an isometry on C with XA where A = [0,1J or 0 the only possible fixed points. However, the action of ST is to map each of these functions onto the other, hence ST is fixed point free on the order interval 0 ::; f ::; l.
Example 4.3 Schechtman's construction. Gideon Schechtman [8] gave a construction which leads to a greater variety of examples and is in some regards somewhat simpler than that of Alspach.
46
Suppose (O,~, J.L) is a measure space for which there exists a measure preserving transformation T : 0 -+ 0 x [0,1]; that is, for any measurable S ~ 0 x [0,1] we have J.L(T-lS) = meas(S) [3]. Then if C is the weak compact convex set
we can define a mapping T : C -+ C by
Tf:= XT-l{(w,t): 0::; t::; f(w)}
Clearly T is an isometry on C and f E C is a fixed point for T if and only if f = XA where A E ~ is such that J.L(A) = ! and irA) := T-l(A X [0,1]) = A a.e.
Thus if T is further chosen so that i is ergodic; that is irA) = A a.e. if and only if A = 0 or A = 1>, then T is an example of a fixed point free nonexpansive mapping on C.
Perhaps the simplest example of an (O,~, J.L) and T suitable for the above construction is the following.
Let 0 = [0, 1J'~o with product Lebesgue measure and define T by
Clearly T is measure preserving, further if A =F 1> and irA) = A, then for any (WI, w2, ... ) E A we see that (t, WI, w2, ... ) E A for any t E [0,1]. Iterating under i gives (tl' t2, ... , tn, WI, W2, ... ) E A for any n E Nand tl, t2, ... , tn E [0,1], and so we have A = O.
An alternative example with 0 = [0,1] is obtained by taking
where tn, On E {O, I} for n = 1,2, ... A good way to view this example is via the correspondence
[0,1] <--t {O, I}No : f ;: <--t (EI, t2, ... ).
n=l
A set specified by prescribing precisely m of the tn'S has measure 112m. From this it is clear that the product of two such sets has measure 1/2ml+m2 where ml + 7112 is also the number of digits prescribed for points in the T- l image of the product. It follows that T is measure preserving. The ergodicity is established by iterating under f and an argument similar to that used for the conclusion of Alspach's example.
Remark 4.4 Schechtman's construction is both simpler and more versatile than that of Alspach and is of course also amenable to Sine's modification. None-the-less, the Alspach example has some advantages. The relatively simple action of the baker transform permits detailed calculations. For example, it is possible to determine the orbit fa, T fa, T2 fa, T 3fo, ... , of certain starting functions fa under T. If fa = !X[O,lj we obtain the iterates depicted below.
Examples 47
• • • I ••••••• I • • • • •••••• • • I I • I • I ••• [lJ [OJ" [UJIill' , , , , , • • • I ••••••• I I • I I I ••••• I I • I •••• I •• • • • I ••• I ••• • • • I • I • I •• I
o o 1 0 1 Tjo
Here we see that the sequence Tnjo = ~(rn + 1) is an orbit under T, where rn is the n'th Rademacher junction. This may be combined with a result of Maurey ([7], also see the chapter entitled Ultra-methods in metric fixed point theory); that reflexive subspaces of LdO, 1] have the fixed point property, to show that the closed convex hull of an orbit of a nonexpansive mapping on a weakly compact convex set need not be invariant. Indeed, define D to be co{TnUo): n EN}. Since the closed linear span of the Rademacher functions is isomorphic to L2[O, 1] [5], D can not be invariant. Indeed, were it invariant, Maurey's result on the reflexive subspaces of Ll[O, 1], would imply that T possessed a fixed point in D and, a fortiori, in C.
5. Notes and Remarks
Example 2.1 is due to Kakutani, the modification presented in example 2.2 is due to Lennard. The presence of the An allow one to compensate for slight perturbations of the en. Thus, the conclusion remains valid if the vectors en are replaced by vectors Xn which are 'asymptotic' to the basis vectors. This allows the example to be transported into spaces containing an 'asymptotically isometric copy' of co, thereby demonstrating that such spaces fail to have the fpp. Similarly, example 3.2 may be exploited to show that spaces containing an 'asymptotically isometric copy' of PI also fail the fpp. Details of these exciting new ideas may be found in the chapter entitled Renormings of PI and Co and fixed point properties.
Example 3.2 is also due to Lennard, the observation that it is in fact a contraction was made by Smyth who also extended it to the following broader result [10]: Let D be an infinite compact Hausdorff topological space. Then C(D)* fails the w* -fpp with an affine contraction.
In our example D is the one point compactification of N, where '00' is the extra point. So we can write nED in the form n = (1,00,2,3, ... ). Now, if for z = (ZI, Z2, Z3, ... ) E c we write
z is a continuous function on D. This is because
lim z(n) = lim Zn = z(oo). n--+oo n-oo
So c=C(D). If we let h act on c by
X(z) = XIZI + X2 lim +X3Z2 + ... , 'ix E h, Z E c n~CXl
then h = c* = C(D)* and T is the affine contraction for which the,w*-fpp fails.
The fpp, w-fpp, or w*-fpp relate to all mappings in a particular class having fixed points. This class of mappings depends on both which mappings are picked out as nonexpansive by the norm and which domains are admissible. Since PI = Co enjoys the
48
w* -fpp in its natural norm, examples 3.1 and 3.2 taken together show that both of these factors are critical. Moving to an equivalent norm varies which mappings are picked out as nonexpansive, but not the admissible domains. On the other hand, for a dual space, changing the pre-dual does not affect the dual norm, nor alter which mapping are nonexpansive, but does change the class of admissible domains. These considerations also show that any characterization of the w* -fpp will necessarily involve a condition on the pre-dual.
Chris Lennard [private communication, 1996] has given a wavelet construction of a fixed point free isometry, similar to that of Alspach, and also on the order interval [0::; f ::; 1] in LdO, 1].
PROBLEMS.
The results of section 4 indicate an intimate connection between fixed point free isometries and ergodic transformations of the underlying measure space. In the true tradition of ergodic theory, we ask:
Is the set of fixed point free isometries on the order interval [0 ::; f ::; 1] residual in an appropriate sense, at least among isometries which map into the set of 0, I-valued functions?
Clearly any space containing an isometric copy of L1 (f.L) also fails the w-fpp. Can one give an intrinsic description of examples demonstrating this failure for the spaces 100 and C[O, I]?
Examples 3.1, 3.2 and those of section 4 also suggest the following question.
If a space X fails the (w, w* )-fpp does it necessarily fail with an isometry?
References
[1[ Alspach, Dale E., A fixed point free nonexpansive map, Proc. Amer. Math. Soc., 82 (1981), 423-424.
[2] Goebel, K. and Sims, B., More on minimal invariant sets for nonexpansive mappings, Nonlinear Anal., to appear.
[3] Halmos Paul R., Measure Theory, Van Nostrand, 1950.
[4] Lim, T.-C., Asymptotic centers and non expansive mappings in conjugate Banach spaces, Pacific J. Math., 90 (1980), 135-143.
[5] Lindenstrauss, Joram and Tzafriri, Lior, Classical Banach Spaces, Springer-Verlag, Lecture Notes in Mathematics, 338 (1973).
[6] Llorens-Fuster, E. and Sims, B., The fixed point property in co, Canad. Math. Bull., 41 1998, 413-422.
[7] Maurey, B., Points fixes des contractions de certains faiblement compacts de Ll, Seminaire d'Analyse Fonctionnelle, Expose No. VIII (1980), 18.
[8] Schechtman, G., On commuting families of nonexpansive operators, Proc. Amer. Math. Soc., 84 (1982),373-376.
[9] Sine, R., Remarks on the example of Alspach, Nonlinear Anal. and Appl., Marcel Decker, (1981), 237-24l.
[10] Smyth, M. A., [1994]' Remarks on the weak star fixed point property in the dual of C(ll), J. Math. Anal. App., 195 (1994), 294-306.
Chapter 3
CLASSICAL THEORY OF NONEXPANSIVE MAPPINGS
Kazimierz Goebel
Maria Curie-Sklodowska University
20-031 Lublin, Poland
W. A. Kirk
Department of Mathematics
The University of Iowa
Iowa City, IA 52242-1419 USA
1. Introduction
Mappings which are defined on metric spaces and which do not increase distances between pairs of points and their images are called nonexpansive. Thus an abstract metric space is all that is ncedcd to define the concept. At the same time, the more interesting results seem to require some notion of topology; more specifically a topology which assures that closed metric balls are compact. This is not a serious limitation, however, because many spaces which arise naturally in functional analysis possess such topologies; most notably the weak and weak' topologies in Banach spaces.
Here we present a brief overview of the classical theory of nonexpansive mappings, concentrating on the topics listed below. We give only passing attention to aspects of the theory which are covered in more detail elsewhere in this work.
• Existence: Classical results.
• The structure of the fixed point sets.
• The approximation of fixed points.
• Abstract Metric Theory
Recognition of fixed point theory for nonexpansive mappings as a noteworthy avenue of research almost surely dates from the 1965 publication of likely the most widely known result in the theory.
49
WA. Kirk and B. Sims (eds.), Handbook of Metric Fixed Point Theory, 49-91. © 2001 Kluwer Academic Publishers.
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Theorem 1.1 If K is a bounded closed and convex subset of a uniformly convex Banach space X and ifT : K -+ K is nonexpansive (that is, IIT(x) - T(y) II ::; Ilx - yll for each x, y E K), then T has a fixed point.
The above theorem was proved independently by F. Browder [15] and D. G6hde [55] in the form stated above, and by W. Kirk [80] in a more general form. (Slightly earlier Browder obtained the Hilbert space version of this result ([14]) as a by-product of the fact that a mapping T in a Hilbert space is nonexpansive if and only if the mapping 1- T is monotone.) Browder and Kirk used the same line of argument - indeed, one which in fact yields a more general result - while the proof of G6hde relies on properties essentially unique to uniformly convex spaces. As a result, G6hde's proof reveals that fixed points in this setting may be obtained as weak limits of sequences of 'approximate' fixed points.
The proof of Theorem 2.1 is contained in the proof of Theorem 2.2 given in the next section. (The proofs included in this article were chosen to illustrate a variety of standard methods in the theory.)
2. Classical Existence Results
2.1. Overview
We begin with a study of nonexpansive mappings in a Banach space setting and take up the purely metric aspects of the theory later. If X is a subset of a Banach space and D ~ X, then a mapping T: D -+ X is said to be nonexpansive if for each x,y E D,
IIT(x) - T(y)11 ::; IIx - YII·
The study of the existence of fixed points for nonexpansive mappings has generally fallen into three categories. We shall say that a Banach space has FPP if each of its nonempty bounded closed convex subsets has the fixed point property (f.p.p.) for nonexpansive self-mappings, the wc-FPP if each of its weakly compact convex subsets has f.p.p., and B-FPP if its closed unit ball (hence any ball) has f.p.p. This latter category is primarily relevant to dual spaces where the unit ball is always compact in the weak* topology relative to any predual, and in these cases the f.p.p. seems invariably to extend to sets which are intersections of closed balls. The classical nonreflexive space £1 provides an example of a space which has B-FPP but not FPP (Karlovitz [69], Lim [106]). Also, Co provides an example of a space which has wc-FPP but neither FPP nor B-FPP (Maurey [112]).
It is only natural that one of the central goals in the theory should be to fully characterize those Banach spaces which have FPP. This goal remains elusive. It is known that essentially all classical reflexive spaces - in particular, all uniformly convex spaces - have FPP, hence wc-FPP, via the presence of a geometric property called normal structure. A Banach space X, or more generally, a closed convex subset K of X, has normal structure if any bounded convex subset H of K which contains more than one point contains a nondiametral point - i.e., there exists a point Xo E H such that
sup{lIxo - xII: x E H} < diam(K) := sup{llx - yll : x,y E H}.
For D c X let
diam(D) = sup{lIu-vll: u,v E D};
Nonexpansive mappings
r", (D) = sup{llx - vii: v E D}; r (D) = inf {r", (D) : XED}.
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If X is reflexive and if D is bounded closed and convex, then weak compactness of closed balls in X yields the fact that the set
C(D) = {z ED: rz(D) = r(D)}
is a non empty closed convex subset of D. The number r(D) and the set C(D) are called the Chebyshev radius and Chebyshev center of D, respectively.
We state the following theorem for reflexive spaces although clearly weak compactness of K suffices.
Theorem 2.1 Let X be a reflexive Banach space and suppose K is a bounded closed convex subset of X which has normal structure. Then any nonexpansive mapping T : K ...... K has a fixed point. In particular, if X has normal structure then X has FPP.
Proof. Let
\V = {D c K: D closed, convex; D =1= 0; T(D) c D}.
Since the members of \V are weakly compact each descending chain in \V has a lower bound (the intersection of its members); hence by Zorn's Lemma \V has a minimal element, say Do. On the other hand,
T (Do) c Do =} T (co (T (Do))) c T (Do) c co (T (Do)).
Thus co(T(Do)) E \V and by minimality, co(T(Do)) = Do. Let u E C(Do); thus r" (Do) = r (Do). But liT (u) - T (v) II ~ lIu - vII ~ r (Do) for all v E Do which implies T(Do) C B(T(u);r(Do)). Therefore
Do = co (T (Do)) c B (T (u) ; r (Do))
which gives rT(u)(Do) = r(Do), from which T(u) E C(Do). In view of minimality of Do this implies C(Do) = Do. Thus diam(Do) ~ r(Do), and since K has normal structure it must be the case that Do consists of a single point which is fixed under T. •
A major obstacle in identifying spaces which have FPP is the fact that it is not known whether FPP is stable under equivalent renormings. There are other hindrances as well. It has been known virtually from the outset that f.p.p. for a bounded closed convex subset K of a Banach space depends strongly on 'nice' geometrical properties of the space or on the set K itself. However, two closed convex subsets Kl, K2 c X may have f.p.p. yet K1 n K2 may fail to have f.p.p.! Indeed, Goebel and Kuczumow [53] have shown how to construct a descending sequence {Kn} of nonempty bounded closed convex subsets of £1 which has the property that if n is odd, Kn has f.p.p., if n is even Kn fails to have f.p.p., and in fact the sequence {Kn} may be constructed so that nKn is nonempty and falls into either category. The space £1 provides the setting for another interesting example. It is possible to construct a family {K.} (E > 0) of bounded closed convex sets in £1 each of which has f.p.p., but which converges in the Hausdorff metric, as E ...... 0, to a bounded closed convex Ko which fails to have f.p.p. ( [49, Example 5.3]).
1. It has been known for some time that even in reflexive spaces normal structure is not essential for FPP. An example is provided by the spaces X{3, f3 > 0, defined by
X{3 = {x E £2: llxll{3 = max{llxlli2 ,f311x ll oo}}·
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Some fact about Xf3 :
(i) Xf3 is reflexive (since it is isomorphic to the Hilbert space £2).
(ii) Xf3 has normal structure {=} j3 < V2.
(iii) Xf3 has asymptotic normal structure {=} j3 < 2.
The concept of asymptotic normal structure was introduced by Baillon and Schoneberg in 1981. A Banach space X has asymptotic normal structure if each nonempty bounded closed and convex subset K of X which contains more than one point has the property: If {xn } C K satisfies IIxn - X n +111 -> 0 then there exists x E K such that
lim inf IIxn - xii < diam(K). n~oo
In [8J Baillon and Schone berg proved the following theorem.
Theorem 2.2 In a reflexive Banach space, asymptotic normal structure ~ FPP.
In the same paper they went on to show that in fact X2 also has FPP, thus proving that asymptotic normal structure is not a necessary condition for FPP. Surprisingly, P. K. Lin proved in 1985 ([107]) that Xf3 has FPP for all j3 > O.
In 1971 it was observed by Day, James, and Swaminathan [28J that every separable spacehas an equivalent norm which has normal structure (also see van Dulst [32]) Thus every separable reflexive space has an equivalent norm which has FPP.
The question of whether reflexivity is essential for FPP remains open, but there is compelling evidence that it might be. First, it is known that some bounded closed convex sets in the classical nonreflexive spaces co and (as noted above) in £1 fail to have FPP. Also, Bessaga and Pelczynski have shown that if X is any Banach space with an unconditional basis, then X is reflexive {=} X contains a subspace isomorphic to Co or £1· Thus all classical nonreflexive spaces can be renormed so that they fail to have FPP.
This raises an obvious question: Can either CO or £1 be renormed so that they have FPP? Recall ([59]) that any renorming of £1 contains almost isometric copies of £1 suggesting, at least for £1, that the answer should be no. If indeed the answer is no, then by the Bessaga-Pelczynski result, in any space with an unconditional basis, FPP~ reflexivity. These matters are discussed in much more detail elsewhere in this volume.
The Space L1. As we have noted £1 (hence L1) fails to have FPP. However, in 1981, Alspach [2J proved much more, namely that L1 fails to have wc-FPP. At the same time, Maurey [112J proved that all reflexive subspaces of L1 do have FPP (hence wc-FPP). There has been another quite recent development. Dowling, Lennard, and Turret (1993) have shown that nonreflexive subspaces of L1 fail to have FPP. Thus: A subspace of L1 has FPP {=} it is reflexive.
Although it remains unknown whether superreflexive spaces have FPP, Maurey proved (also in 1981) that superreflexive spaces have FPP for nonlinear isometries (see [lJ or [36J for a proof). Recall that superreflcxive spaces are ones which have the property that every space which is finitely representable in such a space must itself be reflexive. Such spaces are characterized by that fact that they all have equivalent uniformly convex norms ([37]).
N onexpansive mappings 53
2.2. The Goebel-Karlovitz Lemma
The opening strategy in the proof of Theorem 2.1 was the use of Zorn's Lemma in conjunction with weak compactness to obtain a minimal nonempty closed convex Tinvariant set. Then the normal structure property of the space was used to obtain a contradiction if this minimal set has positive diameter. Such an approach is standard. On the other hand, the proof of Theorem 2.2 requires a refinement of that approach.
Any nonempty closed convex set K of a Banach space which is minimal with respect to being invariant under any mapping T : K -> K must satisfy co(T(K» = K. This has nothing to do with properties of the mapping T. However if T is nonexpansive and if K is also bounded then more can be said. The first is a key observation in the proof of Theorem 2.1.
Lemma 2.3 Each point of K is a diametral point; that is,
sup {llx - yll : y E K} = diam (K)
for each x E K.
Now let K be bounded closed and convex, fix z E K, and let T : K -> K be nonexpansive. Then for t E (0,1) the mapping Tt : K -> K defined by setting
Tt(x) = (1- t)z +tT(x)
is a contraction mapping with a unique fixed point Xt. Since
we have the following.
Lemma 2.4 If K is a bounded closed and convex subset of a Banach space and if T : K -> K is nonexpansive, then there exists a sequence {xn} C K such that
lim Ilxn - T(xn) II = O. n-+oo
This brings us to a fundamental fact which in conjunction with the previous lemma has proved extremely useful in extending fixed point results for nonexpansive mappings beyond those spaces which possess normal structure. It appears to have been discovered independently and at about the same time by Goebel [44] and Karlovitz [68]. This observation is crucial to the proof of Theorem 2.2.
Lemma 2.5 (Goebel-Karlovitz) Let K be a subset of a Banach space X, and suppose K is minimal invariant with respect to being non empty, weakly compact, convex, and T -invariant for some nonexpansive mapping T. Suppose {xn} C K satisfies limn-+oo IIxn - T(xn) II = O. Then for each x E K, limn-+oo Ilx - xnll = diam(K).
Proof. As we have just seen, if diam(K) > 0 then K can have no diametral points. Now suppose there exists {xn} ~ K for which limn-+oo Ilxn - T(xn) II = 0, but for which limn-+oo IIx - xnll = r < diam(K) for some x E K. Let
C = {z E K: lim sup liz - xnll :S r}. n-+oo
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It is easy to see that C is convex. Also C is closed. Indeed, if {Ui} ~ C and if limi~oo Ui = u, then for each i = 1,2,· .. ,
lim sup Ilu - xnll ::; lim sup lIu - uill + lim sup Ilui - xnll ::; lIu - uill + r. n---+oo n--+oo n--+oo
Letting i -+ 00, lim sUPn~oo Ilu - xnll ::; r. Thus U E C and C is closed. Also if U E K,
lim sup IIT(u) - xnll ::; limsupllIT(u) - T(xn)11 + IIT(xn) - xnll] n--+oo n--+oo
::; lim sup IIT(u) - T(xn ) II n~oo
::; lim sup lIu - xnll ::; r. n~oo
This proves that T : C -+ C. Sin,ce any closed convex subset of a weakly compact set is itself weakly compact, we conclude that C = K by minimality of K. However if diam(K) > 0 and if E > 0 is chosen so that r + E < diam(K), then the family {B(x; r + E) : x E K} has the finite intersection property since each such ball contains all but a finite number of terms of the sequence {xn}. Since these balls are weakly compact
nXEK B(x; r + to) 'f 0.
But this implies r(K) ::; r + to < diam(K), contradicting the fact that K cannot have any diametral points. •
2.3. Further Applications of the G-K Lemma
The Goebel-Karlovitz Lemma has been applied extensively in the study of fixed point theory for nonexpansive mappings, both as formulated in Lemma 2.5 and as stated in the language of ultraproducts (see [1] and references cited therein). One of the more recent applications is due to A. Jimenez-Melado and E. Llorens-Fuster in [63] in connection with their introduction and study of a generalization of uniform convexity called orthogonal convexity where, among other things, they used the Goehel-Karlovitz Lemma to prove that weakly compact convex subsets of orthogonally convex spaces have the wc-FPP.
Orthogonal convexity is defined as follows. For points x, y of a Banach space X and >. > 0, let
M>..(x,y) = {z EX: max(llz - xii, liz - yll) ::; ~(1 + >.) Ilx - YII}.
If A is a hounded subset of X, let IAI = sup{lIxll : x E A}, and for a bounded sequence {xn} in X and>' > 0, let
D({xn}) = lim sup (lim sup Ilxi - Xjll); 1---+00 J--+OO
A>.. ( {Xn}) = lim sup ( lim sup IM>..(Xi, Xj)I). 1--+00 ]_00
Definition 2.6 A Banach space is said to be orthogonally convex if for each sequence {xn} in X which converges weakly to 0 and for which D( {Xn}) > 0, there exists>. > 0 such that A>..({xn}) < D({xn}).
Nonexpansive mappings 55
In general it is difficult to test whether a space is OC, but it is noted in [62] that every uniformly convex Banach space is OC. For spaces having a Schauder basis the condition arises more naturally. Other examples given in [62] include Banach spaces with the Schur property (hence Rd, Co, c, and James's classical space J. (Also see [38, 64, 65].)
Theorem 2.7 Let K be a nonempty weakly compact convex subset of a Banach space X, and suppose K is orthogonally convex. Then every non expansive mapping T : K --4
K has a fixed point.
The opening move in the proof of this theorem is to invoke Zorn's lemma and assume K is minimal with respect to being nonempty and T-invariant. Next let {xn} C K satisfy limn~oo IIxn - T(xn) II = 0 and pass to a weakly convergent subsequence. Moreover (by a shift) one may assume {xn} converges weakly to 0 E K. The strategy is to then show that if diam(K) > 0 then K cannot be orthogonally convex. It takes several steps to accomplish this.
Another recent development is due to S. Prus [119] who has introduced a class of spaces which he calls uniformly noncreasy.
To give a precise definition of the concept we need some more notation. We shall use Ex and 5x to denote, respectively, the unit ball and unit sphere of a Banach space X. For any f E 5 x' and 8 E [0, 1], define the slice 5 (j, 8) to be the set
5(j,8) = {x E Ex: f(x) :::: 1- 8}.
Now consider two functionals f,g E Sx' and a scalar 8 E [0,1], and set
S(j, g, 8) = S(j, 8) n S(g, 8).
A Banach space is said to have a crease if there exist two distinct functionals f, 9 E S x' such that diamS(j, g, 0) > O. Since f(x) :::; 1 for each f E Sx and x E Ex, given f E Sx the hyperplane {x EX: f(x) = I} supports the unit ball Ex. Thus to say that X has a crease means that the sphere Sx contains a segment of positive length that lies on two different hyperplanes which support the unit ball Ex.
Uniformly non creasy spaces are characterized by the absence of creases in a relatively strong sense. A space X is called noncreasy if its unit sphere does not have a crease. Any Banach space X with dim(X) :::; 2 is trivially noncreasy, as is any strictly convex space.
Definition 2.8 A Banach space is said to be uniformly noncrcasy if given any E > 0 there is a 8 > 0 such that if f, 9 E X' and Ilf - gil:::: E, then diam(S(j, g, 8)) :::; E.
Every finite dimensional space, every uniformly convex space, and every uniformly smooth space is UNC. Also UNC is a self-dual property in the sense that X is UNC if and only if X' is UNC. Prus also proves that every UNC space is superreflexive.
Regarding FPP, Prus first shows that uniformly noncreasy is a super property; that is, if a Banach space X is uniformly noncreasy then the Banach space ultrapower X of X over any nontrivial ultrafilter 1A is also uniformly noncreasy. Then he shows (via an ultrapower approach) that uniformly noncreasy spaces have the FPP. It is in this
56
latter step that Prus invokes the Goebel-Karlovitz Lemma. Combined, these fact yield the following theorem.
Theorem 2.9 If a Banach space X is UNC, then both X and X have FPP.
2.4. Uniformly lipschitzian mappings.
In 1973 the class of uniformly lipschitzian mappings was introduced to the theory by the present authors in [47J. This class forms a natural extension of the family of nonexpansive maps.
Let M be a metric space with metric {!. A mapping T : M -> M is said to be lipschitzian (satisfies Lipschitz condition, is a Lipschitz map) if there exists k 2 0 such that for all x,yEM,
(!(Tx,Ty) :S ke(x,y).
The smallest k for which the above holds is called the Lipschitz constant of T and is denoted by k(T). For mappings T,8 : M --> M,
k (T 0 8) :S k (T) k (8)
and in particular
n == 0,1,2,···.
Nonexpansive mappings are those with k(T) :S 1. Obviously all the iterates of a nonexpansive map are nonexpansive.
Two metrics {!, ron M are said to be equivalent (uniformly equivalent) if there are two constants a > 0, b > ° such that for all x, y E M
ar(x,y):S {!(x,y):S br(x,y).
If (! and r are equivalent then any e-lipschitzian mapping is r-lipschitzian. Moreover for the relevant Lipschitz constants kg(T) and kr(T) we have
a b bkr (T) :S kg (T) :S ~kr (T) .
It follows that if T is r-nonexpansive, Kr(T) :S 1 then
b sup {kg (T") : n == 0,1,2, ... } :S - < +00.
a
This leads to the following definition.
Definition 2.10 A mapping T : M --> M is uniformly lipschitzian if there exists a constant k 2 ° such that for all x, y E M and n == 0,1,2,· .. ,
In other words for uniformly lipschitzian mappings we have kg(T") :S k for n == 0,1,2, ... . If k is given we say also that T is uniformly k-lipschitzian. Any uniformly lipschitzian mapping generates an equivalent metric r given by
r (x, y) == sup{{! (Tnx, T"y) : n == 0,1,2, ... },
Nonexpansive mappings 57
with respect to which T is nonexpansive. Thus uniformly lipschitzian mappings on a metric space (M, p) are completely characterized as those which are non expansive with respect at least one metric equivalent to g.
In particular all periodic lipschitzian mappings are uniformly lipschitzian. Also if K is a lipschitzian retract of M and R : M --t K is a lipschitz ian retraction satisfying k(R) = k then any nonexpansive mapping T : K --t K admits a natural uniformly lipschitzian extension T : M --t M, T = ToR with k(Tn) = k(Tn 0 R) ::; k.
To state our first fixed point results for uniformly lipschitz ian mappings we recall some basic facts. The modulus of convexity of a Banach space X is the function Ox : [0,2] --t
[0,1] defined by
8x (f) = inf { 1 -II x; y II : Ilxll ::; 1, Ilyll ::; 1, Ilx - yll 2 f} .
The characteristic of convexity of X is the number co(X) = SUp{f : 8x(f) = O}. The space X is said to be uniformly convex if fO(X) = O. Uniformly convex spaces as well as all spaces satisfying fO(X) < 1 have normal structure. Spaces with fO(X) < 2 are superreflexive. For extensive discussion of the properties of the above notions and related topics see the section by S. Prus in this volume.
The first fixed point result for uniformly lipschitz ian mappings was given in [47].
Theorem 2.11 Let C be a nonempty closed, bounded and convex subset of an uniformly convex Banach space X and let T : C --t C be a uniformly k-lipschitzian mapping with k satisfying
k(I-8x(k))<1.
Then T has a fixed point in C.
The above result was obtained by what is now called the asymptotic center technique(see Section 4.4). This technique can also be extended to the case of spaces with EO(X) < 1.
The seminal result in this direction, couched in a general metric space setting, was obtained by E. A. Lifschitz in 1975 ([101]).
Let (M, p) be a complete metric space. The balls in M are said to be c-regular for c 2 1 if the following holds: For any k < c there are numbers /L, a E (0,1) such that for any x, y E M and any r > ° with p(x, y) 2 (1- /L)r, the set B(x, (1 + /L)r) nB(y, k(1 + fl)r) is contained in a closed ball of radius ar.
The balls in any metric space are I-regular, and if the balls are c-regular for some c > 1 then they are d-regular for all d E [1, c].
The number
'" (M) = sup{ c 2 1: the balls in Mare c-regular}
is called the Lifschitz characteristic of M. Obviously ",(M) 2 1, and balls in Mare "'(M)-regular.
For Banach space, the above definition has a simpler and more intuitive formulation: if X is a Banach space and if r(D) denotes the Chebyshev radius of DC X, then
",(X) = sup{c 21: r(B (0, 1) n B (x,c)) < 1 for all Ilxll = I}.
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It is not difficult to verify that K,(H) = V2 if H is a Hilbert space and the same holds for any closed convex subset D of H which has dimension at least 2. For other Banach spaces this is not the case. The value of K,(D) may vary with the set D. In [31] Downing and Turret discussed the following coefficient for an arbitrary Banach space:
K,o (X) = inf {K, (D) : D is a nonempty convex, closed and bounded subset of X}.
The condition K,o(X) > 1 implies reflexivity of X. Also K,o(X) ~ ry, where 1 :"0 ry is the unique solution of the equation
Consequently eo(X) < 1 implies that K,o(X) > 1.
Theorem 2.12 Let (M, (!) be a boundp-d and complete metric space, and suppose T : M --> M is uniformly lipschitzian, with
for x, y E M, n = 1,2,· . ·,where k < K,(M). Then T has a fixed point in M.
Proof. The case K,(M) = 1 is obvious. So suppose K,(M) > 1. Select x E M and let r(x) be defined as
r (x) = inf {r > 0: for some y E M, {! (x, T"y) :"0 r, n = 1, 2, ... } .
Now let fl be the positive number associated with k in the definition of K,(M)-regular balls. For selected x E M there is an integer m such that {!(X,TffiX) ~ (1 - fl)r(x). Also there is a point y E M such that {!(x, Tny) :"0 (1 + fl)r(x) for n = 1,2,· ...
Thus by K,(M)-regularity, the set
D = B (x, (1 + fl) r (x)) n B (Tmx, k (1 + fl) r (x))
is contained in a closed ball centered say at z E M, and having radius ar(x) where a < 1. Next observe that for n > m,
{! (rrnx, Tny) :"0 k{! (x, Tn-my) :"0 k (1 + fl) r (x).
This shows that the orbit {Tny : n > m} is contained in D; hence in B(z,ar(x)). Consequently r(z) :"0 ar(x). Also, for any u E D,
{!(z,x) :"0 {!(z,u) + {!(u,x) :"0 ar (x) + (1 + fl) r (x) = Ar (x)
where A = a + 1 + fl. By setting Xo = x and z(xo) = z, it is possible to construct a sequence {xn} with xn+! = z(xn), where z(xn) is obtained via the above procedure. Thus r(xn) :"0 anr(xo) and (!(Xn+l,Xn) :"0 Ar(xn), and hence {xn} converges to a fixed point. •
The next result is due to E. Cassini and E. Maluta [27]. A Banach space X is said to have a uniform normal structure iffor some hE (0,1) and every convex bounded subset D c X, the Chebyshev radius r(D) :"0 hdiam(D). The normal structure coefficient N(X) of the space X is defined as
. {diam(D) } N (X) = mf r (D) : D C X is closed, bounded and convex .
Nonexpansive mappings 59
Thus X has uniform normal structure if and only if N(X) > 1. In general the normal structure coefficient is difficult to calculate. However, for example, S. Prus [118J has shown that N(IJ') = min{21/p , 21/ Q }, where q = p(p - 1)-1. Theorem 2.13 Let C be a nonempty closed, bounded and convex subset of an uniformly convex Banach space X with N(X) > 1, and let T : C -+ C be a uniformly k-lipschitzian mapping with k satisfying
1
k<N(X)2.
Then T has a fixed point in C.
The preceding three theorems are the most widely known results on existence of fixed points for uniformly k-lipschitzian mappings. We can now formulate the general scheme. We say that a metric space M has the fixed point property for uniformly k-lipschitzian mappings, FPP(k) for short, if all uniformly k-lipschitzian mappings T : M -+ M have fixed points. We now define another metric space characteristic:
'Y(M) = sup{k;::: 1: M has FPP(k)}.
Also, for a Banach space X, we set:
'Yo (X) = inf h (D) : D is a convex, closed and bounded subset of X}
Our theorems give some evaluations for 'Y(M). First, in general, 'Y(M) ;::: K,(M). For any convex closed and bounded subset D of a Banach space X we have the uniform evaluations 'Y(D) ;::: 'Yo(X) ;::: K,o(X); also 'Y(D) ;::: 'Yo(X) ;::: N(X)1/2, and 'Y(D) ;::: 'Y where 'Y satisfies 'Y(1 - ox(1h)) = 1. This implies that for spaces with eo(X) < 1, 'Y(D) > (1 - ox(I))-1 > 1. Mutual relations between such estimations have been discussed by several authors. Also there have been attempts to evaluate 'Yo(X) for concrete spaces. For example Teck-Cheong Lim [105J proved that in IJ' spaces with 2 < p < 00,
'Yo (X) ;::: (1 + (~: :;;11) * , where a is a unique solution of the equation (p- 2)xp - 1 + (p-l)xP- 2 = 1 in the interval (0,1). For the case 1 < p ::; 2, R. Smarzewski [132J found the estimate 'Yo(lJ') ;::: Vii. Tomas Dominguez Benavides [30J presented an interesting improved evaluation involving three constants. For any Banach space X
(X) 1 + Jl + 4N (X) (K,O (X) - 1) 'Yo;::: 2 .
The best evaluation known for any set D in a Hilbert space H is 'Y(D) ;::: K,(D) = K,o(H) = y2. However in this case we know also an evaluation from above. An example was mentioned in the original work of Lifschitz and also independently presented by Jean-Bernard Baillon [6J.
Example 2.14 Let B+ be the positive part of the unit ball in the space 12. Define the mapping T : B+ -+ B+ as
Tx = (cos 7r IIxll) e + _1_ sin 7r IIxll Px 2 1 Ilxll 2 '
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with the natural extension TO = e1, where e1 = (1,0,0,· .. ), and Px = (0, Xl, X2,· .. ) is the shift operator. The mapping T is uniformly I - lipschitzian and fixed point free.
In view of the above example v'2 ::; 'Yo(H) ::; 'Y(B+) ::; 'If /2. The exact value of 'Yo(H) is unknown.
There is another direction in which the study of uniformly lipschitzian mappings has been aimed. Let X be a Banach space having wc-FPP. There is a question whether this property is stable under renorming. We formalize this as follows.
Let B be a Banach space with the original norm II . II. Let /If be the family of all norms equivalent to II· lion B. For two norms II . 111, 11·112 set
( 1IXlb) (1IxI11) K (11·111, 11·112) = ~~~ II x l1 1 ~~~ IIxlb .
The function K is called the Banach-Mazur distance between the equivalent norms. This is not a metric but the family /If may be metrized by taking
iJ (11·111.11 ·112) = InK (11·111.11 . 112).
Suppose now D c X is convex, closed and bounded and T : D -> D is nonexpansive with respect a norm I I E /If. Then obviously T is K(II . II, I . I)-uniformly lipschitzian with respect to the original norm II . II. However I . I-nonexpansivness implies more. We have infxED Ilx - Txll = 0, and moreover the mapping F = ~(I + T) is asymptotically regular (see subsection 4.1).
Suppose now that the space (X, II . II) has wc-FPP. Define the new characteristic
'YN(X) = sup{k: if K(II·II, 1·1) < k then (X,I·I) has wc- FPP}.
Obviously 1 ::; 'Yo(X) ::; 'YN(X). We refer to 'YN(X) as the stability constant. Apart from Schur spaces (spaces for which weak and strong compactness coincide), where 'YN(X) = 00, an exact value of the stability constant is not known for any space. A crucial question for fixed point theory for nonexpansive mappings whether the assertion that all superreflexive spaces have wc-FPP is equivalent to the assertion that 'YN(X) = 00 for all uniformly convex spaces.
A considerable effort has been devoted to finding estimations of 'YN in particular spaces, especially the IP spaces, 1 < p < 00. For example Benavides [29] first proved that
'YN(lP) 2 (1 + 2;)~ where q = p(p - 1)~1. This gives for Hilbert space 'YN(H) 2 ,,13. This result has been improved by A. Jimenez Melado and E. Lorens-Fuster [65J who showed that 'YN(lP) 2 va where C is the smallest solution of
( 'l 'l) E C(C-l)= Cp+(2C-2)p q
which is greater than 1. For a Hilbert space, p = 2, which gives 'YN(H) 2 V2 + v'2 :0::;
1.85. Finally, with the use of nonstandard (ultrafilter) methods, Pei-Kee Lin [109] established the estimate
'YN(H) 2 V(5 + Vl3)/2 :0::; 2.07.
There are various results concerning other coefficients and their applications to the fixed point theory for uniformly lipschitzian mappings scattered in the literature. Interested
Nonexpansive mappings 61
readers should consult the section by S. Prus in this volume, as well as the book by J.M. Ayerbe Toledano, T. Dominguez Benavides and G. Lopez Acedo [4]. Also we call attention the expository article by N.M. Gulevich [56].
We conclude this section with the following observation. Given a Banach space X. The family of all spaces obtained by renorming X can be identified with the space N of norms on X. Coefficients such as EO, "'0, N can be considered as functions of the norm in the space N: Eo(11 . II), "'0(11 . II), N(II . II). It is not difficult to observe that they are continuous. Suppose X is superreflexive. Then there exists a uniformly convex norm I . lEN on X. It is a technicality to prove that for any other norm II . II E N and for any A > 0 the norm II . II>- = II . II + AI . I is uniformly convex. Thus uniformly convex norms form a dense set in N. In view of the continuous dependence of EO on the norms in N, the family of all norms satisfying Eo(11 . II) < 1 is open and dense in N.
Similarly, the sets of norms satisfying "'0(11·11) > 1 and N(II'II) > 1 are open and dense inN.
The above implies that at least in superreflexive spaces the family of equivalent norms for which X has FPP or more general FPP(k) for some k > 1 contains an open and dense subset. As mentioned before it is not known if FPP is stable under renorming, but at least the norms failing FPP are in a topological sense rare.
2.5. Subclasses of nonexpansive mappings
Various subclasses of nonexpansive mappings arise in natural ways. In our standard setting, where D c X is a convex closed san bounded set the family F of nonexpansive mappings T : D --t D can be viewed as convex closed and bounded subset of the space C[D, X] of all continuous mappings of D into X furnished with the natural uniform norm
IITIICID,Xj = sup {IITxllx : XED}.
The subfamily Fo C F of all contractions is dense in F in the topology generated by this norm. Obviously, in view of the Banach Contraction Principle, we can say that D has the fixed point property with respect to Fa but may fail the FPP with respect to the whole family F. There are several other interesting subclasses of F.
The isometries are mappings T : D --t D satisfying
IITx - TYII = IIx - yll . The notion of normal structure, which is so useful in the fixed point theory for nonexpansive mappings, was actually introduced by M. S. Brodskii and D.P. Milman in 1948 [10] Milman to study the fixed points of isometries. They proved that the normal structure plus weak compactness of D implies existence of a common fixed point of all isometries of D. A very deep result for isometries has been obtained by B. Maurey [112].
Theorem 2.15 Let D be a weakly compact convex s'ubset of a superrefiexive Banach space. Then D has the fixed point property relative to the class of isometries.
Another interesting class consists of so called contractive mappings, These are mappings satisfying
IITx - Tyll < Ilx - yll
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for all x, y E D with x oF y. The above condition implies two things. A contractive mapping T has at most one fixed point, and if T is fixed point free and D is weakly compact then T has exactly one minimal closed convex invariant set (see Section 3.2). It is easy to see that D has the fixed point property for contractive mappings if D is weakly compact and such that for any closed convex KeD, there is a point zED (not necessarily z E K) such that
sup {liz - xii: x E K} < diam (K).
Observe that this condition is a modification of normal structure.
In general it is not known if there are convex sets having fixed point property for contractive mappings yet failing FPP for the whole class F.
In 1979 R. Bruck explicitly introduced the following notion. (The concept apparently originates with J. B. Baillon's 1978 Paris VI Thesis.) Let r denote the set of strictly increasing convex functions, : lR+ -t lR+ with ,(0) = o. A mapping T : D -t D is said to be of type r if there exists, E r such that
,(lleTx + (1 - e) Ty - T (ex + (1- e) yll) ::; Ilx - YII- IITx - Tyll·
Three facts about such mappings are easy to observe. Mappings of type rare nonexpansive, affine nonexpansive mappings are of type r and the fixed point sets of mappings of type r are convex. The main result of Bruck [23] was the observation that in uniformly convex spaces any nonexpansive mapping T : D -t D is of type r.
The connection with nonexpansive mappings in uniformly convex spaces is closer than might at first appear. Indeed, Khamsi [76] has shown that for a Banach space X the following are equivalent: (a) X is uniformly convex. (b) There exists, E r such that every nonexpansive mapping T : K -t K with diam(K) = 1 is a mapping of type r for ,. So in these spaces the class of mappings of type r coincides with the whole class F. In less regular spaces there are mappings with nonconvex sets of fixed points (see Section 3.2), and this shows that there are nonexpansive mappings which are not of type r for any,. The class of mappings of type r has proved to be very useful in connection with weak approximation of fixed points (see Section 4.3).
The next class was also introduced by R. Bruck [21]. A mapping T : D -t D is said to be firmly non expansive if for any x, y E D the function
<I>x,y (t) = 11(1 - t) (x - y) + t (Tx - Ty)ll, t E [0,1]
is nonincreasing on [0,1]. Obviously any firmly nonexpansive mapping is nonexpansive. The converse is not true. For example if D is a subset of Hilbert space H, then T: D -t D is firmly nonexpansive if and only if it is of the form T = ~(I + G) where G : D -t H is nonexpansive. However there is an interesting observation. For any xED and a E [0,1) consider the equation
z = (1- a)x+aTz.
Since the right hand side of this equation is a contraction with respect to z, it has exactly one solution, say Za' Since this solution depends of x and a we can define a family of mappings Fa: D -t D, a E [0,1), by putting Fax = Za' It is minor technicality to prove that all mappings Fa are firmly nonexpansive. Moreover for any a E [0,1) the fixed point set FixFa of Fa coincides with FixT. This shows that the fixed point property for firmly nonexpansive mappings coincides with FPP for the whole class F. Finally boundedness of D implies that
lim IWax - TFax11 = O. a->1
Nonexpansive mappings 63
This fact has been used to study the strong approximation techniques (see Section4.2).
Finally we turn our attention to the class of rotative mappings. If T : D -> D is nonexpansive then for any n = 1,2, ... we have
Ilx - T"xll :::; n Ilx - Txll·
The mapping T is said to be (n, a)-rotative with a < n if for all xED we have
IIx - Tnxll :::; a Ilx - Txll .
T is rotative if it is (n, a)- rotative for some n, with a < n. All contractions are rotative, there are also rotative isometries. There are also nonexpansive mappings which are not (n,a)- rotative for any pair (n,a).
In contrast to the general case, the fixed point property for rotative nonexpansive mappings does not depend on the geometry of the set D. Any closed convex set D, regardless of whether it is bounded or not, has this property.
For a more elaborate discussion of this topic, see the section by W. Kaczor and M. Koter elsewhere in this volume.
2.6. The Approximate Fixed Point Property
A convex subset K of a Banach space is said to have the approximate fixed point property (a.f.p.p.) if every nonexpansive T : K -> K satisfies
inf {llx - T (x) II : x E K} = O.
As we have seen, bounded convex sets always have this property. The first observation that some unbounded sets in Hilbert space have this property was made by K. Goebel and T. Kuczumow. They proved in [52] that if K c £2 is a block, i.e., a set of the form K = {x E £2 : l(x,ei)1 :::; Mi} where ei is an orthonormal set in £2, then K has the a.f.p.p. This result was extended by W. Ray in [120] to include all linearly bounded subsets of £p, 1 < p < 00. (A subset K of a normed space is linearly bounded if K has bounded intersection with all lines in X.) Ray went on to prove in [122] that a closed convex subset K of a real Hilbert space has the fixed point property for nonexpansive mappings if and only if it is bounded, from which it follows that K has the a.f.p.p if and only if it is linearly bounded. Subsequently in [125], S. Reich proved that every linearly bounded closed convex subset of a reflexive Banach space has the a.f.p.p. Finally, in order to characterize those closed convex sets which have the a.f.p.p. in general Banach spaces, 1. Shafrir [126] introduced the concept of a directionally bounded set. A directional curve in a Banach space X is a curve 'Y : [0,(0) -> X for which there exists b > 0 such that for each t 2': s 2': 0,
t - s - b :::; Ii'Y (s) - 'Y (t) II :::; t - s. A convex subset K of X is said to be directionally bounded if it contains no directional curves. Note that a line is a directional curve with b = O. Thus directionally bounded sets are always linearly bounded. Shafrir proved two noteworthy things in [126].
(1) A convex subset of a Banach space has the a.f.p.p if and only if it is directionally bounded.
(2) For a Banach space X the following are equivalent: (i) X is reflexive. (ii) Every closed convex linearly bounded subset of X is directionally bounded.
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These facts may be summarized as follows.
Theorem 2.16 For a Banach space X the following are equivalent.
(i) X is reflexive.
(ii) A closed convex subset K of X has the a.f.p.p if and only if it is linear'ly bounded.
3. Properties Of The Fixed Point Set
The investigations of the properties of the sets of fixed points for nonexpansive mappings have their origin in the following observations.
Let K be a convex closed subset of a Banach space X and let T : K --> K be nonexpansive. If the norm of X is strictly convex, the fixed point set F(T) is closed and convex. If X = H is a Hilbert space then any nonempty closed convex subset Ko of K is the range of a nonexpansive retraction r : K --> Ko. The nearest point projection is a standard example of such retraction.
In the general case the above is no longer true. There may exist closed convex subsets of K which are not of the form F(T) for any nonexpansive F (see the example in Section 4.4). Also the fixed point sets and nonexpansive retracts do not need to be convex.
The investigations of the properties of fixed point sets concentrated mostly on the question under which condition the set F(T) is necessarily the range of a nonexpansive retraction.
3.1. Nonexpansive Retracts
There are three principal theorems about the structure of thc fixed point sets of nonexpansive mappings. The first is a classical Banach space result due to R. E. Bruck [22], and the remaining two are metric space results due, respectively, to J. B. Baillon [7] (for hyperconvex spaces, discussed in more detail elsewhere in this work) and M. A. Khamsi [77] (for arbitrary metric spaces, treated in Section 6.3 below).
We first take lip Bruck's result, beginning with the separable case. It is perhaps surprising that in any separable space the following simple assumption is enough to assure the desired result.
Definition 3.1 A bounded closed convex subset K of a Banach space is said to have the hereditary fixed point property (HFPP) for nonexpansive mappings if every nonexpansive mapping f : K --> K has a fixed point in every nonempty bounded closed convex f-invariant subset of K.
If K is a bounded closed convex subset of a Banach space which has FPP the clearly K has (HFPP). Thus by what we have already seen any bounded closed convex subset of a reflexive Banach space which has normal structure has (HFPP) as do any weakly compact convex subsets of orthogonally convex and uniformly noncreasy spaces.
Theorem 3.2 Suppose K is a bounded closed convex subset of a sepamble Banach space X, suppose K has (HFPP), and let T : K --> K be nonexpansive. then the fixed point set F(T) is a (nonempty) nonexpansive retract of K.
Nonexpansive mappings 65
This result is an easy consequence of the following rather deep fact, which is also proved in [22].
Theorem 3.3 Let M be a separable complete metric space and let S be a semigroup of nonexpansive self-mappings of M. Then there exists in S a retraction r of Manto F(S) if and only if one of the following two (equivalent) conditions holds.
(i) Each nonempty closed S-invariant subset of M contains a fixed point of S.
(ii) Whenever x E M then clS(x) n F(S) =10.
Proof of Theorem 3.2. Let KK denote the family of all mappings of K -+ K and put
S = {s E KK : s is nonexpansive and F(s) ;2 F(T)}.
Obviously S is a semigroup on K and F(S) ;2 F(T). Since T E S we conclude that F(S) = F(T). For each x E K the set S(x) = {s(x) : s E S} is clearly nonempty, convex, and T-invariant. Since T is continuous, cl S(x) is a nonempty bounded closed convex T-invariant subset of K, so by (HFPP) cl S(x) n F(T) =I 0. This shows that S satisfies (ii). The conclusion now follows from Theorem 3.3. •
The following is a direct consequence of the preceding theorems.
Corollary 3.4 Suppose K is a, bounded closed convex subset of a separable Banach space, suppose that K has (HFPP), and let J' be a finite family of commuting nonexpansive mappings of K -+ K. then the common fixed point set F(J') ofJ' is a (nonempty) nonexpansive retract of K.
The above corollary actually extends to arbitrary commutative families J'. Rather than pursue this further, however, we take up the question of what can be said in the nonseparable case. The central observation is that the separability assumption on the space can be replaced with reflexivity or, more generally, with the assumption that the domain is weakly compact. The basic ideas here are also due to Bruck [20].
Theorem 3.5 Let K be a nonempty weakly compact convex subset of a Banach space X and suppose T is a family of nonexpansive mappings of K -+ K with a non empty common fixed point set A. Suppose further that
A intersects every nonempty T-invariant closed convex subset of K. (3.1)
Then A is a nonexpansive retract of K.
Proof. As before let KK denote the family of all mappings of K -+ K and let
N = {J E KK : f is nonexpansive and F(f) ;2 A} ,
where F(f) denotes the fixed point set of f. Obviously T ~ lJ1 so N =10. Notice also that by Tychonoff's Theorem K K is compact in the topology of weak pointwise convergence, since this is the product topology on the space IIxEK K induced by the weak topology onK.
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Now for I,!' E N say I ~ I' if /lp - l(x)11 = /lp - I'(X) II for each pEA and x E K, and for lEN let
[/J = {f' EN: I' ~ f}.
Let [NJ = {[/J : lEN} and introduce the partial order ~ on [NJ by saying [/J ~ [gJ if and only if lip - I(x) II ::; lip - g(x)11 for each pEA and x E K. To see that ([N],~) has a minimal element, let {[I",]' a E I} be a descending chain in ([N], ~). Since KK is compact in the topology of weak pointwise convergence the net {fa} has a subnet {f"'E} which converges in this topology to some I E KK. Thus for each x E K,
weak-lim I"'E (x) = I(x). {
Now let x E K and pEA. Since {[/",],a E I} is descending in ~ it is the case that lim{ /lp - la, (x)/I exists, and since norm-closed balls in X are weakly closed, if (' > 0 it must be the case that lip - l(x)11 ::; r + (' for any (' > OJ hence
lip - I(x)/I ::; lim lip - I",,(x) II = lim lip - 1",(x)ll· { '"
This proves that [fJ ~ [f",J for each a E I. Also, for X,Y E K, {f",,(x) - I",,(Y)} converges weakly to I(x) - I(Y) (algebraic operations are always weakly continuous) so
Finally if x E A then I",(x) = x for each a E I so it follows that I(x) = x. Thus F(f) ;2 A and this proves that {[/a], a E I} is bounded below by [fJ in ([N], ~). It is now possible to invoke Zorn's lemma to conclude that ([NJ,~) contains a minimal element [rJ.
Now suppose there exists a point z E K such that r(z) ¢ A. Since lip - r 0 r(x) II ::; IIp- r(x) II for each pEA and x E K, [r 0 rJ ~ [rJ. But since [rJ is minimal, [ro rJ = [rJ. In particular, if Zo = r(z) then for all pEA, lip - r(zo)11 = lip - zoll > O. Let
C = {f 0 r(zo) : lEN}.
Since I,g E N implies AI + (1 - >..)g E N for any>.. E [0,1]' C is convex. To see that C is weakly compact it suffices to show that C is closed. Let {xa } be a net in C for which limaxa = x. Then for each Q there exists la EN such that Xa = la 0 r(zo). But since KK is compact in the topology of weak pointwise convergence {fa} has a subnet {faE}which converges to some I E KK. Arguing as before, I is nonexpansive and F(f) ;2 A. Hence lEN, so 10 r(zo) E C. But
x = weak-lim la, 0 r(zo) = 10 r(zo). {
This proves that x E Cj hence C is closed.
Let lEN. Then F(f) ;2 A so for each pEA,
lip - I 0 r(zo)/I = /l/(p) - I 0 r(zo)/I ::; /lp - r(zo)lI·
Thus [/orJ ~ [rJ. But since [rJ is minimal this implies IIp- lor(zo)1I = IIp-r(zo)11 > 0 for each pEA. Therefore A n C = 0. But as noted at the outset '1" <:;:; '.)1, so s : C --t C for each s E '1". Thus An C i= 0 by our basic hypothesis. This is a contradiction, so we conclude that r(z) E A for each z E K. •
Nonexpansive mappings 67
The following is an obvious special case of the preceding theorem.
Corollary 3.6 Suppose K is a weakly compact convex subset of a Banach space, suppose K has (HFPP), and let T : K -'> K be nonexpansive. Then the fixed point set F(T) f T is a nonempty non expansive retract of K.
Remark. In connection with the foregoing, T. C. Lim [102J proved that X is reflexive and has normal structure, then every left reversible semigroup of nonexpansive selfmappings of a bounded closed convex subset K of X has a common fixed point.
It is interesting, and sometimes useful, to know that the fixed point set of a nonexpansive mapping is a nonexpansive retract of its domain. The following simple facts serve to illustrate this.
Theorem 3.7 Suppose M is a metric space which has the property that the fixed point set of every non expansive mapping of Minto M is a nonempty non expansive retract of M, and suppose T and G are commuting nonexpansive mappings of Minto M. Then F(T) n F( G) =I 0.
Proof. First observe that if x E F(T) then To G(x) = Go T(x) = G(x). This proves G: F(T) -'> F(T). By assumption there is a nonexpansive retraction r of ]v! onto the nonempty set F(T). Thus Go r : M -'> F(T) is nonexpansive and by assumption has a nonempty fixed point set. However, since r is a retraction onto F(T), it must be the case that F(G or) = F(G) nF(T). This proves that F(T) nF(G) =10 (and at the same time that F(T) n F(G) is a nonexpansive retract of M). •
A mapping T : M -> M is said to be eventually nonexpansive if there exists an integer N such that for each n ~ N, Tn is nonexpansive.
Theorem 3.8 Suppose M is a metric space which has the property that the fixed point set of every nonexpansive mapping of Minto M is a nonempty non expansive retract of M, and suppose T : M -> M is eventually nonexpansive. Then F(T) =I 0.
Proof. For N sufficiently large the mappings TN and T N +1 are commuting nonexpansive mappings of Minto M. By the previous theorem F(TN) n F(TN+1 ) =10. But it is trivial to check that F(T) = F(TN) n F(TN +1). •
The above results are not really about nonexpansive mappings since they only require that there is a retraction from the domain of the mapping onto its fixed point set which can be drawn from a class of mappings for which the original domain has the fixed point property. It just turns out that the class of nonexpansive mappings is amenable to this approach.
3.2. General Case
The above results give rise to the opposite question. How "irregular" F(K) can be? Also there is a question concerning the regularity of "approximate" fixed point sets. Let again K be a bounded closed and convex, and for nonexpansive T : K -> K and 10 > 0, let
FE (T) = {x E K: Ilx - T;);II ::; e}.
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R.Bruck [23] has shown
Theorem 3.9 For any K and T as above the set Fe(T) is pathwise connected.
This may suggests that some kind of regularity should be inherited by F(T). However it is not the case.For given Banach space X consider the space X x CO furnished with the max norm
II(x,y)11 = lI(x,(Yl,Y2,···))11 =max{llxllx,IIYllcJ·
Let D c X be a nonempty closed set. Then the mapping T : X x Co -4 X X Co defined by
T (x, y) = T (x, (Yl, Y2, ... )) = (x, (dist (x, D) , Yl,Y2, ... ))
is nonexpansive (even isometric) and we have F(T) = D x {OJ. This proves
Theorem 3.10 Any nonempty closed subset of X is isometric to a fixed point set of an isometry of X x Co into itself
Consequently, since any complete, separable metric space (Polish space) is isometric to a closed subset of C[O, 1], we observe that any such space is isometric to a fixed point set of an isometry of C[O, 1] x co.The above remarks come from [25]. There it is also shown that any nonempty compact subset fa CO is the fixed point set for certain nonexpansive T : Co -4 co. Thus in general case the fixed point sets can be "bizarre". For example; totally disconnected.
Finally, the regularity properties has been studied not only for fixed point sets but also for families of minimal invariant sets. Let us begin with the already mentioned Alspach's example.
Example. Let X = LIlO, 1] and let C be the order segment given by
The baker transformation
(TJ)(t)={ min{2f(2t),1} ifO<;.t<;.! max{2f(2t-1)-1,0} iq<t<;.l
isomet.rically transforms C into itself. Only the two constant functions, 0 and 1, are fixed points of T. The whole set C = UaE[O,l] Ca where
Ca = {f E C : l f = a} are T -invariant convex, closed slices of C by parallel hyperplanes. Since C, is an order segment and therefore weakly compact, so are all the slices Ca. In view of these observations, each of the sets Ca , for 0 < a < 1, contains at least one nontrivial, that is of strictly positive diameter, minimal invariant subset. Further, since each Ca is T-invariant, these are the only minimal invariant sets for T. It is also known that except for a = 0 or 1 each of the Ca are not themselves minimal invariant and that the closed convex hull of an orbit need not be minimal invariant.
Nonexpansive mappings 69
Now let K be a convex weakly compact subset of a Banach space X. For any nonexpansive mapping T : K -> K there exists at least one minimal convex T-invariant subset Ko of K. Alspach's example shows that the family M(T) of such sets may consists of more than one set. It was shown in [46] and [54] that any two sets Ko, Kl in M(T) are metrically parallel in the sense that for any Xo E Ko and Xo E Ko
where H(Ko, Kl) is the Hausdorff distance between sets. This has several consequences. First the family M(T) is metrically convex with respect to H meaning that for any t E [0,1] and anyKO, Kl in M(T) there exists at least one set Kt E M(T) with H(Ko, Kt) = t and H(Kt, K1) = 1 - t. Moreover the sets Kt can be selected to form a continuous path isometric to the interval [0, H(Ko, Kl)]' Finally there exists a linear functional f E X' being constant on each Kt. If f(Ko) = a and f(Kl) = b then f(Kt) = (1 - t)a + tb.
This shows that, in case of the weak compactness, the families of minimal invariant sets show more regular behaviour than fixed point sets in general. It also shows that the situation observed in Alspach's example is typical for general case.
It was also observed that some geometric conditions imposed on the space X may have further influence on M(T). If, for example, X is strictly convex then any minimal invariant set is a shifted copy of any other, if X is a KK-space then all sets in M(T) are of the same diameter.
The structure of fixed point sets of nonexpansive mappings naturally gives rise to questions about the structure of 'approximate' fixed point sets. Let K be a bounded closed and convex, and for T : K -> K and c: > 0, let
FE:(T) = {x E K: IIx-T(x)1I S c:}.
In [24] it is shown that FE:(T) is always pathwise connected.
It seems that little else is known in the general case. Approximate fixed point sets as well a minimal invariant sets deserve more study.
4. Approximation
4.1. Asymptotic Regularity
The concept of asymptotic regularity was formally introduced by Browder and Petryshyn in [19].
Definition 4.1 A mapping f of a metric space (M,d) into itself is said to be asymptotically regular if limn->oo d(r(x), r+1(x)) = ° for each x E M.
In 1976 Ishikawa obtained a surprising result, a special case of which may be stated as follows: Let K be an arbitrary bounded closed convex subset of a Banach space X, T : K -> K nonexpansive, and A E (0,1). Set TA = (1- A)J + AT. Then for each x E K
ii11'(x) -11'+l(x)ii-> 0.
Thus by iterating the 'averaged' mapping TA one can always reach points which are approximately fixed (but on the other hand, these points may not be near fixed points - indeed, it need not be the case that T even have a fixed point).
70
In 1978, Edelstein and O'Brien [35] proved that {T,f(x) - Tf+1(x)} converges to 0 uniformly for x E K, and, in 1983, Goebel and Kirk [48] proved that this convergence is even uniform for T E ~, where ~ denotes the collection of all nonexpansive selfmappings of K. Thus:
Theorem 4.2 Suppose K is a bounded closed convex subset of a Banach space. Then for each E > 0 there exists a positive integer N such that if x E K and T E ~, n 2': N =?
For further information about this topic we refer to the article by Baillon and Bruck elsewhere in this volume. Here we just touch on something new by showing how Ishikawa's result can be obtained by embedding the problem in a larger space. The following special case of Theorem 4.2 suffices to illustrate the method.
Theorem 4.3 Let K be a bounded convex subset of a Banach space X and let T : K ---+ K be nonexpansive. Fix a E (0,1) and define fOt : K ---+ K by setting fOt(x) =
(1 - a)x + aT(x), (x E K). Then fOt is asymptotically regular on K.
The preceding result is quite significant. In particular, since the fixed point sets of f and T coincide, it effectively reduces the study of fixed point properties of nonexpansive mappings in convex sets to the corresponding study for asymptotically regular nonexpansive mappings.
We base the proof given here on the following simple metric space identity.
Lemma 4.4 Let (M, d) be a metric space and a E (0,1), and suppose {xo,"" XN} and {YO, ... , YN} are finite subsets of M satisfying:
(i) d(Yn,xn) == r 2': 0, (0 ~ n ~ N);
(ii) d(xnx n+1) = ar and d(xn+1, Yn) = (1 - a)r (0 ~ n ~ N - 1);
(iii) d(Yn+1,Yn) ~ d(Xn+l,Xn) (0 ~ n ~ N -1);
(iv) d(Yn,Xi) ~ (1- a)d(Yn,Xi_l) + ad(Yn,Yi-Il (0 ~ i ~ n ~ N).
Then d(YN, xo) = (1 + aN)r.
Lemma 4.4 can be proved using a straightforward induction argument on N.
Let K be a bounded convex subset of a Banach space X and denote by foo(X) and eo(X) the Banach spaces obtained by substitution of X into foo(l~) and eo(N). Let X denote the quotient space foo(X)/eo(X). For x = (Xl,X2,"') E foo(X), let x or [(Xn)] denote the equivalence class x + eo(X). The quotient norm is given by
Ilxllq = lim sup IIxnll. n-->oo
Then k := rr~=l K/eo(X) is a bounded closed convex subset of X. Using this embedding, Lemma 4.4 gives a very quick proof of Theorem 4.3.
Proof. Define T: D(T) c k -> k by setting for x = (Xl,X2,"') E k
T(x) = [(T(Xl), T(X2J.- .. )]
Nonexpansive mappings 71
whenever {T(xn)} is bounded, and define f analogously. Now let x E K and set
xo = [(f(x), j2(x), . . " r(x),· .. )].
Since {fn(x)} is bounded, Xo E k. Also, as we have seen in the previous argument, the sequence (1Ir(x) -T(r(x)lI) is nonincreasing. Thus since {r(x)} is bounded {T(r(x))} is also bounded. Suppose
Ilxo - T(xo)ll q = li;.n.:~p IIr(x) - T(r(x))11 = r ~ O.
Then if Xl := (p(.r), j3(x), . . " r+1(x), . .. )
IIXI - T(Xl)ll q = li~~p Ilr+1(x) - T(fn+1(x)) II = r.
Continuing, if Xk := (fk+1(x), jk+2(x), . . " jk+n(x),· .. ), then
Also notice that for each n,
Ilr(x) - T(r(x))11 = a Ilr(x) - r+1(x)11 = ar.
Thus if Yn := T(xn) we conclude Ilxn - Ynllq = r, n = 0, 1,2,' . '. Also
Xn+l = (1 - a)xn + aT(xn), n = 0, 1" . '.
These observations give (i) and (ii) of Lemma 4.4, and (iii) follows from the fact that Tis nonexpansive and Yn+1 = T(xn+1) with xn+1 E seg(xn, Yn). Since (iv) holds in any norrned space we conclude that for each n EN,
llYn - xoll q = (1 + an)r.
Since k is bounded, this can only happen if r = O. • The approach just discussed is expanded on further in [88].
4.2. Strong Convergence
It has been known for some time that even in a uniformly convex setting the averaged sequence of the previous section need not actually converge to a fixed point ([40]). On the other hand, in 1971 Kaniel [67] described a rather complicated discrete convergence procedure for approximating fixed points of nonexpansive mappings in such spaces. For a refinement of Kaniel's method, see Molony [113].
The following theorem due to Reich [124] provides what is perhaps the nicest (strong) convergence result.
Theorem 4.5 Let X be a uniformly smooth Banach space, K a bounded closed convex subset of X, and T : K -+ K nonexpansive. For fixed y E K and t E (0,1), let Yt denote the unique fixed point of the contraction mapping
TtO = (1 - t)y + tT(·).
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Then limt~l- Yt exists and is a fixed point ofT.
Baillon [6] first proved that T always has a fixed point under the above assumptions. Later Turett [137] proved that uniformly smooth spaces are both superreflexive and have normal structure (and thus have FPP).
Below we give an elegant proof of Theorem 4.5 due to Reich which makes use of Banach limits.
Recall that a Banach space is uniformly smooth if and only if its norm is uniformly Frechet differential. Thus if J : X --> 2x ' is the normalized duality mapping defined by
J(x) == {j E X* : (x,j) == Ilx11 2 , and Iljll == Ilxll}
(whose existence is always assured by the Hahn Banach Theorem) then J is single valued and uniformly continuous on any bounded subset of X which is bounded away from O. Indeed, J(z) == IlzllDz where Dz is given by
Dz (x) == lim h-1 [lIz + hxll - Ilzll] h~O
where the limit exists uniformly for IIxll == Ilzll == l.
Recall also that a Banach limit is a continuous linear functional F defined on £00 which satisfies for all {~n} E £00 :
(i) liminfn~oo~n::; F({~n})::; limsuPn~oo~n.
(ii) F({~n+d) == F({En}).
The existence of Banach limits is assured by the Hahn Banach Theorem.
Proof of Theorem 4.5. Since T always has at least one fixed point we may assume K is bounded. Fix a sequence {tn} C (0,1) with tn --> 1 and let Yn == Ytn . Note that Yn - TYn --> O. Now let F be a Banach limit and define f : X --> lR.+ by
f(x) == F ({llYn - xI12}).
Since f is a convex and continuous function and since X is reflexive f attains its minimum on K. Let C c K denote the set of minimizers of f. Then if u E K
f(T(u))=F({IIYn- T (u)11 2 })
==F({IIT(Yn)-T(u)1I 2 })
::;F({IIYn-UI12}) ==f(u).
Thus C is closed convex and invariant under T. Another application of Baillon's theorem yields a fixed point v of T which lies in C. Since T is nonexpansive
((I - T) (u) - (I - T) (v) , j (u - v)) ~ 0
from which (u - T (u), J (u - v)) ~ 0, u E K. Also
Nonexpansive mappings 73
hence
Thus if v is a fixed point of T,
( 4.1)
Now we use the fact that J is the subgradient of the function 'P(') = ~11·1I2. Therefore for X,z EX,
2(x, J (z)) ::; Ilx + zl12 - IIzl12 .
It follows that for all x E X and t > 0,
2t(x, J (Yn - V - tx)) ::; llYn - V - tx + txl1 2 - llYn - V - txl1 2 .
Let c > 0 and x E X. Uniform continuity of J implies that for all t > 0 sufficiently small
from which
(x,J(Yn-v)) <c+(x,J(Yn- v - tx ))
::; c + (lj2t) [llYn - vl1 2 - llYn - V - tXI12].
Therefore, since V is a minimizer of f,
F ((x, J (Yn - v))) ::; c + (lj2t) (J (v) - f (v + tx))
::; c.
Since c > 0 is arbitrary F( (x, J(Yn - v))) ::; 0 for all x E X, and in particular
F ((y - v, J (Yn - v))) ::; O.
Combined with (4.1) this gives F( {llYn - vI1 2}) = O. This implies that some subsequence of {Yn} converges strongly to v, say Ynk --+ Vi as k --+ 00. Then (4.1) implies
Adding gives IIvi - v211 2 = 0, so Vi = V2. It follows that limn-+oo Yn exists and, in turn, limt--> 1 - exists. •
4.3. Weak Convergence
There are three classical weak convergence results. The first involves an aspect of Gohde's original proof which inspired Browder to formulate the so-called demiclosedness principle.
Theorem 4.6 Let K be a closed and convex subset of a uniformly convex Banach space X, and let T : K --+ X be nonexpansive. Then the mapping (I - T) 'is demiclosed on K. Specifically, if {Uj} is a sequence in K which converges weakly to U and if {(I - T( Uj))} converges strongly to w E X, then U E K and (I - T(u)) = w.
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Browder's original proof of the above theorem (see [17]) actually reveals a little more. First it shows that in a uniformly convex space the following is true: if {Vj} and {Wj} are bounded sequences in a convex set K for which both {Vj - T( Vj)} and {Wj - T( Wj)} converge (strongly) to 0, and if Zj for each j is a point of the segment joining Zj and Wj then {Zj - T(zj)} also converges to O. This fact makes crucial use of the uniform convexity of the space. However once this fact is established the remainder of Browder's argument, which consists of a subtle thinning of the original sequence {Uj}, may be carried out in any Banach space using only continuity of T.
It is also trivial to see that the demiclosedness principle also holds in spaces which satisfy the so-called Opial's condition (i.e., the condition
Xn --> Xo weakly =? liminf Ilxj - xoll < liminf Ilxj - xii for all x =1= xo.) n---+oo n---+oo
Uniformly convex spaces which possess weakly sequentially continuous duality maps satisfy this condition, an observation due to Opial [114]. (For a complete proof see Browder [18]; also see, e.g., [49] p. 108.) In particular Opial's condition is independent of uniform convexity since the £p spaces satisfy this condition for 1 < p < 00, while it fails for the Lp spaces, p =1= 2. In fact, spaces satisfying Opia]'s condition need not even by isomorphic to uniformly convex spaces ([99]).
There have been a number of attempts to extend the demiclosedness principle. It is known ([83]) that the demiclosedness principle does not hold in the space (£2 EIl]Rl )00' Some extensions and related problems are also discussed in [108]. On the positive side, it is shown in [108] that if X is uniformly convex and if Y has the Schur property, then (X Ell Y)p satisfies the demiclosedness principle for 0 < p < 00.
In [39] it is shown that if e is an closed convex subset of an arbitrary Banach space and if T : e --> X is norm continuous and Q-almost convex for a continuous strictly increasing Q : ]R+ --> ]R+ with Q(O) = 0, then (I - T) is demiclosed at O. The mapping Tis Q-almost convex on e if for each x, y E e and A E [0,1]
Jy (AX + (1- A) y) ::; Q (max {JT (x), h (y)})
where Jr(x) = Ilx - T(x)lI. Earlier Khamsi [73] had obtained the same result for nonexpansive mappings of 'convex type' ; that is, for mappings T : e --> X which satisfy: if Jr(xn) --> 0 and Jr(Yn) --> 0 then Jr(~(Xn + Yn)) --> O. Clearly mappings of convex type are Q-almost convex.
Several people have considered the demiclosedness principle (at 0) for asymptotically nonexpansive mappings. Perhaps the strongest such result is due to Xu [141] who proved that if e is a bounded closed convex subset of a uniformly convex Banach space and if T : e --> e is asymptotically nonexpansive (thus there exists a sequence {kn } with kn --> 1 for which IIrn(x)-Tn(y) II ::; knllx-YII, X,y E e), then (I -T) is demiclosed at O. However the usefulness of the demiclosedness principle for asymptotically nonexpansive mappings is severely limited by the fact that it is not known in general whether an asymptotically nonexpansive mapping T : e --> e always has an approximate fixed point sequence - see [94] for a discussion.
There is another classical weak convergence that makes use of the fact that nonexpansive mappings in uniformly convex spaces are of type r (see Section 2.5).
As we have seen, if K c X is bounded and if T : K --> K is nonexpansive, then Ilr(x) - r+l(x) II --> 0 for each x E K, where f = ~(I +T). Therefore if X is uniformly
Nonexpansive mappings 75
convex, demiclosedness of (I - T) implies that any weak subsequential limit point of {r(x)} is a fixed point of T. The following result due to Reich [124] shows that, if X is also uniformly smooth, then in fact {In(x)} always converges weakly. The significance of this result is that it extends a fact known already for the I!p spaces, 1 < p < 00, ( [114]) also to the corresponding Lp spaces.
Theorem 4.7 Let X be uniformly convex and uniformly smooth, and let K c X be bounded closed and convex. Suppose {en} c [0,1] satisfies 2:::"=1 en(l - en) = +00. Then for any Xl E K the sequence {xn} defined by
xn+1 = (1 - cn) Xn + enT (Xn) ,
converges weakly to a fixed point of T.
n = 1,2,···
Proof. We only consider the case en == ~. Thus let X E K, and with f defined as above let r+1(x) = ~(r(x) + T(r(x))), n = 1,2,· ... Minor technical modifications yield the general result.
Let WI and W2 be two weak limit points of {xn }, and for each t E [0,1]' let
and
bn,m = Ilr (tr (x) + (1 - t) wI) - (tr+m (x) + (1 - t) WI) II·
Then
IIWI - W2 + tun+m(x) - WI)II
= IIWI - W2 + r(tr(x) + (1 - t)WI) - r(tr(x) + (1 - t)WI) + t(r+m(x) - WI)II
:S IIWI - r(tr(x) + (1 - t)WI) + t(r+m(x) - WI)II + IIr(tr(x) + (1 - t)WI) - r(W2) II
:S IIr(tfn(x) + (1- t)wI) - (tfn+m(x) + (1- t)WI) II + IIWI - W2 + t(r(x) - wI)II·
Thus
m,n= 1,2,· ...
Let M = IIx - wIiJ. Since fm is of type r there exists 'Y E r such that
'Y (bn,m) = 'Y (lir (tr (x) + (1 - t) WI) - (tr un (x)) + (1- t) r (wI))I!)
:S Ilr (x) - wlil-Ilr+m (x) - wIll·
(4.2)
Since {lIr(x) - wIll} is monotone decreasing, limn .... oo 'Y(bn,m) = 0 uniformly in m. In view of (4.2),
lim sup an :S lim inf an
so limn .... oo an(t) = a(t) exists for all t E [0,1).
Now assume WI i= W2 and let
n .... oo
Since X is uniformly smooth, if t: > 0 there exists t = t(t:) such that
Ian (t) It - dnl < t:.
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From this
limsupdn :<=; aCt) +c, n--->oo t
liminf dn > a (t) - C. n--+oo - t
Therefore limn--->oo dn exists, and since some subsequence of {fn(x)} converges weakly to WI this limit must be O. Since the corresponding limit with WI and W2 interchanged must also be 0,
Ilwi - w2112 = (WI - W2, J (WI - W2))
= (WI - r (x), J (WI - W2)) + (W2 - r (x), J (W2 - WI));
hence WI = W2. • The third classical weak convergence result is the mean ergodic theorem for nonlinear nonexpansive mappings. This fact asserts that if K is a bounded closed convex nonempty subset of a Banach space X whim is both uniformly convex and has a Frechet differentiable norm, and if T : K -> K is nonexpansive, then the Cesaro means of {Tn(x)} always converge weakly to a fixed point of T. This fact was first established by Baillon for the Lp spaces, 1 < p < 00, and a modification of his original proof establishes the more general fact. A simple direct proof of this fact can be obtained via the following result of Bruck [23].
Theorem 4.8 Suppose K is weakly compact, X has a Prechet differentiable norm, and T : K -> K is a nonexpansive mapping for which rn is of type, E r for each n. Then for each x E K, the sequence
converges weakly as n -+ 00 to a fixed point ofT.
4.4. Asymptotic center technique
The sequences of iterates for nonexpansive mappings as well as asymptotic fixed point sequences may determine or localize the fixed points in other way then being convergent. A useful tool has been introduced in 1972 by M. Edelstein (see [34]).
For x E X and a bounded {xn} C X define the asymptotic radius of {xn} at x as the number
r (x, {xn}) = lim sup IIx - xnll. n--->oo
If {xn } is fixed, rex, {Xn }) is a nonnegative, continuous and convex function of x. Moreover r(x, {xn }) = 0 if and only if x = limn--->ooxno
For a subset K of X, the number
and the set
Nonexpansive mappings 77
are called respectively, the asymptotic radius and asymptatic center af {xn} relative to, K. Asymptotic center A K ( {xn}) in general case may be empty. The standard assumption that guarantees that all bounded sequences have nonempty asymptotic centers is weak compactness of K. Obviously the convexity of K implies convexity of AK({Xn}) for all bounded sequences {Xn}.
Let T : K --> K be nonexpansive. For Xo E K put Xn = Tnxo and let Kl = A K( {xn}). In view of the nonexpansivness of T , for any z E Kl we have
r (Tz, {xn}) = lim sup IITz - Tnxall S lim sup liz - Tn-1xoll = r (z, {Xn}). n---l>OO n---+oo
implying Tz E K 1 • Thus the asymptotic center relative to K of any sequence of iterates of T is T invariant.
We observe the same effect for any asymptotic fixed point sequence. Indeed if
lim llYn - TYnl1 = 0 n-->oo
then for any z E AK ( {Yn})
r(Tz,{Yn})=limsupIITz-Ynll slimsupIlTz-TYnll+ lim IITYn-y,,1I n---+oo n----+oo n-----).oo
S lim sup liz - Ynll = r (z, {Yn}) . n-->oo
This implies that A K ( {Yn}) is T invariant.
Using both facts one can construct a sequence K :J Kl :J K2 :J K3 ... of T invariant sets. Each Kn+1 can be taken to be the asymptotic center relative to Kn either of a sequence of iterates or an asymptotic fixed point sequence. Obviously if K is convex and weakly compact so are all K i .
We can continue this procedure using the intersections and transfinite induction. If the procedure terminates on a set consisting of one point, this must be a fixed point of T.
Some geometrical regularity conditions of the space X or the set K itself force the above scheme to stabilize "quickly". For example if the space X is uniformly convex, or uniformly convex in every direction, then the asymptotic center of any bounded sequence is a singleton. Thus Kl consists of exactly one point, a fixed point of T. Further, for spaces with the characteristic of convexity c:o(X) = sup{c: : 8x (c:) = O} < 1 we have (see [49])
diamAK({Un }) sc:o(X)r(K,{un })
for any sequence {un} C K. This implies that limn-->oo diam(Kn) = 0 and consequently n;;"=l Kn is a singleton, a fixed point of T. Also k-unifarmly rotund spaces of Sullivan [134] show nice properties. Asymptotic centers of bounded sequences are in this spaces compact in norm topology. So in this case we can stop at Kl and then, for example, search for fixed points as norm cluster points of asymptotic fixed point sequences.
Asymptotic center technique has some other applications. It proved to be useful for investigations of fixed points of nonexpansive multivalued mappings. This will be discussed in the next section. We end with an observation connected to Section 3. It was mentioned that in Hilbert space case all convex subsets of K C H are of the form F(T) for certain nonexpansive T : K --> K. This is no longer true in general case even for uniformly convex spaces and even for spaces isomorphic to H.
Consider the space IK ( the space ]R3 furnished with the norm 1
II(Xl,X2,X3)1I = (IXIIP + IX21P + IX 3IP)p)
78
for 1 < p < 00, and let E denote the triangle {x = (Xl,X2,X3) : Xi 2: O,i = 1,2,3, Xl + X2 + X3 = 1}. The norm is uniformly convex. All spaces l~ are isomorphic to l~. However for p of 2 the set E is not the nonexpansive retract of l~ nor is of the form of F(T) for certain nonexpansive T : l~ -> l~. Indeed, suppose that E = F(T). Consider the periodic sequence of unit vectors {e\e2,e3,el,e2,e3, ... }. Let Z be the unique point of the asymptotic center of this sequence relative to the whole space. It can be calculated that z = (c, c, c) with
c = 1+ 2p-1 ( 1 )-1
Since our sequence consists of fixed points of T ( thus it is an asymptotic fixed point sequence) we should have z = Tz and z E E(T). However it is not true unless p = 2 and we have a contradiction. The example carries over to the case of T being a nonexpansive self mapping of any closed convex set K containing E in its interior.
5. Set-Valued Nonexpansive Mappings
The principal result in this direction is due to T. C. Lim [103J.
Theorem 5.1 Let X be a uniformly convex Banach space (or, more generally, a reflexive space which is uniformly convex in every direction), let K be a bounded closed convex subset of X, and let T : K -+ C!:(K), where C!:(K) denotes the collection of all nonempty compact subsets of K. Suppose T is nonexpansive relative to the Hausdorff metric on c!:(K). Then there exists a point x E K such that X E T(x).
The proof of Theorem 5.1 uses the so-called asymptotic center method which was discussed in the previous section. Here we make use of the fact that if X is uniformly convex in every direction then AK ( {xn}) is a singleton for each bounded sequence {xn} in X. Also, such a sequence is said to be regular (relative to K) if rK( {un}) = rK( {xn}) for each subsequence {un} of {xn}. The following fact was noticed independently by Goebel [45J and Lim [104J. Since K is fixed throughout, we drop the subscripts.
Lemma 5.2 Every bounded sequence {xn} in X has a regular subsequence.
Proof. For any bounded sequence {un} let
f ({un}) = inf {r ({zn}) : {zn} is a subsequence of {Un}}.
Now let {y;,} = {xn} and, given {y~}, let {y~+l} be a subsequence of {y~} for which
Then if {yd is a subsequence of the diagonal sequence {y;;}
proving r( {Yk}) = r( {y;;}). • Proof of Theorem 5.1. By choosing fixed points of set-valued contraction mappings which uniformly approximate T and passing to a subsequence, it is possible to obtain
Nonexpansive mappings 79
a regular sequence {xn} in K for which dist(xn,T(xn)) ---> O. Let r = r({xn}) and let {v} = A({xn}). For each n select Yn E T(xn) such that llYn - xnll ---> 0, and select Zn E T(v) such that
where H denotes the Hausdorff distance on I!:(K). Since T(v) is compact some subsequence {znk} of {zn} converges to an element w E T(v). Therefore
Ilxnk - wll S IIxnk - Ynk II + llYn. - znk II + IIznk - wll and
IIYnk - znk II s IIxnk - vII . It follows that limk sup IIxnk - w II s r, proving w E A( {xn}) = {v}. Therefore v = w E T(v). •
Earlier versions of Theorem 5.1 were obtained by Markin [111J in Hilbert spaces, by Browder [18J for spaces possessing weakly continuous duality mappings. In each these instances the mapping is assumed to have compact convex values. It has since been shown in [83J that under this additional assumption about the values of T Theorem 5.1 holds in an even wider class of spaces.
Theorem 5.3 Let X be a Banach space and let K be a bounded closed convex subset of x. Suppose K has the property that the asymptotic center of each sequence in K (relative to K) is nonempty and compact. Suppose T : K --+ I!:I!:(K) is nonexpansive, where I!:I!:(K) denotes the collection of all non empty compact convex subsets of K endowed with the Hausdorff metric. Then there exists a point x E K such that x E T(x).
Spaces which satisfy the assumptions of the above theorem include, for example, all the k-uniformly rotund spaces of Sullivan [134J. The initial step in the proof involves showing that every sequence in K has a subsequence with the property that each of its subsequences has the same asymptotic radius and asymptotic center. The proof also has a topological ingredient in that it invokes the Bohnenblust-Karlin extension ([9]) of a well known fixed point theorem of Kakutani [66J for upper semicontinuous set-valued mappings.
As with the demiclosedness principle, Theorem 5.1 also holds for spaces satisfying Opial's condition. This fact is due to E. Lami Dozo [99J.
Theorem 5.4 Let K be a weakly compact convex subset of a Banach space X which satisfies Opial's condition and let T : K --+ I!:(K) be nonexpansive. Then there exists x E K such that x E T(x).
6. Abstract Theory
6.1. Introduction
The axiomatic approach in modem mathematics which flourished in the early part of the twentieth century has had a lasting impact. The study of nonexpansive mappings in a 'purely' metric space context (indeed, in an even wider context) appears well suited to this approach via the use of convexity structures. The advantages accrue from the elegance and generality achieved, and the generality leads in tum to a unification of applications within more concrete settings.
80
General set-theoretic convexity in the study of fixed point theory of nonexpansive mappings made its first explicit appearance in the work of J-P. Penot [117], although it is implicit in the much earlier studies of hyperconvex metric spaces ([3]). The idea involved is to proceed only with those essential properties of a metric structure needed for the development of a well grounded theory. In addition to the usual assumptions involved in the study of abstract convexity structures, it turns out that in this study an additional assumption is indispensable; namely the the closed metric balls of the underlying space must be included in the structure. Indeed, the presence of the closed balls and their intersections in the structure lies at the heart of the approach.
It is only fair to acknowledge, however, that there is one major disadvantage to this more abstract approach. As seen in the previous section, a standard tool in many concrete results is the use of the so-called 'asymptotic center' technique. Unfortunately this technique involves assumptions which are not appealing in a more abstract framework.
The convexity structure approach described below has been the subject of a vigorous study in its own right (see, e.g., van de Vel [138]). However one needs to be cautious because the terminology is not always consistent. For example the term 'normal' as it is used below is different from its topologically motivated usage in [138].
6.2. Convexity Structures
Metric convexity structures. If one were to describe the behavior of the collection C of all convex sets in a Banach space X the following three properties stand out:
(A-I) Both X and 0 are in C.
(A-2) C is closed under intersections, that is, if {K,,} is any subcollection of C then n"K",EC.
(A-3) If {K",} is any chain (relative to set inclusion) of members of C, then U",K", E C.
If only the behavior of closed sets is under consideration it is necessary to drop (A-3), or at the very least modify (A-3) so that the closure of U",K", is a member of C. Since this involves the explicit introduction of a topological concept, we drop (A-3).
Definition 6.1 Let X be a set and let C be a family of subsets of X. The pair (X,C) is called a closed convexity structure if
(A-I) Both X and 0 are in C.
(A-2) C is closed under intersections, that is, if {K",} is any sub collection of C then n"K", EC.
If the metric properties of the space X are of primary concern an additional axiom is needed. The correct one seems to be the one given below.
Definition 6.2 Let X be a metric space and let C be a family of subsets of X. The pair (X,C) is called a metric convexity structure if
(A-I) Both X and 0 are in C.
Nonexpansive mappings 81
(A-2) C is closed under intersections, that is, if {Ka} is any sub collection of C then naKa E C.
(A-4) C contains the closed balls of X.
In the presence of (A-2), (A-4) leads to another fact. For each x E X,
{x} = nB(x; lin) E C.
Given a metric convexity structure (X, C) there is a sub collection of C that is of special interest. These are the so-called admissible subsets of X. A set DEC is said to be an admissible set if D = cov(D) where
cov(D) := n{B : B is a closed ball in X and B :;2 D.
Notice that if A(X) denotes the family of all admissible subsets of X, then (X,A(X)) is itself a metric convexity structure.
We also denote
co(D) := n{F : F E C and F :;2 D}.
To summarize, D E A(X) iff D = cov(D) and DEC iff D = co(D). In general co(D) <::: cov(D), but in considering only the structure (X, A(X)) there is no need for a distinction.
We note the following fact for later reference.
Proposition 6.3 Let (X, C) be a metric convexity structure and suppose DEC is bounded. Then diam(co(D)) = diam(cov(D)) = diam(D).
Proof. Let d = diam(D). Obviously diam(cov(D)) ~ diam(co(D)) ~ d. On the other hand, if xED and y E cov(D) then since
D <::: cov(D) <::: B(x; rx(D)),
d(x,y) :s; rx(D) :s; d. This implies ry(D):S; d and thus
D <::: cov(D) <::: B(y; d).
Therefore if z E cov(D), d(z,y) :s; d. • Definition 6.4 Let X be a metric space and let A(X) denote the collection of all admissible subsets of X. Then the pair (X, A(X)) is called an admissible convexity structure.
We now formalize the basic definition of the mappings considercd.
Definition 6.5 A mapping f : M --t N from a metric space (M, d) into a metric space (N,p) is said to be nonexpansive if p(f(x),f(y)) :s; d(x,y) for all x,y E M.
The central problems treated in the ensuing sections are the following. Let (X, C) be a metric convexity structure, suppose DEC is nonempty, and suppose f : D --t D is a nonexpansive mapping.
82
Problem 6.6 What properties on (X,C) insure that I has at least one fixed point?
Problem 6.7 What properties on (X,C) insure that
Fix(f):= {x ED: I(x) = x} E C?
A note of caution. If (X,A(X)) is an admissible convexity structure and if DE A(X), then (D, C) is a metric convexity structure where
C:= {DnA: A E A(X)}.
However in general, (D,C) f= (D,A(D)). On the other hand, it may well be that (D,C) inherits nice properties of (X,A(X)).
Compact convexity structures. If (X, C) is a convexity structure, the family C is always a subbase for a topology on X, and by Alexander's subbase theorem [72] X is compact in this topology if (X,C) is compact in the following sense.
Definition 6.8 A convexity structure (X, C) is said to be compact [resp., countably compact] if every family [resp., countable family] of C which has the finite intersection property has nonempty intersection.
Examples of compact metric convexity structures are abundant. Suppose the underlying space X is taken to be a bounded closed convex subset of a Banach space E. For reflexive E, the convexity structure (X,C) where C is the family of all closed convex subsets of X is compact. If E is a conjugate space and if X is a weak· compact convex subset of E, then (X, A(X)) is an admissible compact convexity structure. The same is true if X is a bounded hyperconvex metric space.
Compact convexity structures (often countable compactness suffices) permits immediate passage to a minimal invariant set in the following sense.
Proposition 6.9 Let (X, C) be a compact convexity structure and let I : X -> X. Then there exists DEC such that D is minimal with respect to being nonempty and invariant under I. For such a minimal D it is the case that D = co(f(D)).
Proof. The existence of a minimal such D is a direct consequence of Zorn's Lemma. The last statement follows from the fact that l(co(f(D))) ~ I(D) ~ co(f(D)) ~ D. Since co(f(D)) E C, D = co(f(D)) by minimality. •
Normal convexity structures. denote
For a bounded subset A of a metric space M we
Also set
diam(A) = sup{d(x,y) : x,y E A}; rx(A) = sup{d(x,y) : YEA} (x E A);
rCA) = inf{rx(A) : x E A}.
C(A) := {x E A : rx(A) = rCA).
Nonexpansive mappings 83
The number rCA) and set C(A) (which may be empty) are called, respectively, the Chebyshev radius and Chebyshev center of A.
In view of Proposition 6.9 a standard tactic in addressing Problem 6.6 becomes this. Find conditions on C which insure that any minimal nonempty invariant DEC must be a singleton. The following condition is in some sense canonical.
Definition 6.10 A metric convexity structure (X, C) is said to be normal if reD) < diam(D) for every DEC with diam(D) > O.
This brings us to the fundamental theorem for nonexpansive mappings.
Theorem 6.11 Let X be a bounded metric space and suppose (X, C) is a normal compact metric convexity structure. Then every nonexpansive T : X ~ X has at least one fixed point.
Proof. By Proposition 6.9 there exists DEC which is minimal with respect to being nonempty and invariant under T. Let r = reD). By the definition of r each of the sets
is nonempty, and clearly Cn(D) E C, n = 1,2,· ... Therefore
is nonempty and in C. Now let x E C(D). Then D ~ B(xir). Since T is nonexpansive
T(D) ~ B(T(x)ir).
This implies B(T(x)ir) :2 co(T(D)) = D. Hence rT(x)(D) :::; r, from which T(x) E C(D). Therefore T : C(D) ~ C(D) and by minimality, C(D) = D. But if r > 0, diam(C(D)) :::; r < diam(D). It follows that r = 0 and this in turn implies that diam(D) = O. •
By strengthening the normality condition it is possible to replace explicit reference to compactness with completeness.
Definition 6.12 A metric convexity structure (X,C) is said to be uniformly normal if there exists c E (0,1) such that reD) :::; c diam(D) for every DEC.
Theorem 6.13 Let X be a bounded complete metric space and suppose (X,C) is a uniformly normal metric convexity structure. Then every nonexpansive T : X ~ X has at least one fixed point.
The proof of this theorem requires two technical lemmas. The first can be proved by translating its Banach space counterpart (see [49, p.4I]) into a metric space framework.
Lemma 6.14 Let X be a bounded metric space and suppose (X, C) is a metric convexity structure. Suppose T : X ~ X is nonexpansive, and let
~:= {D E C : D =f. 0 and T: D ~ D}.
84
Then for each DE;)' there exists iJ E ;)' such that
. 1 diam(D) ::; 2(diam(D) + r(D)).
Compactness has not really been entirely eliminated in Theorem 6.13 because of the following fact. This observation is due to Khamsi [75]. Another proof is given by replacing II . II with d and 'co' with 'co' in the proof of Theorem 4.4 of [49].
Lemma 6.15 . Let X be a bounded complete metric space and suppose (X,C) tS a uniformly normal metric convexity structure. Then (X, C) is countably compact.
In view of Lemma 6.15, Theorem 6.13 is a consequence of the following extension of Theorem 6.11.
Theorem 6.16 Let X be a bounded complete metric space and suppose (X,C) is a normal countably compact metric convexity structure. Then every non expansive T : X --> X has at least one fixed point.
Proof. Let;)':= {D E C : D =10 and T: D --> D} and define {j:;)' --> lR+ by
o(D) = inf{diam(F) : FE;)' and F <:;; D}.
Select Dl E ;)" and with Dl,· .. , Dn given, select Dn+l E ;)' with Dn+l <:;; Dn such that
Now let C := n~=l Dn. Then C E C, and by countable compactness C =I 0. Since T: C --> C, C E;)'. Also for each n E N,
diam(C) - .!. ::; diam(Dn+l) - .!. ::; o(Dn) ::; 0(6) ::; -21 (diam(C) + r(C)). n n
Letting n --> 00 gives diam(C) = r(C); hence C is a singleton. • Remark. Kulesza and Lim [97] have recently made a very interesting observation in connection with Lemma 6.15; namely:
Theorem 6.17 Let M be a bounded complete metric space and suppose A(M) is countably compact and normal. Then A(M) is compact.
6.3. Khamsi's Structure Theorem
The following concept is the key to the structure of fixed points sets of nonexpansivc mappings in metric spaces M for which A( M) is compact and normal. It also paves the way to the proof that in such a setting commuting families of nonexpansive mappings have a nonempty common fixed point set.
Definition 6.18 A subset A of M is said to be a I-local retract of M if for each family {Bi};EI of closed balls centered at points of A and for which niEIBi =10 it is the case that An (niE1Bi ) =10.
Nonexpansive mappings
It is easy to check that every nonexpansive.retract of M is a I-local retract of M.
We now describe several fundamental properties of I-local retracts.
85
Proposition 6.19 If M is a metric space for which A(M) is compact, then M is complete.
Now we collect several additional properties.
Proposition 6.20 Let M be a metric space and let A be a nonempty subset of M. Then:
(1) cov(A) = n{B(x;rx(A)) : x EM};
(2) rx(A) = rx(cov(A)) for every x EM;
(3) r(cov(A)) :.:::: r(A);
(4) diam(cov(A)) = diam(A).
Proof. (1) This is immediate since B(x; rx(A)) is the smallest ball centered at x which contains A.
(2) Since A c cov(A) it follows from the definition of rx that
rx (A) :.:::: rx (cov (A)).
On the other hand, (1) implies rx(cov(A)) :.:::: rx(A).
(3) This is immediate from the definition of rand (2).
(4) It obviously suffices to show that diam(cov(A)) :.:::: diam(A). Let z E cov(A). Then Z E B(x; rx(A)) for each x E M. In particular, d(x, z) :.:::: rx(A) :.:::: diam(A) for each x E A; thus A c B(z; diam(A)). Hence
coy (A) C B (z; diam (A)).
It follows that diam(cov(A)) :.:::: diam(A). • In the case of I-local retracts, normal structure is hereditary.
Proposition 6.21 Let M be a metric space and suppose A(M) is compact and normal. Suppose N is a given subset of M which is a I-local retract of M. Then A(N) is compact and normal.
The key step in proving the above is the following lemma.
Lemma 6.22 Under the assumptions of Proposition 6.21, r(cov(A)) = r(A) for each A E A(N).
Proof. Suppose diam(A) > O. Proposition 6.20 implies r(cov(A)) :.:::: r(A) so we only need to establish the reverse inequality. By assumption A E A(N) so A is of the form
86
for some family {Xi} C N. Also, cov(A) c niB(xi;ri). Choose Z E cov(A) and let r = rz . Then
Z E S := (nXEAB (x; r)) n (niB (Xi; ri)).
Thus S is nonempty and S E A(M). Since N is a I-local retract of M and S is the intersection of balls centered in N, SnN =10. Let wE SnN. Then wE (niB(Xi; ri))nN, that is, wE A. On the other hand, w E nXEAB(x; 1'), so rw :s; r. This implies
I' (A) :s; r:S; rz (cov (A)).
Since Z was an arbitrary element of cov(A) the proof is complete. • Proof of Proposition 6.21. The definition of a I-local retract assures that A(N) is compact. To see that A(N) is normal, let A E A(N) and note that by Proposition 6.20 diam(cov(A)) = diam(A). By Lemma 6.22, r(cov(A)) = rCA). Since A(M) is normal, if diam(A) > 0, rCA) = r(cov(A)) < r(diam(A)). •
It is now possible to prove the following.
Theorem 6.23 Let M be a bounded metric space for which A(M) is compact and normal, and let T : M -'> M be nonexpansive. Then the fixed point set F(T) of T is a non empty I-local retract of M. Moreover, A(F(T)) is compact and normal.
Proof. The fact that F(T) =I 0 follows from previous results. To see that F(T) is a I-local retract of M let {B;} be a family of closed balls centered at points of F(T) for which S := niBi =10. Then since T is nonexpansive, T : S -'> S. Also, since S E A(M), A(S) is compact and normal. Therefore T has a fixed point in S; i.e., F(T) n S =I 0. The final assertion follows from Proposition 6.21. •
The above result extends as well to arbitrary commutative families.
Theorem 6.24 Let M be a bounded metric space for which A(M) is compact and normal. Then every family W of commuting non expansive mappings of M -'> M has a nonempty common fixed point set F(W). Moreover, F(W) is a I-local retract of M.
If W is finite, say W = {TI ,' . " Tn} the proof is easy. By Theorem 6.23 A(F(Ti)) is compact and normal for each i. Since TI and T2 commute, T2 : F(TIl -'> F(TI)' Thus F(TI) n F(T2) =10. Also F(TIl n F(T2) is a I-local retract of F(Td, hence of M. The result for finite W now follows by induction. Full details in the infinite case may be found in [77].
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Chapter 4
GEOMETRICAL BACKGROUND OF METRIC FIXED POINT THEORY
Stanislaw Prus
Department of Mathematics
Maria Curie-Sktodowski Univer8ity
20-031 Lublin, Poland
1. Introduction
The interplay between the geometry of Banach spaces and fixed point theory has been very strong and fruitful. In particular, geometrical properties play key roles in metric fixed point problems. In this text we discuss the most basic of these geometrical properties. Since many fixed point results have a quantitative character, we place special emphasis on the scaling coefficients and functions corresponding to the properties considered. The material we cover is far from exhaustive, in particular we do not consider applications. These are treated elsewhere in the Handbook. The interested reader may also consult [5], [44] and [1].
We consider Banach spaces over the real field only. Unless otherwise stated metric and topological notions are related to the metric induced by a norm. Our notation and terminology are standard. They coincide for instance with that of [5]. However, for the convenience of the reader we recall some notation. Let X be a Banach space. By Ex and Sx we denote the closed unit ball and the unit sphere, respectively. Next, let A be a nonempty subset of X. Then co(A) stands for the convex hull of A and span(A) denotes the linear subspace spanned by A. Given x E X, by d(x, A) we denote the distance of x to A.
2. Strict convexity and smoothness
Convexity is the most elementary property of a norm. In the simplest case when we deal with points of a line in a space X we can actually apply the well-known facts concerning convex functions of a real variable.
Remark 2.1 For any points x, y of a Banach space X the function t I--T Ilx + tyll is convex on JR.
In some applications it is necessary to impose conditions stronger than convexity on a norm. Strict convexity is one of these conditions.
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W.A. Kirk and B. Sims (eds.), Handbook of Metric Fixed Point Theory. 93-132. © 2001 Kluwer Academic Publishers.
94
Definition 2.2 A Banach space X is strictly convex (or rotund:) if
IIX;yll < 1
whenever x,y E Sx, x 1= y.
Condition 3 of our first theorem gives a simple geometric interpretation of strict convexity.
Theorem 2.3 Let X be a Banach space. The following conditions are equivalent.
1. X is strictly convex.
2. If 1 < p < 00 and x,y E X, x 1= y, then p.x + (1- AMP < AllxliP + (1- A)lIylIP for every 0 < A < 1.
3. The sphere Sx does not contain a nontrivial segment.
4. If x, Y E X and IIx + yll = IIxll + Ilyll, then x = 0 or y = 0 or y = ox for some 0> O.
5. Ifx* E X*, x* 1= 0, then x*(x) = Ilx*11 for at most one point x E SX.
Proof. Let 1 < p < 00. Considering the function t >-+ tP , we see that
( a +b)P 1 -2- < 2(aP + bP) (2.1)
for all a, b ;::: 0, a 1= b. Assume that X is strictly convex and x, y EX, x 1= y. If Ilxll = Ilyll, then II (x + y)/211 < IIxli. Consequently,
II x; y liP < ~(lIxIIP + lIyln
From (2.1) we see that this inequality holds also in the case when IIxll 1= Ilyli. Take now 0 < A < 1. If 0 < A ::; ~, then
The proof is similar if ~ < A < 1. This shows that 1 =? 2.
The implication 2 =? 3 is obvious. To prove that 3 =? 4, suppose IIx + yll = Ilxll + Ilyll for some nonzero vectors x, y E X such that y 1= ox for alIa> O. We can assume that 0 < IIxll ::; lIyll = 1. Consider the functions f(t) = Iltxlllxll + yll and g(t) = t + 1, t ;::: O. Then f(O) = g(O), f(lIxll) = g(lIxll), and f(t) ::; g(t) for every t ;::: O. From Remark 2.1 we see that f(t) = g(t) for every t ;::: 0 which shows that the segment with the endpoints x/llxll, y is contained in Sx.
To show that 4 =? 5, assume there is a functional x* E X*, x* 1= 0 for which x* (x) = x*(y) = IIx·11 where x,y E Sx, x 1= y. Then
Ilx·11 =x· (x;y)::; IIx·IIIIX;yll::; IIx·ll.
Geometrical background 95
This gives us the equality IIx + yll = 2 = Ilxll + Ilyll, which contradicts condition 4.
Assume now that there exist x,y E Sx, x =I y, such that II (x + y)/211 = 1. Choose a functional x* E Sx* so that x*((x +y)/2) = lI(x + y)/211 = 1. It is easy to show that x*(x) = 1 = x*(y), and 5 =? 1 is proved. •
In some cases it is more convenient to use metric segments instead of affine ones (see for instance [1]). Let X,y be points of a Banach space X. The set 1(x,y) of all z E X such that liz - xII ~ (l-t)lIy- xII and Ilz-YII ~ tlly -xII for some t E [0,1] is called the metric segment with the endpoints x, y. It is easy to see that equality actually holds in the above inequalities. Setting t = 1/2 in this definition, we obtain the so-called metric midpoint of x,y. Obviously, 1(x,y) contains the affine segment co({x,y}). In general those two sets may be different but in strictly convex spaces they coincide.
Proposition 2.4 A Banach space X is strictly convex if and only if 1(x, y) = co( {x, y}) for any X,y E x.
Proof. Assume that X is not strictly convex. By Theorem 2.3 the sphere Sx contains a segment co( {x, y}) with x =I y. Then x, yare in the metric segment 1(0, x + y) but not in co( {O, x + y} ). Suppose in turn that X is strictly convex and z E I (x, y) where x =I y. Then liz - xII = (1 - t)lly - xII and liz - yll = tlly - xII for some t E [0,1]. Consequently,
liz - xII + liz - yll = IIY - xII = II(z - x) + (y - z)lI·
From Theorem 2.3 we see that z = x or z = y or y - z = a(z - x) for some a > O. In each case z E co({x,y}). •
We will show two more characterizations of strict convexity. The first of them is related to fixed point theory. Let C be a nonempty subset of a Banach space X. A mapping T : C ---> X is called nonexpansive if
IITx - TYII ~ Ilx - yll
for all x, y E C. By Fix(T) we denote the set of all fixed points of T, i.e. points x E C such that Tx = x. Clearly, Fix(T) may be empty.
Theorem 2.5 A Banach space X is strictly convex if and only if for every nonempty convex set C C X and every nonexpansive mapping T : C ---> C the set Fix(T) is convex.
Proof. Assume that X is strictly convex and x, y E Fix(T) for some nonexpansive mapping T : C ---> C. If z = tx + (1 - t)y where t E [0,1], then IITz - xII ~ liz - xii = (l-t)lIy-xll and similarly, IITz-YIl ~ tlly-xll. Hence Tz E 1(x,y) and from Theorem 2.4 we see that Tz E co({x,y}). It is easy to check that Tz = tx+(l-t)y, so Tz = z.
To show the converse, we assume that X is not strictly convex. By Theorem 2.3 the sphere Sx contains a segment co({x,y}) with x =I y. Consider the convex set
{ x+y x-y } C = a-2- + f3-2 - : a E [-1,1], f3 E [0,1] .
Let x* E Sx' be such that x*((x + y)/2) = 1. Then x*(x) = 1 = x*(y) and in particular x*(u) E [-1,1] for every u E C. Given such u, we put Tu = x*(u)x if
96
x*(u) 20 and Tu = x*(u)y if x*(u) < O. It is easy to see that T(C) c C is not convex and Fix(T) = T(C). In order to show that T is nonexpansive we take U,V E C such that x*(u) 2 0 and x*(v) < O. Since co({x,y}) C Sx,
IITu - Tvll = IIx*(u)x - x*(v)YII = x*(u) - x*(v) ::; Ilx - YII.
• Another characterization of strict convexity is connected to approximation theory. Let A be a nonempty subset of a Banach space X. Given x E X, by PAX we denote the set of all points in A nearest to x, i.e. points yEA such that Ilx - YII = d(x, A). Clearly PAX may he empty. The set-valued mapping PA is called the metric projection of X onto A. As an easy consequence of Theorem 2.3 we obtain the following result.
Theorem 2.6 A Banach space X is strictly convex if and only if for each nonempty convex set A c X and each X E X the set PAX contains at most one point.
We now pass to the property which is partially dual to strict convexity.
Definition 2.7 A Banach space X is smooth if for each xES x there is a unique functional x* E Sx* such that x*(x) = 1.
From Theorem 2.3 we derive the relations between smoothness and strict convexity.
Theorem 2.8 Let X be a Banach space. Then
1. If X* is smooth, then X is strictly convex.
2. If X* is strictly convex, then X is smooth.
Of course, if X is reflexive, then also the opposite implications hold, but in general this is not so (see [5J p. 61). It is also worth while to mention that each separable space X admits an equivalent norm such that X is both strictly convex and smooth with respect to this norm. Moreover, the same is true for reflexive spaces (see [27], p. 148).
Definition 2.7 means that for each X E Sx there is only one hyperplane supporting the ball Ex at x. This is strongly connected to differentiability of the norm. Let X be a Banach space. A function f : X --> ~ is Gateaux differentiable at x E X if for each hEX the limit
f'(x)(h) = lim f(x + th) - f(x) t~O t
exists and f' (x) E X*. Then f' (x) is called the Gateaux derivative of f at x. If in addition the above limit is uniform in h E Sx (uniform in x, hE Sx), we say that f is Frechet differentiable at x (resp. uniformly Frechet differentiable). In this case f'(X) is called the Frechet derivative of f at x.
As an example consider the function f(x) = Ilxll. We say that the norm is Gateaux differentiable if f is Gateaux differentiable at each nonzero point u EX. Frechet differentiability is defined in a similar way. Notice that by homogenity it is enough to check these conditions for u E S x. By Remark 2.1 f has one-sided directional derivatives at each nonzero point u. This leads us to the following observation.
Geometrical background 97
Remark 2.9 A norm II· II of a space X is Gateaux differentiable if and only if
lim ~(Ilu + tyll -Ilull) = lim ~(Ilu + tyiI - Ilull) t~O- t t~O+ t
for any vectors u, y EX, u i= 0.
Example 2.10 1. Let X be a Hilbert space and <I>(t) = t 2 for t :::: 0. Then it is easy to check that the function x f--7 <I>(llxll) is uniformly Fnlchet differentiable with <I>'(llxll) = 2x for every x E X. Consequently, the function f(x) = IIxll is also uniformly Fn\chet differentiable and f'(x) = x/llxli for each x i= 0. Here we identify X* with X.
2. Consider the space LJ'(n) where 1 < p < CXJ and n is a measure space. Put <I>(t) = tP for t :::: 0. Then the function <I>(llxll) is uniformly Frechet differentiable with <I>'(llxlll = plx(w)IP-l sgnx(w) for every x E LJ'(n) (see [26], p. 184). It follows that the function f(x) = Ilxll is also uniformly Frechet differentiable with f'(x) = IlxI11-Plx(w)IP-2x(w) for every x i= 0.
There is a tool that can be used without any differentiability assumption. Let f : X --+
lR be a convex function and x EX. The set
iJf(x) = {x* E X* : x*(y - x) :::; f(y) - f(x) for all y E X}
is said to be the subdifferential of f at x. Elements x* of iJf(x) correspond to hyperplanes supporting the epigraph epif = ((y,t) E X x lR : f(y) :::; t}. Namely, the condition x* E iJ f (x) means that epi f is entirely on one side of the hyperplane H = {(y,t) EX xlR: x*(y)-t = x*(x)-f(x)} and intersects H at the point (x,f(x)). If f is continuous, then iJf(x) i= f/J for every x E X. Moreover, f is Gateaux differentiable at x if and only if iJf(x) is a singleton. In this case iJf(x) = {.f'(x)} (see [98]).
Given a function ¢ : [0, +00) --+ [0, +00) and x E X, we put
Jq,x = {x* E X* : x*(x) = Ilx*llllxll, Ilx*11 = ¢(llxll)}· The function ¢ is said to be a weight if it is continuous, strictly increasing, ¢(O) = 0, and limt~+oo ¢(t) = +00. In this case the set-valued mapping x f--7 Jq,x is called a duality mapping with the weight ¢. This concept is strongly related to subdifferential.
Theorem 2.11 Let X be a Banach space and ¢ : [0, +00) --+ [0, +00) be a continuo'us nondecreasing function. Then
iJ<I>(lIxll) = Jq,x
for every nonzero x E X where <I>(t) =.r~ ¢(s)ds, t:::: 0.
Proof. Clearly, <I>'(t) = ¢(t) for all t :::: 0, so the function <I> is convex. Let x E X, x i= ° and assume that x* E X* satisfies the conditions x*(x) = IIx*llllxll, IIx*11 = ¢(lIxlll. We shall show that x* E iJ<I>(llxll) , i.e. x*(y - x) :::; <I>(llyll) - <I>(llxlll for every y E X. If Ilyll > Ilxll, then
Ilx*11 = ¢(llxll) = <I>'(llxll):::; <I>(II~~~ = :;::x ll ).
Hence <I>(llylll - <I>(llxll) :::: Ilx*II(llyll-lIxll) :::: x*(y) - x*(x). The proof in the other case is similar.
98
Assume now that x' E DcI>(llxll). Then x*(y) ::; x*(x) for every y E X with lIyll = Ilxli. It follows that x*(x) = Ilxllllx'll. Next,
cI>(tllxll) - cI>(llxll) 2: x*(tx - x) = (t - 1)llxllllx'll
for every t > 0. If t > 1, we therefore get
Ilx*11 < cI>(tllxlll- cI>(llxll) - tllxll - Ilxll
which shows that Ilx*11 ::; cI>'(lIxll) = ¢(llxll). Similarly we obtain the opposite inequal~ . If ¢ is a weight, then both sets Jpx and DcI>(llxll) are equal to {o} at x = 0. In this case it is therefore not necessary to exclude this point. On the other hand, the subdifferential of the norm at zero is the whole ball Bx. Theorem 2.11 with ¢(t) = 1 for all t 2: ° gives us the formula for this sub differential at nonzero points.
Corollary 2.12 Let X be a Banach space and f(x) = Ilxll for x E X. Then
Df(x) = {x' E X* : Ilx'll = 1, x*(x) = IIxll}
for every nonzero x EX.
As a consequence we obtain the following characterization of smoothness (for the direct proof see [9]' p. 179).
Theorem 2.13 A Banach space X is smooth if and only if its norm is Gateaux differentiable.
The function cI> in Theorem 2.11 is differentiable. From Theorem 2.13 we therefore see that if X is smooth, then Jq,x is a singleton for every nonzero x.
3. Finite dimensional uniform convexity and smoothness
Several uniform versions of strict convexity and smoothness can be found in the literature. They usually correspond to some moduli defined as the least upper or/and greatest lower bounds of some quantities depending on norms of elements of a space. The following remark describes the construction and basic properties of such moduli in a more precise way.
Remark 3.1 Let a> ° and f : [0, a) --t [0, +(0) be a function obtained by taking the least upper or/and greatest lower bounds of a family consisting of convex uniformly bounded functions 9 : [0, a) --t [0, +(0) such that g(O) = 0. Then f is Lipschitzian on each interval [O,bj with b < a and the function t >-> f(t)/t is nondecreasing.
Of course, the function f need not be convex. Nevertheless, one can construct a convex function which is equivalent to f. Let 1= [0, aj, where 0< a < +00, or 1= [0, +00). Given a function f : I --t [0, +00) with f(O) = 0, by 1* we denote its dual Young's function, i.e.
r(t) = sup{st - f(s) : s E I}
Geometrical background 99
for all t 2: O. Clearly, f* is a convex nonnegative function with f*(0) = O. Moreover, the restriction of f** to I is the maximal convex function minorizing f (see [35], Theorem 1.4.1).
Lem'~:d 3.2 Let us assume in addition that the function t I-t f(t)/t is non decreasing. Then
1. f(t/2) ::; f**(t) ::; f(t) for every tEl.
2. lims-+of(s)/s = sup{t 2: 0: f*(t) = O}.
Proof.
1. Consider the function j : I -7 [0, +00) given by the formula
J(t) = rt f(s) ds. Jo s
It is easy to see that j is convex and f(t/2) ::; J(t) ::; f(t) for every tEl. Consequently, J(t) ::; f**(t) ::; J(t) for all tEl.
2. Let a = lims .... of(s)/s and t 2: 0 be such that f*(t) = O. Then f(s) 2: st for every s E I. Hence a 2: t which shows that 0: 2: sup{t 2: 0 : f*(t) = O}. Next, observe that f(s) 2: as for every s E I. Therefore f*(a) ::; O. But f(O) = 0 which implies that f* is nonnegative. It follows that 0: ::; sup{ t 2: 0 : f*(t) = O}.
• We review those moduli which are most frequently applied in metric fixed point theory.
Definition 3.3 Let X be a Banach space. Given Z E Sx and € E [0,2], we put
8x (Z;E) = inf {l-IIX;YII: X,y E Ex, IIx-yll2: E, x -y = tz for some t 2: O}. The space X is called uniformly convex (or uniformly rotund) in every direction (UCED for short) if 8x (z; E) > 0 for every z E Sx and every E E (0,2).
The class of all UCED spaces is quite large. For instance every separable Banach space X admits an equivalent norm with respect to which it is UCED (see [109]). Here we will establish some properties of the modulus 8x (z; E).
Proposition 3.4 Let X be a Banach space with dim(X) 2: 2, Z E Sx and E E [0,2). Then
Proof. Take 0 < E < 2. It suffices to show that for any x, y E Bx with IIx - yll 2: E
and x - y E span({z}) there exist u,v E Sx such that u - v = EZ and II (x + y)/211 ::; lI(u+v)/211. We can clearly suppose that x-y = EZ. Moreover, translating x anruy in the direction of (x + y)/2, we can also assume that x E Sx. Then we choose norm-one vectors u, v in a two dimensional subspace containing x, y for which u - v = x - y and
100
Y, u, v are contained in one of the half planes determined by the line joining x with -x. Then ),(x + y)/2 = f3u + (1 - f3)x for some), ;::: 1 and f3 ;::: O. Observe that
), 2(u + v) = (f3 + ),)u + (1 - f3 - ),)x,
so the points u, ),(x + y)/2, ),(u + v)/2 lie on the same ray with the initial point x. Moreover, Ilxll = 1 = Ilullo )'1I(u+v)/211 ;::: 1 and ),(x+y)/2Iies between x and ),(u+v)/2 on the ray. From Remark 2.1 we therefore see that II (x + y)/211 ::; II(u + v)/211. •
Example 3.5 1. Let X be a Hilbert space. Using the parallelogram law, one can easily show that
8X (ZjE) = 1- )1- Gr for every Z E Sx and every E E [0,2].
2. Let X be the plane ]R2 with the norm:
where (x,y) E ]R2. Direct computation gives
if 0::; E::; 1,
if 1 < E ::; ~, iq <E::;2
where Z = (1,0). This example shows that 8x(zj E) need not be a convex function.
Our next result shows that the modulus 8x(Zj E) fits the general scheme described at the beginning of this section. Consequently, it satisfies the conclusion of Remark 3.1.
Proposition 3.6 Let X be a Banach space with dim (X) ;::: 2 and Z E Sx. Then there exists a family F of convex functions f : [0,2] -> [0,1] such that f(O) = 0 and
8x (Zj E) = inf{f(E) : f E F}
for every E E [0,2].
Proof. Given U E Sx, E E [0,2]' we set 8(u, Zj E) = inf{1 - II(x + y)/211} where the infimum is taken over all x, y E Ex such that x + y = sz for some s ;::: 0 and x - y = tz for some t ;::: O. One can easily check that the family F of all functions 8(u, Zj E) where u E Sx has the desired properties. •
UCED can be also characterized without using the modulus 8x (zj E).
Theorem 3.7 A Banach space X is UCED if and only if limn~oo Ilxn - Ynll = 0 whenever (xn ), (Yn) are sequences in X such that limn~oo Ilxnll = 1, limn~oo IIYnl1 = 1, limn~oo Ilxn + Yn II = 2 and there is Z E Sx with Xn - Yn E span( {z}) for each n.
Proof. Suppose that a space X is UCED and there exist sequences (xn), (Yn) in X with the following properties: limn~oo Ilxnll = 1, Iimn~oo IIYnl1 = 1, limn~oo IIxn + Ynll = 2,
Geometrical background 101
lim infn_ oo Ilxn - Ynll > c > 0 and there is Z E Sx such that Xn - Yn E span( {z}) for each n. Given 0 < "(, we can assume that Xn, Yn E (1 + "()Ex, lI(xn + Yn)/211 :::: 1 - "( and IIxn - Ynll :::: c for all n. Then
8x(z._c )<1 __ 1 Ilxn+Ynll<~. '1+,,( - 1+,,( 2 -1+,,(
Since "( > 0 is arbitrary and 8x (z; E) is continuous, 8x (z; c) = 0, contrary to our assumption.
Assume now that X is not UCED. Then 8x(z; E) = 0 for some z E Sx and E > O. From the definition of 8x (z; E) we obtain sequences (xn), (Yn) in Ex such that Ilxn - Ynll :::: E and 1- lin::; II(xn + Yn)/211 for every n. Clearly,
2 2 - - ::; IIxnll + IIYnl1 ::; IIxnll + 1 ::; 2
n
for all n. This shows that limn_ oo Ilxnll = 1. Similarly, limn--->oo IIYnl1 completes our proof.
1, which
• The modulus 8x (z; E) has local character since it depends on the direction z. Taking the infimum over z, we obtain the global modulus of convexity. This is however a special case of more general idea of measuring convexity of balls. To describe this idea we need to recall concept of a k-dimensional volume. Let X be a Banach space. Given Xl, ... ,Xk+l E X, we put
{I", 1 }
xiexI) ... xi(Xk+l). • A(Xl"",Xk+l)=SUP det : : : :XI,,,,,XkEEx* ..
Xt,(XI) ... X A', (Xk+l)
(3.1)
Clearly, A(XI,X2) = IlxI - x211 for any XI,X2 EX. Moreover, in the case when X is the k-dimensional Euclidean space, irA(XI,,,,,Xk+l) is equal to the volume of CO({XI"" ,Xk+l}) and A(XI, ... ,Xk+l) is equal to the volume of the parallelepiped drawn on the vectors Xl, ... , Xk+l' Formula (3.1) can be therefore seen as a definition of k-dimensional volume of such parallelepiped for an arbitrary Banach space. We put
J.L'X = sup{A(xl, ... ,Xk+l) : X!, ... ,Xk+l E Ex}.
Let now X be a Banach space with dim (X) :::: k + 1. This will be the standing assumption when dealing with the following modulus of k-convexity of X.
Definition 3.8
k . { II 1 k+l II } 8X(E) = mf 1- k + 1 ~Xi : Xl,··· ,Xk+l E Ex, A(XI, ... ,Xk+l) 2:: E
where E E [0, J.L'X). The space X is sa.id to be k-uniformly rotund (k-UR for short) if 8'X(E) > 0 for every 0 < E < JL'X.
Additionally, we say that all spaces of dimension less than k+ 1 are k-UR. The modulus with k = 1 was introduced as the first one and in this case slightly different notation and terminology is used. Namely, we write 8x or just 8 instead of 8ir. Clearly,
102
where € E [0,2]. The function 8x is called the modulus of convexity of X and instead of I-uniform rotundity terms uniform convexity (UC for short) or uniform rotundity (UR for short) are used.
Remark 3.9 1. In the definition of 8~(€) the inequality in "A(X1,'" ,Xk+l) 2: €" can be replaced by the equality.
2. The function 8~ : [0, JL~) ---> [0, 1] is nondecreasing.
3. We have 8~ (€) = inf 8M €) where the infimum is taken over all subspaces E of X with dim(E) = k + 1.
The same reasoning as in the proof of Proposition 3.4 applies to the modulus 8.
Proposition 3.10 Let X be a Banach space with dim(X) 2: 2 and € E [0,2]. Then
8(€) = inf { 1-11 x; y II : X,y E Sx, IIx - yll = €} .
The modulus of convexity 8 need not be a convex function (see [78] or [44]). On the other hand, the idea of the proof of Proposition 3.6 can be extended even to 8~. Indeed, let a> ° and kEN. We say that a function f : [0, a) ---> [0,1] is k-convex if the function € ..... f(€k) is convex. In [71] the following result was proved.
Proposition 3.11 Let X be a Banach space. There exists a family F of k-convex functions f : [0, JL~) ---> [0,1] such that f(O) = ° and
8~(€) = inf{f(€) : f E F}
for every € E [0, JL~).
Notice that in general 8 need not be continuous at 2 (see [44]). Nevertheless, Proposition 3.11 and Remark 3.1 enable us to establish continuity of 8~ inside its domain.
Corollary 3.12 The function 8~ is Lipschitzian on each interval (a, b) where ° < a < b < JL'X and the function 8~(€k)/€ is non decreasing. In particular, 8~ is continuous on [0, JL'X).
Example 3.13 1. Let X be a Hilbert space. In [11] it was shown that
k . _ _ (_ k 2/k) 1/2 8x (€)-1 1 (k+1)H1/k€
for every € in the domain of this function. If k = 1, this is just an easy consequence of the parallelogram law (compare to Example 3.5.1).
2. Let 1 < P < DC and X = V'([O, 1]). In this space we have the following Clarkson's and Hanner's inequalities (see [22], [50]). If P 2: 2, then
2(11xliP + lIyllP) ::; Ilx + yilP + Ilx - ylIP::; (lIxll + Ilyll)P + Illxll-llylllP for all X,y E X. Moreover, in case 1 < P::; 2 the opposite inequalities hold. From these inequalities one can derive the following formulae (see [50]). If P 2: 2, then
( (€)P)l/P 8x (€) = 1- 1- 2 ' (3.2)
Geometrical background 103
and if 1 < p :S 2, then
(1-6X(E)+~r +11-6X(E)-~IP =2 (3.3)
where E E [0,2J.
Analysis of the proof in [50J shows that in the above example the interval [0, 1J can be replaced by any measure space 0 such that LP(O) is at least two dimensional. Notice that LP(O) is n dimensional only if the measure is purely atomic with n atoms (see [5], p. 30). Then this space is isometrically isomorphic to ]Rn with the lP-norm.
There is a general way of showing that two spaces have the same modulus of convexity. Let X and Y be Banach spaces. We say that Y is finitely representable in X whenever for every finite dimensional subspace F of Y and every 'Y > ° there exists a linear one-to-one operator T : F -> X such that IITIIIIT-111 < 1 + 'Y.
Remark 3.14 Let 1 :S p < 00 and 0 be a measure space such that LP(O) is infinite dimensional. Then each of the spaces LP(O), LP([O, l]) is finitely representable in the other one (see [56J, p. 60).
It is easy to see that if Y is finitely representable in X, then 6!HE) :S 6}(E) for every k and every E E [0, {t}). Remark 3.14 gives us therefore another way of extending formulae (3.2) and (3.3) to the space X = LP(O) where 1 < p < 00 and 0 is an arbitrary measure space such that LP(O) is infinite dimensional.
By Dvoretzky's theorem (see [34] or [75]), 12 is finitely representable in any infinite dimensional Banach space X. Consequently,
6~(E) :S 6~(E) = 1- (1- (k + 1~1+1/kE2Ik r/2
In case k = 1 this estimate holds even under the weaker assumption that dim (X) :::: 2. This result is known as the Day-Nordlander theorem (see [87J or [27], p. 60).
A Banach space X is said to be superreflexive if each Banach space finitely representable in X is reflexive. In particular, every superreflexive space is reflexive. Moreover, a space X is superreflexive if and only if X· has the same property (see [26], p.152). The coefficient
E~(X) = SUp{E: 6~(E) = O}
is called the characteristic of k-convexity of the space X. If k = 1, we denote it by EO(X) and call simply the characteristic of convexity of X. The condition EO(X) < 2 characterizes the so-called uniformly nonsquare spaces (see [53]). The following result was proved in [12J.
Theorem 3.15 Let X be a Banach space and kEN. If E~(X) < 2k, then X is superreflexive.
This shows that uniformly convex spaces and even uniformly nonsquare spaces are superreflexive. On the other hand, superreflexive spaces can be renormed in such a way that they become uniformly convex (see [36]). Actually, for each superreflexive space X there exist p :::: 2, K > 0, and an equivalent norm II . II in X such that
104
for any E E [0,2]' where 5 is the modulus corresponding to the norm II . II (see [92] or [26]' p. 154).
A modification of the proof of Theorem 3.7 gives us its counterpart for k-UC.
Theorem 3.16 A Banach space X is k-UC if and only iflimn~oo A(x~, ... ,x~+l) = 0 whenever (x~), ... , (x~+1) are sequences in X such that limn->oo Ilx~11 = 1 for i =
1, ... , k + 1 and limn-><x> Ilx~ + ... + x~+111 = k + 1.
Clearly, UCED spaces are UC spaces and Theorem 2.3 shows that UC spaces are strictly convex. By Theorem 3.16 with k = 1 and the compactness argument, UC is actually equivalent to strict convexity for finite dimensional spaces. We now pass to a result which gives a uniform version of condition 2 in Theorem 2.3.
Theorem 3.17 A Banach space X is UC if and only if for any 1 < p < 00 and r > 0 there is a convex strictly increasing function gr : [0, +(0) -> [0, +(0) such that gr(O) = 0 and
for all X,Y E rBx and all 0:::::,\::::: 1.
Proof. Assume that X is UC and let 1 < p < 00. Clearly, it is enough to prove the conclusion for r = 1. From (2.1) it follows that
for every 0 < t ::::: 2.
Suppose that Jlo(t) = 0 for some t > O. Then there exist sequences (xn ), (Yn) in Bx such that Ilxn - Ynll ::::: t for all nand
lim 2p- 1Ulxn ll P + IIYnII P ) - Ilxn + Ynll P = O. n->oo
Passing to subsequences, we can assume that the limits
a = lim IIxnll , b = lim IIYnl1 and c = lim Ilxn +Ynll n----+oo n----+oo n-+oo
exist. For these values of a and b equality in inequality (2.1) holds. Consequently, a = b > 0 and so cP = 2PaP . From Theorem 3.16 we see that limn->oo Ilxn - Ynll = 0, which is a contradiction. This gives us the inequality Jlo(t) > 0 for every 0 < t ::::: 2.
We set
where the infimum is taken over all x, y E B x with II x - y II ::::: t and 0 < ,\ < 1. A similar reasoning as in the proof of Theorem 2.3 shows that fL(t) ::::: 21- PJlo(t) > 0 for every 0 < t ::::: 2 and it suffices to take as gl the double dual Young's function Jl".
Assume now that (3.4) is satisfied. If x, Y E Bx, IIx - YII = E, then
Thus we obtain the estimate 5(E) 2: ~91(E) which shows that the space X is UC. •
Geometrical background 105
Consider the case when X is a Hilbert space. Then the generalized parallelogram law
p,x + (1 - A)yII2 = Allxl1 2 + (1 - A) IIyll2 - A(l - A) Ilx _ Yl12
holds for all x, y E X and 0 :s; A :s; 1. We can therefore put gr(t) = t2 for every T.
A different general way of measuring convexity of balls in Banach spaces was developed by Milman [84]. Here we recall only its particular case. Let X be a Banach space with dim(X) ~ k + 1 and Ek be the family of all k dimensional subspaces of X.
Definition 3.18 b..1-(E) = inf inf sup Ilx + Eyll - 1
xESx EEt:k yESE
where E ~ O. A Banach space X is called k-uniformly convex (k-UC for short) if b..1-(E) > 0 for every E > O.
In addition we say that all spaces X with dim(X) < k + 1 are k-UC. Using Remarks 2.1 and 3.1, one can easily obtain the following properties of the above modulus.
Remark 3.19 Let f denote the function b..1-. Then
1. f(O) = 0 :s; feE) for every E ~ O.
2. The function E f-+ f(E)/E is nondecreasing on (0,+00).
3. f satisfies the Lipschitz condition with the constant 1.
The next two facts are also worth noting. We have
b..1-(E) = inf b..~(E)
for every E ~ 0 where the infimum is taken over all subspaces E of X with dim(E) ~ k + 1. Secondly, b..1-(E) :s; b..~+1(E) for every E ~ 0 and consequently k-UC implies (k + l)-UC. Of course,
b..k(E) = inf max{llx + EYll,llx - Eyll} - 1 x,yESx
for every E ~ O. It turns out that there is a simple relation between this modulus and ox. Namely,
b..1 ( E ) _ OX(E) X. 2(1 - OX(E)) - 1- Ox (E)
(3.5)
for every E E [0,2). This formula was established in [37] (see also [28]). It gives us a special instance of the following general result proved in [72].
Theorem 3.20 Let kEN. A Banach space X is k-UR if and only if X is k-UC.
In order to characterize the property dual to k-UC we need to introduce one more modulus. Let X be a Banach space with dim (X) ~ k + 1.
Definition 3.21 The modulus f3'X of k-uniform smoothness is given by the formula
f3'X(t) = sup sup inf ~(llx + tyII + IIx - tyII) - 1 xESx EEt:. yESE 2
106
where t 2': O. A Banach space X is called k-uniformly smooth (k-US for short) if
lim f3'X(t) = o. t->O t
In addition we say that all spaces X with dim(X) < k+l are k-US. The function f = f3'X
has the properties listed in Remark 3.19. Moreover, f3~+1(t) :s; f3~(t) for every t 2': 0 and consequently k-US implies (k + I)-US. It is also clear that f3x(t) = inf f3~(t) for every t 2': 0 where the infimum is taken over all subspaces E of X with dim(E) 2': k + 1. The following duality theorem can be found in [5] (see also [80]).
Theorem 3.22 Let kEN and X be a Banach space. X (X*) is k-UC if and only if X* (resp. X) is k-US.
From Theorems 3.22 and 3.15 we see that k-US spaces are superreflexive. In case k = 1 a different notation and terminology is used. Namely, instead of 131 we write px or simply p. We called this function the modulus of uniform smoothness of the space X and instead of I-uniform smoothness we just use the term uniform smoothness (US for short). Clearly,
1 px(t) = sup -(llx + tyII + Ilx - tyll) - 1.
x,yESx 2
Let u, v be nonzero vectors of a Banach space X. The function f(t) = Ilu+tvll+llu-tvll is convex on R This shows that the modulus Px is also a convex function. Moreover, f( -t) = f(t) for every t E R Consequently, f is nondecreasing on the interval [0, +00). It easily follows that in the definition of px one can replace the condition "x, y E Sx" by "x,y E Bx".
Example 3.23 1. Let X be a Hilbert space with dim(X) 2': 2. Then
px(t) = J1+t2 - 1
for every t 2': O.
2. Let 1 < p < 00 and 0 be a measure space such that the space X = IJ'(O) is at least two dimensional. Using Clarkson's and Hanner's inequalities and Remark 3.14, one can show that if 1 < p :s; 2, then
and if p > 2, then
( (l+t)P+II-W)l/P px(t) = 2 - 1
for every t 2': 0 (see [74]).
The proof of Theorem 3.22 does not give any satisfactory relations between moduli of convexity and smoothness. Such relations, known as Lindenstrauss' formulae, were obtained for k = 1.
Theorem 3.24 Let X be a Banach space and t 2': O. Then
Geometrical background
(i) px.(t) = sup{tEj2 - tiX(E) : 0::; E::; 2},
(ii) px(t) = sup{tEj2 - 8X.(E) : 0::; E::; 2}.
Proof. We will prove only formula (i). We have
2px·(t) = sup{llx* + ty*11 + IIx* - ty*ll- 2: x*,y* E Sx'}
= sup{x*(x) +ty*(x) +x*(y) - ty*(y) - 2: X,y E Sx, x*,y* E Sx'}
= sup{llx + yll + tllx - yll - 2: x, y E Sx}
= sup{llx+yll +tE- 2: X,y E Sx, Ilx - yll = E, 0::; E::; 2}
= SUp{tE - 28x(E) : 0::; E::; 2}.
Equality (ii) can be obtained in a similar way.
The coefficient (X) I· px(t)
Po = Im--t~O t
107
•
is called the characteristic of smoothness of a Banach space X. Lindenstrauss' formulae means that 2px' and 2px are dual Young's functions of 28x and 28x " respectively. By Lemma 3.2 we therefore obtain the following equalities.
Corollary 3.25 Let X be a Banach space. Then 2po(X*) EO(X*).
EO(X) and 2po(X)
This and Theorem 3.15 show that if po(X) < 1, then X is superreflexive. From Lindenstrauss' formulae and the corresponding theorems for the modulus of convexity we can also deduce the next two results. If dim (X) 2 2, then
px(t) 2 Ji+t2 - 1 (3.6)
for every t 2 o. Moreover, for each superreflexive space X there exist an equivalent norm II . II and constants C > 0, q ::; 2 such that p(t) ::; Ctq for every t 2 0 where the modulus p is evaluated with respect to II . II (see [26], p. 157).
Uniform smoothness is strictly related to differentiability of a norm.
Theorem 3.26 A Banach space X is US if and only if the norm of X is un~formly Frechet differentiable.
Proof. Let X be a Banach space and f(x) = Ilxll for x E X. Clearly,
1 1 px(t) 0::; t(j(x + ty) - 1) - -t (j(x - ty) - 1) ::; 2-t - (3.7)
for all x, y E Sx and t > O. In light of Remark 2.9 this shows that if X is US, then the norm of X is Gateaux differentiable. Moreover, from Remark 2.1 we see that the directional derivative 1'(x)y lies between the divided differences which appear in (3.7). It follows that they tend to l' (x)y as t -t 0 uniformly with respect to x, yES x.
To show the opposite implication we observe that
px(t) 11 'I 11 )) '() I 2--::; sup -(j(x + ty) - 1) - f (x)y + sup -(f(x - ty - 1 - f x Y t x,yESx t x,yESx -t
for every t > O. If f is uniformly Fnlchet differentiable, then the right-hand side expression tends to 0 as t -t O. Consequently, limt~o px(t)jt = O. •
108
4. Infinite dimensional geometrical properties
In the sequel T will denote a Hausdorff topology on a Banach space X. We will always assume that
1. scalar multiplication is sequentially continuous with respect to T, i.e. if a sequence of scalars (tn) converges to t and a sequence (xn) converges to x in T, then (tnxn) converges to tx in T,
2. if a sequence (xn) converges to x in T and y E X, then (xn +y) converges to x +y in T,
3. the norm of X is sequentially lower semicontinuous with respect to T, i.e.
Ilxll s: liminf Ilxnll n-->oo
whenever (x n ) converges to x in T.
Of course, if a topology T is linear, then it satisfies conditions 1 and 2. The basic examples of Hausdorff topologies which satisfies 1, 2, and 3 are T = w, i.e. the weak topology of X and T = w', i.e. the weak' topology if X is a dual space. Another important example is the topology of convergence locally in measure (elm for short) in LP spaces. Namely, let 11 be a (T-finite measure space with a measure JL and 1 s: p < 00.
We fix a partition {l1n} of 11 into subsets of positive finite measures. The elm topology can be defined via the metric
_ ~_1_ r If-gl d d(f, g) - ~ 2nJL(l1n) .Inn 1 + If _ gl JL
where f, 9 E LP(I1). Each elm-convergent sequence contains a subsequence that converges a.e. to the same limit. This and Fatou's lemma show that the norm of LP(I1) is elm-sequentially lower semicontinuous. It is also easy to see that dm-topology is weaker than the norm topology in LP(I1).
If JL(I1) < 00, then instead of d we can consider the simpler metric
r If -gl do(f, g) = .In 1 + If _ gl dJL.
In this case the elm topology reduces to the topology of convergence in measure. The set N with the counting measure is another special case. Then we obtain the space lP and elm convergence of a sequence is just coordinatewise convergence. For bounded sequences it is therefore equivalent to weak convergence if p > 1 and to weak' convergence if p = 1 and 11 is seen as the dual space of Co.
Definition 4.1 A Banach space X (or its norm) has the Kadec-Klee property with respect to a topology T (KK(T) for short) provided that if (xn) is a sequence in Bx converging to x E X in T and lim infn-->oo Ilxn - xII > 0, then Ilxll < 1.
This definition means that if a sequence (xn ) in Sx converges to x E Sx in the topology T, then it converges to x in norm. In particular, if a space X has KK(w) property, then weak convergence of a sequence in Sx to a limit in Sx is equivalent to convergence in norm.
Geometrical background 109
We will define a uniform version of KK(T) property. For brevity the limit of a sequence (xn) with respect to a topology T will be denoted by T-limn~oo X n. We say that (xn) is T -null if the limit is equal to O. Next, by N1 ( T) (N1 ( T )) we denote the set of all T - null sequences (xn) such that IIxnll :::: 1 (resp. Ilxnll ::; 1) for all n. The condition N1 (T) = 0 characterizes the spaces for which the convergence of sequences with respect to T is equivalent to the norm convergence. In case T = w this is the so-called Schur property. Obviously all finite dimensional spaces have this property and by Schur's theorem the same is true for 11. We set VX,T = sup infnEN Ilxn - xii where the supremum is taken over all sequences (Xn) E Bx with x = T-limn~oo xn.
Definition 4.2 Let X be a Banach space with N1(T) i= 0. Given E E [0, VX,T), we put
KX,T(E) = inf{l - Ilxll}
where the infimum is taken over all points x such that x = T-limn~oo Xn for some sequence (xn) in Bx with Ilxn - xii :::: E for all n. The space X has the uniform Kadec-Klee property with respect to T (UKK(r) for short) if KX,T(E) > 0 for every E E (O,VX,T)'
Note that since N1 (T) i= 0, there exists a T-null sequence in Sx. It follows that 1 ::; VX,T and the set over which we take the infimum in the formula for KX,T(E) is not empty. In addition, we say that spaces X with N1(T) = 0 also have UKK(T) property. In case T = W we drop the name of the topology in our notation. Thus KK, UKK and the modulus Kx refer to the weak topology.
Clearly, UKK(T) property implies KK(T) property and in general they are not equivalent. For instance each separable space and each reflexive space ean be renormed to have KK property (see [26]) which is not the case for UKK property (see [93]).
Proposition 4.3 There exists a family F of convex functions f : [0, VX,T) -+ [0,1] such that f(O) = 0 for every f E F and
KX,T(E) = inf{f(E) : f E F}
for every E E [O,VX,T)'
Proof. Take u E Sx and a r-null sequence (un) in Sx. Having E E [O,VX,T)' by K(E;U, (un)) we denote the infimum of all numbers t E [0,1] for which there is a scalar sequence (sn) such that 11(1- t)u + snunll ::; 1 and Sn :::: E for all n. It is easy to check that K(E;U,(Un )) is a convex function of E and KX,T(E) = infK(E;u,(un )) where the infimum is taken over all U E Sx and all T-nllll sequences (un) in Sx. •
Corollary 4.4 The modulus KX,T satisfies the conclusion of Remark 3.1.
Our next aim is to establish a relation between Kx and 8*. For this purpose we need the following lemma.
Lemma 4.5 Let (xn) be a weakly null sequence in a Banach space X and (x;J be a bounded sequence in X*. For every E > 0 there exists an increasing sequence (nk) such that IX~k(xnJI < E ifi i= k.
Proof. We can assume that X is a separable space. Passing to a subsequence, we can also assume that (x~) converges weak' to some y* E Bx'. The desired subsequence
110
can be chosen by induction. Given ~ > 0, we find nl so that ly*(xn)1 < ~/2 for all n:::: nl· Next, having nl < ... < nk-l, we pick nk > nk-l with IX~i(xnk)1 < E and I(X;'k - y*)(xnJI < E/2 for i = 1, ... , k - 1. Then IX~k(xnJI < E. •
Theorem 4.6 Let X be a Banach space and kEN. Then
JOT every ~ E [0, vx).
Proof. We fix ~ E [0, vx) and take a sequence (xn) in Ex such that it converges weakly to x and infnEN Ilxn - xii :::: ~. For each n there is x~ E Sx* such that x~(xn - x) = Ilxn -xii·
Take now 'Y > 0. In view of Lemma 4.5 we can assume that Ix~(xm - x)1 < 'Y if m i= n. Then
whenever nl < ... < nk. Hence
Using Corollary 3.12, we see that 81 (£k) S 1-lIxll. This clearly gives us the conclusion .
• Theorem 4.6 shows that k-UR spaces have UKK property. However, UKK property is much weaker than k- UR. Spaces with UKK property need not be even reflexive (see Example 4.23.2). One may therefore look for a more direct infinite dimensional generalization of uniform convexity.
Let (xn) be a sequence in a space X. The number
sep(xn) = inf{llxm - xnll : m i= m}
is called the separation constant of (xn).
Definition 4.7 A Banach space X is neaTly unifoTmly convex (NUC for short) provided that for every ~ > ° there is 8 > ° such that if (xn) is a sequence in Ex with sep(xn) :::: ~, then
inf{llxll : x E CO({xn})} S 1- 8.
NUC implies reflexivity. This is a consequence of a result due to James. Before formulating this theorem, we need to recall some additional terminology. A sequence (Xn) in a Banach space X is called a (Schauder) basis of X if each x E X has a unique expansion
00
x = LOnXn
n=l
Geometrical background 111
for some scalars aI, a2, .. .. We say that (xn ) is a basic sequence if it is a basis of the closed linear span of the set {xn} (see [76]). The following result can be derived from [54].
Theorem 4.8 If a Banach space X is not reflexive, then for every ° < , < 1 there exists a basic sequence (xn) in Bx such that sep(xn) 2': , and
inf{lIxll : x E cot {xn})} 2': ,.
Now we are in a position to establish a relation between NUC and UKK property.
Theorem 4.9 A Banach space X is NUC if and only if X is reflexive and has UKK property.
Proof. Assume that a Banach space X is NUC. Theorem 4.8 shows that X is reflexive. Let (xn) be a sequence in Bx converging weakly to x for which liminfn-->oo Ilxn - xii 2': E > 0. Using weak lower semicontinuity of the norm, one can easily find a subsequence (xnk ) so that sep(xnk) 2': E/2. Since X is NUC, we can assume that there exist elements Yk E co({x;}7~~~+1)' k = 1,2, ... , such that IIYkll ::; 1- 8 for every k, where 8 > ° depends only on Eo Clearly, (Yk) also converges weakly to x, so Ilxll ::; 1- 8. This shows that X has UKK property.
Let now X be a reflexive space with UKK property. Take a sequence (xn) in Ex such that sep(xn ) 2': E > 0. Passing to a subsequence, we can assume that (xn ) converges weakly to some x and lim infn-->oo IIxn -xii 2: E/2. Then Ilxll ::; 1-KX(E/2) and Mazur's theorem gives us Y E co({xn}) for which lIylI ::; 1- KX(E/2)/2. •
From Theorems 4.6 and 4.9 we immediately obtain the following corollary.
Corollary 4.10 If a Banach space X is k-UC for some k, then X is NUC.
We introduce another modulus corresponding to UKK(T) property. Let X be a Banach space with Nl (T) oF 0. Given E 2': 0, we put
dX,T(E) = inf {liminf Ilx + EXnll-l} n-->oo
where the infimum is taken over all x E X with Ilxll 2': 1 and all sequences (xn) E Nl(T). The function dX,T is nonnegative and satisfies the Lipschitz condition with the constant Ion [0,(0). For the moduli dX,T and KX,T we have a formula similar to (3.5).
Theorem 4.11 Let X be a Banach space. Then
for every E E [0,1).
Proof. Given E" E (0,1), we find x E Sx and (Yn) E Nl(T) such that sUPnEN Ilx + EYnl1 - 1 < dX,T(E) + ,. We put u = (1 + dX,T(E) + ,)-lx, Un = (1 + dX,T(E) + ,)-lEYn and Zn = u+un for n E N. Then (zn) is a sequence in Bx converging to U with respect to T and Ilzn - ull 2': E(l + dX,T( E) + ,)-1 for every n. Therefore
112
Since KX,T is continuous,
(4.1)
From Corollary 4.4 we see that if 0 < € < VX,T' then KX,T(€) < 1. We fix 0 < 'Y < 1 - Kx,T(e) and choose a sequence (zn) in Bx so that (zn) converges to x with respect to T, IIzn - xii 2: € for all nand 1 -lixll < Kx,T(e) + 'Y. Put v = (1 - Kx,T(e) - 'Y)-lx, Vn = Cl(Zn - x) for n E N. Then IIvll > 1, IIvnll 2: 1 for every nand (vn) is T-null. Hence
dXT ( € ) <supllv+ e vnll-l< KX,T(€)+'Y. , 1 - Kx,T(e) - 'Y - nEI'I 1 - Kx,T(e) - 'Y - 1 - Kx,T(e) - 'Y
It follows that
(4.2)
for every € E [0, VX,T).
To show the inequality opposite to (4.2) we set ¢(€) = e/(1- KX,T(€» where e E [0,1). By (4.1),
dx,T(¢(e» 2: (1 + dx,r{¢(e»)Kx,T C + d~~:~¢(€») . But from (4.2) we see that
Using Corollary 4.4 again, we obtain
This is the equality opposite to (4.2). • Corollary 4.12 A Banach space X has UKK(T) property if and only if dx,T(e) > 0 for every € > o.
We will study the property dual to NUC. For this purpose we need the next definition.
Definition 4.13 A Banach space X is nearly uniformly smooth (NUS for short) provided that for every € > 0 there is 1) > 0 such that if 0 < t < 1) and (xn ) is a basic sequence in B x, then there exists k > 1 for which
(4.3)
Notice that if a space X is NUS, then only finitely many elements of a given sequence (xn) may not satisfy condition (4.3). NUS can be therefore characterized in the following way.
Remark 4.14 An infinite dimensional space X is NUS if and only iflimt~O bx(t)/t = 0 where
bx(t) = sup {lim sup IIXl + txnll- 1} n~oo
Geometrical background 113
and the supremum is taken over all basic sequences (xn) in Ex.
Recall that if a weakly null sequence (xn) does not converge to zero in norm, then there is a subsequence (xnk ) which is a basic sequence. Moreover, every basic sequence in a reflexive space is weakly null (see [76]). It follows that in the case when X is reflexive the supremum in the definition of bx can be taken over all sequences (xn) E Nl(w). This leads us to a more general modulus.
Let T be a topology in a Banach space X and t :::: O. We put
bX,T(t) = sup {lim sup Ilx + tXnll- I} n""'oo
where the supremum is taken over all x E Ex and all (xn) E Nl(T). Notice that bX,T is a nonnegative convex function with bX,T(O) = o.
Definition 4.15 A Banach space X is nearly uniformly smooth with respect to T
(NUS(T) for short) if
lim bX,T(t) = O. t....,o t
Proposition 4.16 A Eanach space X is NUS if and only if X is NUS(w) and reflexive.
Proof. If X is reflexive, then the moduli bx,w and bx are equal. To complete the proof it is therefore enough to show that NUS implies reflexivity. Assume that a space X is NUS but. not reflexive. By Theorem 4.8 for every 0 < B < 1 there exists a basic sequence (xn) in Ex such that
B < 11_1 Xl + _t Xkll - 1+t l+t
for ~ll 0 < t < 1 and k. Hence B(1 + t) ::; 1 + bx(t) and since 0 < B < 1 is arbitrary, t::; bx(t). This contradicts our assumption. •
Recall that 1* denotes dual Young's function of f.
Theorem 4.17 Let X be a reflexive space. Ey band d we denote the moduli bx,w and dx ' ,w, respectively. Then
2b* G) ::; d(E) ::; b*(2E)
for every E E [0,1).
Proof. Fix E E (0,1). Next, take arbitrary x* E X* with IIx*11 :::: 1 and a sequence (x~) E Nl(w). We find elements X,Yl,Y2,'" E Ex so that x*(x) = Ilx*11 and x~(Yn) = Ilx~1I for every n. We can assume that (Yn) converges weakly to some y E Ex. Then (xn) E Nl(w) where Xn == ~(Yn - y) for all n. Moreover,
limn....,oox~(xn) == ~limn....,oo Ilx~ll. Hence
liminf Ilx' + Ex~lllimsup Ilx + sxnll :::: liminf(x' + EX~)(X + SXn) n----.oo n~CX) n..--.-J.oo
== X*(X) + ES lim x~(xn) n""'oo
ES > 1+- 2
114
for every S 2:: O. This shows that
€S 1 + "2 ::; (1 + b(s))(1 + d(E)) ::; 1 + b(s) + d(€) + €b(s).
Therefore Es/2 - 2b(s) ::; d(E) which gives us the inequality 2b*(E/4) ::; d(E).
Take now x E Ex and (xn) E N 1(w). Put s = d(E)/E. For each n there is y~ E Sx' such that IIx + sXnll = y~(x + sXn). Passing to a subsequence, we can assume that (y~) converges weakly to some x' E Ex. and the limit TJ = limn--->oo Ilx~1I exists where x~ = y~ - x'. Consider two cases.
1. I" < TJ. Then Ellx*II/TJ < 1. Assuming that X* =I 0, from Remark 2.1 we get
1 -lIx*1I 2:: IIx* + x~1I - 1 2:: EII~'II (1Ix* + EII~*II x~ll- 1) .
for every n. Hence
12:: IIx*11 + ~l~~~f (II 11:*11 x' + ~x~ll-l) 2:: ~d(E) so IIx'll ::; 1- STJ. Consequently,
lim sup Ilx + sXnll = x*(x) + S limsupx~(xn) ::; 1- sTJ + STJ = 1. n--->oo n--->oo
Observe that if x* = 0, then TJ = 1 and the last inequality also holds.
II. € 2:: TJ. Then lim sup Ilx + sxnll ::; 1 + sTJ ::; 1 + SE.
n--->oo
In both cases we obtain b(s) ::; SE. Hence d(E) = SE ::; 2SE - b(s) ::; b*(2E). • Theorem 4.17 and Lemma 3.2 enables us to establish duality between NUC and NUS.
Corollary 4.18 A Banach space X (X*) is NUS if and only if X* (resp. X) is NUC.
From Theorem 3.22 and Corollaries 4.10 and 4.18 we see that if a space X is k-US for some k, then X is NUS. The next geometrical infinite dimensional property we want to discuss is not so strongly related to convexity or smoothness as the previous ones.
Definition 4.19 A Banach space X satisfies the Opial property with respect to a topology T if
liminf IIxnll < liminf IIx + xnll n-+oo n-+oo
for every T-null bounded sequence (xn ) in X and every x =I O.
Replacing the strict inequality "<" with "::;", we obtain the definition of the so-called nonstrict (or weak) Opial property with respect to the topology T. Of course, in this case it is not necessary to exclude the point x = O. If T = W or T = w*, then T
convergent sequences are bounded so the word "bounded" can be dropped from the definition.
Example 4.20 Let cp be a periodic function with period 271' such that
cp(t) = {I -2
if 0::; t ::; ~7r, if ~7r < t < 271".
Geometrical background 115
Put <I>n(t) = <p(nt), n E N. Then (<I>n) is a weakly null sequence of Rademacher-like functions in lJ'([0, 27r]) for every 1 < p < 00. Consider the function
where c E lR is treated as the constant function. Since
(21r 4 A~(O) = -p Jo 1<p(t)IP- J sgn(<p(t)) dt = 37rp(2P-2 - 1),
A~(O) # 0 whenever p # 2. It follows that Ap(O) is not a minimal value of Ap, except for the case p = 2. Consequently, lJ'([0,27f]) does not satisfy even the nonstrict Opial property for any p # 2.
Definition 4.21 Let X be a Banach space with NJ(T) # 0. The Opial modulus of X with respect to the topology T is the function rX,T : [0,00) --> lR given by the formula
rx,T(c) = inf{liminfllx+xnll-l} n-->oo
where c 2: 0 and the infimum is taken over all elements x E X with Ilxll 2: c and all sequences (xn) E NJ(T). The space X has the uniform Opial property with respect to T if rx,T(c) > 0 for every c > O.
In case T = w the name of the topology is omitted from the terminology concerning Opial properties. The uniform Opial property with respect to a topology T implies the Opial property with respect to T. Moreover, for spaces with NJ (T) # 0 the condition rX,T(O) 2: 0 characterizes the nonstrict Opial property with respect to T. Notice that
for every t > O. Consequently, X has the uniform Opial property with respect to T if and only if dX,T(t) > t - 1 for all t > O.
We will describe the general method of evaluating some of the considered moduli. Let T
be a topology in a Banach space X and p : [0,00) x [0,00) --+ [0,00) be a nondecreasing with respect to each variable continuous function. We say that X has property L(p, T) if
l~~j;;f Ilx + xnll = p (l~j;;f Ilxnll, Ilxll)
whenever x E X and (xn) is a bounded T-null sequence. In spaces with property L(p, T) we easily get the following formulae.
Proposition 4.22 Let X be a Banach space with property L(p,T). Then
dX,T(t) = bxAt) = p(t, 1) - 1, rX,T(t) = p(l, t) - 1
for every t 2: O.
Example 4.23 1. Let 0 be a measure space with a measure {L and 1 ::; p < 00. We will show that if f E lJ'(O) and (fn) is a bounded sequence in lJ'(O) converging to zero a.e., then
lim IIf + fnll P = IlfilP + lim IlfnllP n---i-CXl n-4OO
(4.4)
116
provided that the above limits exist. First observe that for every E > 0 there is C > 0 such that
for all a, b E R Hence
for any n E N, wEn. Let 9n denote the nonnegative part of the function Ilf + fnlP - IfnlP - IflPI - Elfnlp· Then 9n :s: (C + l)lflP for every n and the sequence (9n) converges to zero a.e. By the dominated convergence theorem, limn~oo In 9n dfL = O. Moreover,
Passing to the limit with n -+ 00, we see that
where M = limsuPn~oo Ilfnllp. Since € > 0 is arbitrary, this gives us formula ( 4.4).
Let now n be a-finite. Each elm-null sequence in X = Il'(n) has a subsequence converging to zero a.e. It follows that X has property L(pp, elm) with
By Proposition 4.22
1
dX,clm(t) = bX,clm(t) = T'X,clm(t) = (1 + tP)p - 1
for every t ::::: O. In contrast to the case of weak topology, for each 1 :s: p < 00
the space Il'(n) has therefore the uniform Opial property with respect to the elm topology. It has also UKK(elm) property and if p > 1, then it is NUS(elm).
2. Let 1 < p < 00 and (Xn) be a sequence of Banach spaces with the Schur property. Consider the space
X= (fxn) , n=l ["
i.e. X is the space of all sequences x = (x(k)) such that x(k) E Xk for every k and
1
Ilxll = (~IIX(k)ll~k);; < 00.
We put Pmx = (x(l), ... ,x(m),O,O, ... ) where x = (x(k)) with x(k) E Xk for every k. This formula gives us the projection Pm : X -+ X and we set 11m = Id-Pm. Take an element x E X and a weakly null sequence (xn ) in X. Since weak convergence is equivalent to norm convergence in each Xn , limn~oo IIPmxn11 = 0 for every m. Consequently,
liminf Ilx + xnllP =liminfliminf(llPm(x + xn)IIP + IIRm(x + xn)IIP) n-too m-too n-tOQ
=liminf (11Pmx11P + liminf l111mx + xnllP) m---too n-too
Geometrical background 117
=IIXIIP + liminf IlxnllP. n~oo
This shows that X has property L(pp,w).
Consider now some particular cases. If Xn = [1 for every n, then X is not reflexive but it has UKK property in the nontrivial way. If in turn Xn is the space lE!.n endowed with the maximum norm, then 100 is finitely representable in the resulting space X. Consequently, X is not superreflexive. On the other hand, X is reflexive, so it is NUC.
3. Let X be a Hilbert space, x E X and (xn) be a weakly null sequence in X. Then limn~oo(xn,x) = 0, so
This shows that X has property L(p,w) with p(s,t) = (s2 +t2)1/2.
Let J¢ be a duality mapping on a space X. We say that J¢ is sequentially T-continuous if J¢ is single-valued and sequentially continuous as the function from X with the topology T to X* with the weak' topology.
Proposition 4.24 Let X be a Banach space with a sequentially T-continuous duality mapping J¢. If x E X and (xn) is a bounded T-null sequence, then
liminf <I>(llx + xnll) = <I>(llxll) + liminf <I>(llxnll) n-+oo n~oo
where <I>(t) = J~ ¢(s) ds, t 2': o.
Proof. Take x 1= 0 and a bounded T-null sequence (xn ). Next, fix n and consider the function g(t) = <I>(lltx + xnll) where t E lEI.. By our assumption and Theorem 2.11, 9 is differentiable and g'(t) = J¢(tx + xn)(x) for every t E lE!.. Since <I> is convex and increasing on [0,00), 9 is also convex. Hence g' is continuous and consequently,
<I>(IIX + xnll) - <I>(lIxnll) = l J¢(tx + xn)(x) dt. (4.5)
From our assumption and the dominated convergence theorem we obtain
1n1 i'l 11 lim J¢(tx + xn)(x) dt = J¢(tX) (x) dt = Ilxll ¢(lltxll) dt = <I>(llxlI)·
n-+oo. 0 0 0
This and equality (4.5) give us the desired formula. • Corollary 4.25 Let X satisfy the assumption of Proposition .4-24. Then X has property L(p, r) with p(s, t) = <I>-I(<I>(s) + <I>(t)) ,
dX,T(t) = bx,T(t) = rx,T(t) = <I>-1(<I>(1) + <I>(t)) - 1
for every t 2': 0 and
for all 0 < c < 1.
118
Proof. The first part of the conclusion is a direct consequence of Propositions 4.22 and 4.24. In order to obtain the last formula we take a sequence (xn ) in Bx with T-limn-->oo Xn = x and IIxn - xII 2: E for all n. By Proposition 4.24,
iJ>(lIxll) + iJ>(E) ~ iJ>(lIxll) + liminfiJ>(llxn - xiI) ~ iJ>(1). n-->oo
Hence 1-lIxll 2: 1-iJ>-1(iJ>(1) -iJ>(f)) which shows that KX,T(f) 2: 1-iJ>-1(iJ>(1) -iJ>(f)).
Take now Y E Sx and a T-null sequence (Yn) such that IIYnll = E for every n. Since E < 1, there exists s > 0 such that lim infn-->oo IIsy + Ynll = 1. Given, > 0, we can therefore assume that IIsy + Ynll ~ 1 +, for all n. Then
iJ>(s) = iJ>(lIsyll) = Hminf iJ>(lIsy + Ynll) - iJ>(f) = iJ>(1) - iJ>(E). n-->oo
On the other hand,
K (_E_) < 1 _ Ilsyll X.T 1 +, - 1 +,
which shows that KX,T(E) ~ 1 - s. • Corollary 4.26 Let X satisfy the assumption of Proposition 4. 24- Then X has UKK(T) property, the uniform Opial property with respect to T and is NUS(T ).
Corollary 4.25 gives us another way to obtain the formulae for the moduli of the space X = IJ'(!1) in the case when 1 < p < 00 and T is the clm topology (see Example 4.23). In particular,
1
KX,clm(E) = 1 - (1 - EP)P
for every f 2: O.
5. Normal structure
Normal structure plays essential role in some problems of metric fixed point theory, especially those concerning nonexpansive mappings (see [44] and [64]). Before formulating the definition, we need to set up some additional notation. Let A be a nonempty bounded set in a Banach space X. We put
rCA) = inf {sup IIx - yll : x E A} . yEA
This number is called the Chebyshev radius of A and the set ZeAl of all points x E A for which the infimum is attained is called the Chebyshev center of A (with respect to A). In general zeAl may be empty. Clearly, ~ diam(A) ~ rCA) ~ diam(A). A set A is said to be diametral if rCA) = diam(A) > O.
Definition 5.1 Let B be a nonempty subset of a Banach space X and let :F be a nonempty family of subsets of B. The family :F is said to be normal provided that
rCA) < diam(A)
for every bounded set A E :F with diam(A) > O. If there exists a constant 0 < c < 1 such that
rCA) ::; c diam(A)
Geometrical background 119
for every bounded set A E F with diam(A) > 0, then F is said to be uniformly normal.
There are two basic special instances of normal families in a subset B of a space X.
1. The family F of all bounded closed convex subsets of B. In this case we say that B has normal structure, which we abbreviate to NS, (uniform normal structure which we abbreviate to UNS) whenever F is normal (resp. uniformly normal).
2. Let T be a topology in the space X. Take F to be the family of all bounded convex T-sequentially compact subsets of B. Then B is said to have normal structure with respect to T (T-NS for short) if F is normal. Normal structure with respect to the weak topology is called weak normal structure (WNS for short). Clearly, NS implies WNS and for reflexive spaces these two properties are equivalent.
Theorem 5.2 Let X be a Banach space. A set B c X does not have NS if and only if there is a bounded sequence (xn) in B such that
lim Ilxn - xii = diam( {xn}) > 0 n-->()()
(5.1)
for every x E co( {xn}).
Proof. Assume that B contains a sequence (xn) for which condition (5.1) holds. Then the closure of co({xn}) is diametral, so B does not have normal structure.
Suppose in turn that B contains a bounded convex closed diametral subset A. Put d = diam(A). By induction we construct a sequence (xn) in A so that Ily-xnll ::::: d-2/n for every n and every y E co( {xk}~:D. As Xl we take a arbitrary element of A. Next, having Xl, ... , xn-l, we choose a lin-net C = {Yl, ... , Ym} in co( {xk}~:D and set
1 m Z= - LYi.
m i=l
We can clearly assume that m ::::: n. Since A is diametral, there exists Xn E A such that liz - xnll ::::: d - 11m2 • Then
for each k = 1, ... , n. This gives us the inequality IIYk - xnll ::::: d - 11m::::: d - lin. Consequently, lIy - xnll ::::: d - 2/n for every Y E co( {xd~:~)·
Take now Y E CO({Xk}). Then
d::; liminf Ily - xnll ::; lim sup lIy - xnll ::; diam({xk}) ::; d n-->()() n-->()()
which shows that (xn) satisfies (5.1). • The sequence (xn) constructed in this proof does not have a Cauchy subsequence. It follows in particular that compact sets have NS.
120
Proposition 5.3 Let B be a subset of a Banach space X such that every T-sequentially compact set A c B is separable. If B does not have T-NS, then there is a bounded sequence (xn) in B such that (xn) converges to x with respect to T and
lim IIxn - xii = lim Ilxn - xmll > 0 n--+oo n--+oo
for every m. In case T = w the separability assumption is not necessary.
Proof. To show the first part assume that A is a bounded convex T-sequentially compact diametral subset of B. Let a countable set {Uk} C A be dense in A. We repeat the construction in the proof of Theorem 5.2 with the net C replaced by {Ul, ... , un}. This gives us a sequence (xn) which satisfies (5.1) for every x E A and it suffices to take its T-convergent subsequence. The second part follows directly from the proof of Theorem 5.2 and Mazur's theorem. •
Our next theorem is a simple consequence of Proposition 5.3.
Theorem 5.4 Let X be a Banach space in which every T-sequentially compact set is separable. If X has the Opial property with respect to T, then X has T-NS. In case T = w the separability assumption is not necessary.
ueED is another geometrical property which implies NS.
Proposition 5.5 Let A be a bounded convex subset of a Banach space X with
diam(A) > O.
Then there exists Z E S X such that
rCA) ::; (1- 8x(z; 1)) diam(A).
Proof. Put d = diam(A). Given 0 < E < d, we choose x, yEA so that IIx - yll :::: d - E
and set z = (x - Y)/lix - YII. For each u E A we have Ilu - xii::; d, lIu - yll ::; d and (u - y) - (u - x) = IIx - yllz. Hence
Ilu - x; Y II ::; d (1 -8x ( z; d ~ E) ) .
It follows that rCA) ::; d(l - 8x(z; 1 - E/d)). In view of continuity of 8x(z; E) this completes the proof. •
Proposition 5.5 shows that if 8x(z; 1) > 0 for every z E Sx, then X has NS. In particular, we obtain the following result.
Corollary 5.6 If a Banach space X is UGED, then X has NS.
Several coefficients related to normal structure were defined. We will discuss some of them. Let (xn) be a bounded sequence in a space X. Using Ramsey's theorem (see [33], p. 235), one can find a subsequence (xnk ) such that the double limit of IIxnk - xn; II over k,i --+ 00, k of i, exists. We denote this limit by limk#i IIxnk - xn;lI·
Geometrical background 121
Definition 5.7 Let X be a Banach space withN1(r) # 0. The r-convergent sequences coefficient of X is defined as
where the infimum is taken over all bounded r-null sequences in X such that both limits exist and limn-+co IIxnll # o.
In the particular case when r = w we obtain the weakly convergent sequences coefficient WCS(X). Clearly, 1 :::; rCS(X) :::; 2 and Proposition 5.3 gives us the following result.
Proposition 5.8 Let X be a Banach space in which every r-sequentially compact set is separable. If 1 < rC SeX), then X has T-NS. In case r = w the separability assumption is not necessary.
Obviously,
TCS(X) = inf {lim IIxn - xmll} n;em
where the infimum is taken over all bounded T-null sequences in X such that the above limit exists and limn-+co Ilxnll = 1. This observation shows that if. a space X has L(p, T) property, then TCS(X) = p(l, 1). In particular, we have rCS(X) = q>-1(2q>(1)) in the case when X satisfies the assumption of Proposition 4.24. We see for instance that WCS(lP(r)) = 21/ p for every infinite set r and every 1 < p < 00. As the next example we take X = £P(n) where n is a IT-finite measure space and 1 :::; p < 00. Then clmCS(X) = 21/ p . As we shall see in Theorem 5.19 the space £P(n) has UNS whenever 1 < p < 00. Consequently, it has clm-NS. In [69] it was shown that clm-sequentially compact sets in L1(n) are separable. From Proposition 5.8 we therefore deduce that also L1(n) has clm-NS. On the other hand, the closed convex hull of the Rademacher functions is a diametral set in L1([0, 1]) (see the proof of Theorem 5.19). It follows that L1([0,1]) does not have even WNS.
Assuming the uniform Opial property, we can strengthen the conclusion of Theorem 5.4. Namely, we have the following obvious estimates.
Proposition 5.9 Let X be a Banach space. Then
1 + dx,T(l) = 1 + rX,T(l) :::; rCS(X).
Corollary 5.10 Let X be a Banach space. If X has UKK(T) property or the uniform Opial property with respect to T, then rCS(X) > 1.
This and Proposition 5.8 show in particular that UKK property implies WNS. From Theorem 4.9 we therefore see that NUC spaces have NS.
Let X be a Banach space. We set
N(X) = inf { di:~:) } ,
where the infimum is taken over all bounded convex closed sets A c X with diam(A) > O. The number N(X) is called the normal structure coefficient of X. Clearly, the condition N(X) > 1 characterizes spaces X with UNS.
122
Proposition 5.11 Let X be a Banach space. Then N(X) :s: WCS(X).
Proof. Let (xn) be a weakly null sequence in X such that the limits d = limn#m Ilxn -xmll and c = limn~oo Ilxnll > 0 exist. Given 0 < "( < c/2, we can assume that Ilxn - xmll :s: d + "( and Ilxnll ~ c - "( for all n, m. We find functionals x~ E Sx. so that x~(xn) = Ilxnll for every n. By Lemma 4.5 we can assume that Ix~(xm)1 < "( whenever m =I n. If x E CO({Xk}), then Ix~(x)1 < "( for n sufficiently large. Hence
This shows that r( A) ~ c - 2"( where A is the closure of co( {xd ). Consequently,
N(X) < diam(A) < d + "( - r(A) - c - 2"(
which gives us the inequality N(X) :s: die. Now we can conclude the proof by taking the infimum. •
A reasoning similar to that in the proof of Proposition 5.5 leads us to the following estimate.
Theorem 5.12 Let X be a Banach space. Then
1 N(X) ~ 1 _ 8x(I)
Theorem 5.12 shows that if 8x(l) > 0 (or equivalently EO(X) < 1), then X has UNS. This result was extended in [3J.
Theorem 5.13 Let X be a Banach space and kEN. Then
N(X) ~ (max { 1 - \~k E, 1 - 8~(E)} r1
In particular, if E~(X) < 1, then X has UNS.
To obtain an estimate for the normal structure coefficient one can also use methods of convex analysis. A useful criterion for a point to be a minimum point of a convex function can be given in terms of subdifferentials (see [98]).
Theorem 5.14 Let A be a nonempty convex subset of a Banach space X and f : X ->
lR be a continuous convex function. The function f attains its minimal value on A at a point Xo E A if and only if there is a functional x* E 8f(xo) such that
x*(x-xo)~O
for every x E A.
Let A =I 0 be a bounded convex subset of a space X. Given YEA, we put fy(x) = Ilx-YIl for x EX. The function fy is convex, continuous, and weakly sequentially lower semicontinuous. Consequently, the same is true for the function f(x) = sup{fy(x) : YEA}. Assume that A is weakly compact. Since f is weakly sequentially lower semicontinuous, it attains its minimal value on A. Clearly, this minimal value equals
Geometrical background 123
r(A) and it is attained at the points of the Chebyshev center Z(A). We therefore see that if A is weakly compact, then Z(A) =1= 0. This is in particular the case when A is compact. We need now a formula for the subdifferential of the function f.
Theorem 5.15 Let T be a compact metric space and X be a finite dimensional Banach space. Assume that {ft}tET is a family of convex functions defined on X such that for each x E X the function t ...... ft(x) is continuous on T. Put f = SUPtET It. Then
8f(xo) = co ( U 81t(xo)) tET(xo)
for every Xo E X where T(xo) = {t E T : ft(xo) = f(xo)}.
We can return to the problem of estimating the normal structure coefficient. From Theorems 5.14 and 5.15, and Corollary 2.12 we obtain the following result.
Theorem 5.16 Let A be a nonempty compact convex subset of a finite dimensional Banach space X and Xo E A. If Xo E Z(A), then there exist elements Y1, ... ,Yn E A, functionals xi, ... ,x~ E B x', and nonnegative scalars t1, ... ,tn such that L:~=1 ti = 1,
for i = 1, ... , n, and
for every x E A.
xt(xo - Yi) = Ilxo - Yill = r(A)
n
L tiXi(X - xo) ~ 0 i=l
It turns out that the assumptions appearing in Theorem 5.16 are not restraining in our case. Indeed, if a space X is not reflexive, then N(X) = 1 (see [79] or [6]) and for reflexive spaces we can modify the formula for N(X).
Lemma 5.17 If a space X is reflexive, then
(5.2)
where the infimum is taken over all convex hulls K of finite subsets of X such that diam(K) > O.
Proof. Let X be a reflexive space and N'(X) denote the right hand side expression in (5.2). We take a bounded convex closed set A c X with diam(A) > O. If 0 < r < r(A), then n B(y,r) nA = 0
yEA
where B(y, r) = {x EX: Ilx - YII ::; r}. Since B(y, r) n A are weakly closed subsets of the weakly compact set A, there is a finite set F c A such that
n B(y,r) nco(F) c n B(y,r) nA = 0. YEco(F) yEF
124
It follows that r ::; r(co(F)) and consequently,
N'(X) < diam(co(F)) < diam(A). - r(co(F)) - r
But 0 < r < r(A) is arbitrary, so we get the inequality N'(X) ::; diam(A)/r(A). This gives us the estimate N'(X) ::; N(X) and the opposite one is triviaL •
We will calculate the values of N(LJ'(n)) and WCS(LJ'(n)). For this purpose we need the following result from [105J.
Lemma 5.18 Let 1 < p < 00 and l/p+l/q = 1. ffYl, ... , Yn E LJ'([O, 1]) andtl, .. ·, tn are nonnegative numbers with L:~=1 ti = 1, then
where s = max{p, q}.
Notice that if n is a purely atomic measure space, then LJ'(n) is isometrically isomorphic to IP(r) for some set r. As we have already observed wcs(lP(r)) = 21/ p if 1 < p < 00
and the set r is infinite.
Theorem 5.19 Let 1 < P < 00 and n be a measure space such that the space X = LJ'(n) is infinite dimensional. Then
N(X) = min { 2~, 21-*} and WCS(X) = N(X) if P 2: 2 or n is not purely atomic.
Proof. We will first establish the formula for N(X). Since X is reflexive, in view of Lemma 5.17 and Remark 3.14 it suffices to consider the case when X = LJ'([O, 1]). Let A be a convex hull of a finite subset of X such that diam(A) > o. Then A is contained in a finite dimensional subspace of X. In particular, A is compact, so there is Xo E Z(A). From Theorem 5.16 we obtain elements Yl, ... ,Yn E A, functionals xi, ... ,x~ E Ex -, and nonnegative scalars tl, ... , tn such that L:~=1 ti = 1, xi{xo -Yi) = Ilxo -Yill = r(A) for i = 1, ... , n, and
By Lemma 5.18
where s = max{p,q}, l/p + l/q = 1. On the other hand,
t till ~ tkYk - Yill2: t tiX; (~tkYk - Yi) 2: t tiX;(XO - Yi) = r(A).
Consequently, diam(A)/r(A) 2: 21/ 8 • In light of Lemma 5.17 this shows that N(X) 2: 21/ 8 •
Geometrical background 125
Clearly, X = .D'([0,1]) contains a subspace isometrically isomorphic to [P. Hence N(X) ::; WCS(X) ::; WCS(lP) = 21/ p , so N(X) = WCS(X) = 21/ p if p 2': 2. Assume now that 1 < p < 2. To estimate N(X) from below we consider the sequence (rn) of the Rademacher functions. It is a normalized weakly null sequence with IIrn - rmll = 21/ q
whenever m # n. Therefore diam(A) = 21/ q where A is the closure of co{rn }. Recall that (rn) is an orthonormal system in L2([0, 1]). Hence
lim inf Ilrn - fll 2': lim r1 rn(t) (rn(t) - f(t)) dt = 1 n--+oo n-+oo io
for every f E span({rn}). This shows in particular that sup{lIf - rnll : n E N} 2': 1 for every f E A and consequently, r(A) 2': 1. It follows that N(X) ::; 21/ q and finally N(X) = 21/ q if 1 < p < 2.
To complete the proof it suffices to evaluate WCS(X) for X = .D'(n) where 1 < p < 2 and n is not purely atomic. This last assumption implies that an infinite Rademacher system (rn) can be found in X (see [5], p. 32). Hence WCS(X) ::; limm#n IIrm - rnll = 21/q. This and Proposition 5.11 show that WCS(X) = N(X) = 21/ q if 1 < p < 2. •
Every infinite dimensional Hilbert space is isometrically isomorphic to a space /2(r) for some infinite set r. Theorem 5.19 gives us therefore the values of considered coefficients for such a space.
Theorem 5.20 Let H be an infinite dimensional Hilbert space. Then
N(H) = WCS(H) = .../2.
These formulae can be also obtained in a more direct way (see [99) and [79]). From Dvoretsky's theorem, Lemma 5.17, and Theorem 5.20 we see that N(X) ::; N(/2) = .../2 for any infinite dimensional Banach space X. We recall two more estimates for the normal structure coefficient. The first of them was established in [3) but it can be also proved with help of the subdifferential technique (see [90]). It shows that all finite dimensional spaces have UNS.
Theorem 5.21 If X is a Banach space with dim(X) = n, then N(X) 2': 1 + l/n.
The second estimate is given in terms of the modulus of smoothness.
Theorem 5.22 Let X be a Banach space with dim (X) 2': 2. Then
N(X) 2': (inf { 1 + px(t) - ~ : t 2': O} ) -1 (5.3)
Proof. Given a space X, we put 7/>x(t) = 1+px(t)-t/2 and a(X) = inft~o7/>(t). Easy calculation gives a(/2) = .,;3/2. Let now X be a Banach space X with dim (X) 2': 2. From (3.6) we see that
a(X) 2': a(Z2) > 2/3. (5.4)
Moreover, if t > 4/3, then
:....px_(:....:t)_---'p:...,x---"'(~'"_) > _3p_x_(~_) > _3p_12_(~_) = ~ t-~ - 4 - 4 2
126
and consequently 'l/Jx(t) 2: 'l/Jx(4/3). This shows that a(X) = inf099/3'I/JX(t).
If X is not reflexive, then N(X) = 1 = 1/'l/Jx(0) 2: 1/a(X). We can therefore assume that X is reflexive. Let A be a convex hull of a finite subset of X with d = diam(A) > O. We will show that d/r 2: 1/a(X) where r = r(A). In view of (5.4) this is true if d/r > 3/2. Suppose now that d/r ~ 3/2 and take xo E Z(A). Theorem 5.16 gives us corresponding elements Y1, ... ,Yn E A, functionals xi, ... ,x~ E B x', and nonnegative scalars t1, ... ,tn' We put Zi = Xo - Yi for i = 1, ... ,n. Then
2(px(t) + 1) 2: 11~(Zi - Zj) + ~Zill + 11~(Zi - Zj) - ~z;11 > ~x~(zo - zo) + !x~(zo) - ~x*(".o - zo) + !x*(zo) -d" J r" dJ""" J r J '
= 2~ + t - ~xi(Zj) + G -D xj(z;)
for any i,j and every t 2: O. Hence
if 2/ d 2: t/r. Since d/r ~ 3/2, the last condition holds for all t E [0,4/3]. Consequently, d/r 2: inf099/3 'l/Jx(t) = 1/a(X). In light of Lemma 5.17 this gives us the desired estimate. •
Theorem 5.22 can be given a form analogous to Theorem 5.12. Indeed, Theorem 3.24 (ii) shows that 6(E) = sup{tE/2 - px(t) : t 2: O} is double dual Young's function of 8x· and estimate (5.3) can be written as
1 N(X) 2: 1 _ 6(1)'
Using Theorem 5.22, we can show that if po(X) < 1/2, then X has UNS. Form Corollary 3.25 and Theorem 5.12 we therefore see that each of the conditions po(X) < 1/2 and EO(X) < 1 gives UNS both in X and X*. In contrast to Theorem 5.13 multi-dimensional uniform smoothness does not imply even NS.
Example 5.23 Consider Bynum'S spaces 12,1 and 12,00 (see [19]). The first of them is the space 12 endowed with an equivalent norm given by the formula
where x E 12 , II . II stands for the standard norm in 12, x+ denotes the positive part of x and x- denotes the negative part of X. Similarly, 12,00 is the space 12 with the norm
The space 12,00 is dual to 12,1.
In [102] it was proved that 12,1 is 2-UR. Theorem 3.22 therefore shows that the space 12,00
is 2-US. It does not however have normal structure. Indeed, let (en) be the standard basis of 12. It is easy to see that the closure of the set co( {en}) is diametraJ.
Geometrical background 127
6. Bibliographic notes
Books [9], [27] and [24] are our general references for geometry of Banach spaces. The basic notions of this theory, i.e. strict convexity and uniform convexity were introduced by Clarkson [22]. The monograph [52] is devoted to detailed study of the first property, but it also contains sections on the second one. Theorem 2.5 was obtained by Khamsi [59] and our proof simplifies his idea.
Mazur [83] obtained the formula for the derivative of the norm in V spaces. The notion of smoothness of a space was in turn introduced by Krein [65]. Various aspects of smoothness and differentiability are discussed in the book [26]. Duality mappings were studied for the first time in [14]. Theorem 2.11 is due to Asplund [4]. The book [21] provides the systematic treatment of duality mappings and further references.
There are numerous uniform versions of convexity in the literature. Many of them can be found in [27], [52], [26] and the survey papers [49], [86]. Uniform convexity of Clarkson is the most extensively studied and applied one. Clarkson [22] established also the inequalities, which are now named after him, and proved that the spaces V([O, 1]) are UC if 1 < p < 00. The modulus of convexity was in turn introduced by Day [23]. He also introduced uniform smoothness and obtained Theorem 3.22 with k = l. This result was next improved by Lindenstrauss. Namely, he defined the modulus of smoothness and proved Theorem 3.24 (see [74]). The exact values of the moduli of convexity and smoothness for V spaces were given by Hanner [50] and Lindenstrauss [74]. Our method of proving continuity of moduli has its origin in [43] (see also [44]). Part 1 of Lemma 3.2 was established in [82].
Several different scaling functions corresponding to UC and US were defined (see, for instance, [2] and [7]). In particular, the modulus considered in [10] is related to both these properties. The concept of UC can be extended to some metric spaces. Counterparts of UC for the hyperbolic metric on the open unit ball of a Hilbert space were successfully apply to the theory of fixed points of holomorphic mappings (see [47], [45] and [66]). A general approach to UC in metric spaces was given in [100]. Theorem 3.17 is a special case of a more general result due to ZaJinescu [108] (see also [106]). Uniform convexity in every direction was introduced by Garkavi [39] in connection with his study of Chebyshev centers.
Milman [84] was the first to undertake systematic study of multi-dimensional uniform convexity and smoothness. Another approach to finite dimensional uniform convexity was found by Sullivan [103] who defined the modulus of k-convexity. Theorem 3.20 was proved by Lin [74], but its partial cases with k = 1,2 had been earlier obtained in [37] and [40]. The reader should be warned that there are different definitions of convexity,k-uniform in the literature (see, for instance, [52], p. 73). The same is true for the Kadec-Klee property (see [26], p. 42). Moreover, some authors use the terms Radon-Riesz property or property (H) instead of the Kadec-Klee property (see [24]). In our terminology we follow Huff [51] who also introduced the uniform Kadec-Klee property and nearly uniform convexity. It should be noticed that independently of Huff a property equivalent to NUC was introduced in [46] under the name of noncompact uniform convexity. Lennard [69] extended Huff's concepts to an abstract topology T
(see also [13]). He used them to obtain a fixed point theorem in the space L1 (0) with the elm topology. Here we use the terminology from [55] which is slightly different from that of Lennard.
Corollary 4.10 was obtained independently in [107] and [63]. Our proof bases on Lemma 4.5 which was shown in [94]. Theorem 4.11 is a direct generalization of an analogous
128
result obtained for the weak topology in [81]. Two different definitions of nearly uniform smoothness can be found in the literature (see [93] and [8]). Here we follow [93]. Corollary 4.18 was also obtained in that paper, but the present proof is patterned on a reasoning from [42]. A similar result was obtained in [30].
The Opial property originates in a fixed point theorem proved by Opial in [88] which is also the proper reference for Example 4.20. The uniform Opial property with respect to the weak topology was defined in [96] and the Opial modulus was introduced in [73]. The formula for rX,T given in Corollary 4.25 was also essentially obtained in [73]. Kadec-Klee and Opial properties were extended to the abstract hyperbolic metric setting in [60]. Refined versions of Opial properties can be found for instance in [101], [17] and [18]. Property L(T,p) has its origin in [70] (see also [60] and [32]). Formula (4.4) is a special case of a more general result from [15].
The notion of normal structure was introduced by Brodskii and Milman in [16]. They also proved Theorem 5.2 and found the first application of normal structure to fixed point theory. Uniform normal structure was in turn introduced in [41]. It is an open problem whether UNS implies superrefiexivity. In [49] a positive solution to this problem was announced but the proof turned out to be erroneous. Generalizations of NS to metric spaces were found in [61] and [89] (see also [58] and [64]). Normal structure with respect to an arbitrary topology was introduced in [62]. Here we follow the terminology of [55] and [32].
The idea of the proof of Theorem 5.2 was developed by Landes who obtained various characterizations of normal and weak normal structure and studied their hereditariness properties (see [67] and the survey paper [68]). More details on normal structure, its numerous modifications and applications can be found in survey papers [49], [86] and [104]. Theorem 5.4 has its origin in [48] and Corollary 5.6 was obtained independently in [25] and [109].
The coefficient WCS(X) and normal structure coefficients were introduced by Bynum [20]. He established their relations to normal structure and calculated the value of WCS(lP). Theorem 5.12 is also due to Bynum. Several different formulae for WCS(X) were found (see [5], p. 120) and the definition of TCS(X) bases on one of them. This definition is contained in [55] and [32] where relations of TCS(X) to another quantities appearing in metric fixed point theory are studied.
Lemma 5.17 and Theorem 5.21 were proved by Amir [3]. The method used by him to obtain the second of these results is different from ours. We apply a technique due to Pichugov [90]. Theorem 5.19 was essentially proved in [94] (see also [91]). A different proof was given in [29]. Relation between the coefficient po(X) and UNS was studied in [57] but estimate (5.3) was given in [95].
Although multi-dimensional uniform smoothness does not imply normal structure, coefficients related to the modulus bx T turned out to be useful in some fixed point problems (see [38], [31] and [32]). '
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Chapter 5
SOME MODULI AND CONSTANTS RELATED TO METRIC FIXED POINT THEORY
Enrique Llorens Fuster
Department d'Analisi Matematica
Facultat de Matematiques
Universitat de Valencia
Doctor Moliner 50, 46100 Burfassot, Spain
Enrique.Llorens. uv.es
1. Introduction The classical modulus of convexity introduced by J.A. Clarkson in 1936 to define
uniformly convex spaces is at the origin of a great number of moduli defined since then [1].
Indeed, there are a lot of quantitative descriptions of geometrical properties of Banach spaces. The most common way for creating these descriptions, is to define a real function (a "modulus") depending on the Banach space under consideration, and from this define a suitable constant or coefficient closely related to this function. The moduli and/or the constants are attempts to get a better understanding about two things:
• The shape of the unit ball of a space, and
• The hidden relations between weak and strong convergence of sequences.
One might well ask: Are there too many moduli for these purposes? Maybe! In part this is because many of these moduli involve very difficult computations, and, often there are intricate links between them. Moreover, it is not unusual to find some moduli defined in (seemingly) different ways, depending on the preferences of the writer.
The aim of this Appendix is to give only a brief summary of a few of these properties that are in some ways related to Metric Fixed Point Theory. It is not a comprehensive list of the geometrical properties of Banach spaces that have been described using moduli and/or constants. Among other things, such a list would be too long, and have many unavoidable overlaps.
Further, similar coefficients, for example those which were given in ([69]), have not been included as independent items. We have only concerned ourselves with coefficients and moduli defined for general Banach spaces. For example, we have not listed those defined in [56J or [53J for Banach spaces with the Schauder finite dimensional decomposition. For the sake of brevity, we have also considered only those constants depending on the weak topology, and of course, on the norm topology of the space. The author apologizes in advance for any omissions.
133
w.A. Kirk and B. Sims (eds.), Handbook of Metric Fixed Point Theory, 133-175. © 2001 Kluwer Academic Publishers.
134
The excellent survey by Prus [79J, as well as the chapter in this volume on the Geometrical Background for Metric Fixed Point Theory, has helped guide the preparation of this summary.
We shall assume throughout this paper that (X, 11·11) is a Banach space with dimension at least two. We will use Bx and Sx to stand for the unit ball and the unit sphere of X, respectively. Wherever possible we have retained the original notation used for the various moduli and constants, however a few exceptions were necessary to avoid confusion.
2. Moduli and related properties
2.1. Clarkson's Modulus of Convexity, [19]
It is the function 8x : [O,2J --+ [0, 1J given by
8x (c) := inf {l-II~(X +y)11 : X,y E Bx, Ilx - yll ~ c}.
Related coefficient. Characteristic of convexity of (X, II . II)
co(X):= sup{c E [O,2J : 8x(c) = OJ.
Geometrical properties in terms of this modulus and/or this coefficient.
1 Definition 2.1 The Banach space (X, II ·11) is uniformly convex (UC) whenever 8x (c) > 0 for 0 < c ~ 2, or equivalently if co(X) = O.
2 The Banach space (X, II . II) is strictly convex (SC) if and only if 8x (2) = l.
3 The Banach space (X, 11·11) is uniformly nonsquare if and only if co(X) < 2.
4 James, Enflo.
For a Banach space (X, II . II) the following statements are equivalent
a) X is superrefiexive.
b) X has an equivalent uniformly nonsquare norm.
c) X has an equivalent uniformly convex norm.
5 ([45]) If 8x (1) > 0 then (X, II . II) is reflexive and has uniform normal structure.
6 (Gao-Lau [43]) If 8x (~) > ! then (X, II . II) has uniform normal structure.
7 (Gao-Lau [43]) If 8x (c:) ~ !c: for some c E (0, ~J then (X, II . II) has uniform normal structure.
8 ([76]) If 8x (c) > 8h(c) := max {O, ';1} then (X, 11·11) has uniform normal structure.
Facts about this modulus and/or this coefficient.
1 In the above definition 'Bx' and '~' can be replaced respectively by 'Sx' and '='. (For a proof see, for instance, [21]).
Moduli and constants
2 8 X is a monotone increasing function on [0, 2].
3 If Y is a closed subspace of X, 8y (c) 2: 8x (c) for 10 E [0,2].
4 For X,y,P E X, R> 0, and r E [0,2R]
Ilx -pll::; R } Ily - pil ::; R => lip - ~(x + y)11 ::; (1 - 8x (-~)) R. Ilx - yll 2: r
In particular,
Ilxll::;R }=>II!(x+Y)II< (1-8 (1Ix-yll))R. Ilyll ::; R 2 - x R
5 For all u,v E Ex and c E [0,1],
2min{c, 1- c}8x(llu - vii) ::; I-lieu + (1- c)vll·
(See [46], p. 111).
6 If (H, II . II) is a Hilbert space,
8H (c):= 2-~. 2
135
7 (Nordlander, [73]) The highest possible value of 8x is attained in Hilbert spaces, that is, for any Banach space (X, II . II)
8 () 2 - V4 - 102 Xc::; 2 .
8 8x is continuous in [0,2). (See [46] or [24J for a proof). (Also at 10 = 2 if (X, 11·11)
is (UC), [1]).
9 There are examples for which 8x is not a convex function. See Goebel-Kirk, [46J, p.60.
10 (Hanner, [50]. See also Prus, [79]).
11
Let (0, JL) be a measure space such that JL takes at least two different values. For p 2: 2,
For 1 < p < 2 then
This shows in particular that the spaces IJ'(O,JL) are (UC) whenever 1 < p < 00.
Clearly this covers the case of the spaces £p.
lim 8x(f) = 1 - !fO(X). €~2- 2
See [46J.
136
12 For all c: E [c:o(X), 2],
1 8x (2(1 - .sx(c:))) = 1 - 2C:o(X).
See [46].
13 See [24], p.125. .sx(c:Il < .sX(C:2)
C:l C:2
whenever 0 < C:I < C:2 :s: 2.
14 (D.J. Downing and B. Thrret, [35]).
If J1 is a measure, C:o (IJ'(J1,X)) = max{c:o(cP),c:o(X)}.
15 Let). ;::: 1 and let X A denote the space obtained by renorming the Hilbert space (£2, II· II) by means of
Then
c: (X ) = { 2().2 - 1) ~ ).:s: \1'2 o A 2 ). ;::: \1'2.
16 ([79]) Let E be the space ]R2 endowed with the norm
II(x,y)1I := max{lxl, IYI, Ix - YI}·
We have
OE(C:) = max {o, ~(c: - I)} =} c:o(E) = 1.
17 ([47]) Let (X, 11·11) be a uniformly convex Banach space, and let C be a convex, bounded and closed subset of X. If T : C --+ C is uniformly lipschitzian with constant k and k is less than the (unique) solution of the equation
then T has a fixed point in C.
18 ([35]) Stability of condition c:o(X) < 1.
Let X be a Banach space and let XI := (X, II . 111) and X2 := (X, II ·112), where II . 111 and II . 112 are two equivalent norms on X satisfying for a, f3 > 0,
for all x E X.
If C:o(Xd < 1 and if
where k := ~, then C:O(X2) < 1.
Moduli and constants 137
Equivalent statement: [35],Th. 6. Let (X, 11·11) be a Banach space with c:o(X) < l. Let Y be a Banach space isomorphic to X. If k is the unique solution of the equation
k (1 -Dx G) ) = 1, and d(X, Y) < k, then c:o(y) < l.
19 ([57]) If c:o(X) = 1 then every nonempty bounded closed convex subset of X has the fixed point property for asymptotically nonexpansive mappings.
2.2. Lovaglia local modulus of convexity, [68)
For x E Sx, the modulus of convexity of X at the point x, is the function
8x(x,c:):= inf {l-II~(x+Y)11 :X,y E Ex, Ilx -yll2: c:}.
Related coefficient. Characteristic of convexity of (X, II . II) at x E Sx
c:o(X, x) := sup{c: E [0,2] : Dx(X,c:) = OJ.
Geometrical property in terms of this modulus and/or this coefficient.
Definition 2.2 Lovaglia [68]. The Banach space (X, II . II) is locally uniformly rotund (LUR) whenever 8x(x, c:) > ° for all x E Sx and c: E (0,2].
Separation of this property.
1 Obviously, every (UC) Banach space is (LUR). The converse is not true:
Example 2.3 M.A. Smith, 1978. [86]
Let £1 be endowed with the norm
IIxll := (1lxlli + IIxll§)!·
Then for all x E £1 IIxlh ::; Ilxll ::; v2llxll1
and (£1,11·11) is (LUR) but it does not admit a uniformly convex norm (otherwise it would be a reflexive Banach space).
2 (LUR) Banach spaces are strictly convex. The converse is not true again.
Example 2.4 M.A. Smith, 1978. [86]
Let £2 be endowed with the norm
IIxll w := (1Ixll~ + IITxll§)~·
where IIxlis := max{lx11, 11(0, X2,··· )1I2}, and
138
Then the norm II . IIw is equivalent to II . 112 but (£2, II . IIw) is reflexive (SC) but not (LUR).
3 Example 2.5 See [72]. Consider the classical space Co endowed with Day's norm
[00 1 ]~ IIxll := sup ~ 2j (x(aj))2 ,
where the supremum is taken over all permutations a of the positive integers.
One has that Z := (CO, II . II) is (LUR) and hence (SC). Nevertheless, the well known mapping T : B-; ---> B-; given by
T(Xl,X2, ... ):= (1,Xl,X2, ... )
is II . II-nonexpansive and lacks fixed points. Thus, (LUR)~(FPP).
2.3. Smulian's modulus of weak uniform rotundity
For x· E Sx., the modulus of convexity of X with respect to x·, is the function given by
8x (x·,.): [0,2]-- [0,1]
6x(x',.o) := inf ({l} U {1 -11~(x + y)11 : x,y E Sx, Ix'(x - y)l ~ .o}) . The reason for specifically including 1 in the set whose infimum defines this modulus is to avoid the following particular situation. When x· is a non-norm attaining functional there are no points x, y in S x such that Ix' (x - y) I 2: 2. So 6 x (x' , 2) would not be well defined.
Geometrical property in terms of this modulus and/or this coefficient.
Definition 2.6 Smulian, 1939. The Banach space (X, 11·11) is weakly uniformly rotund (WUR) whenever 8x(x', c) > ° for all x· E Sx and .0 E (0,2].
Separation of this property.
1 Neither of the conditions (LUR) and (WUR) implies the other. (See [86]).
2 Every (UC) Banach space is (WUR). See, for example, the book of RE. Megginson, p.465, [70].
The converse is not true: The space (£2, II 'lIw) given in the M.A. Smith's paper [86] is (WUR) but not (UC).
3 (WUR) Banach spaces are strictly convex. See again the book of R.E. Megginson, p.465, [70] for a proof.
The converse is not true.
Example 2.7 M.A. Smith, [86]. Recall the equivalent norm defined on Co by M.M. Day: For u in CO enumerate the support of u as (nk) in such a way that lu(nk)1 ~ lunkHI· Define Du E £2 by
Du(n) := {U~'1k) n = nk o otherwise
Moduli and constants 139
and define Illulll = IIDuI12. For x E £2 let
1 ~ IlxilL := 111(21IxI12,xl,x2,x2, ... ,Xj,Xj, ... Xj, ···)111
Then (h II· IlL) is (LUR) and hence rotund, but not (WUR).
2.4. V.L. Smulian modulus of weak* uniform rotundity
For x E Sx, the modulus of convexity of X* with respect to x, is the function
bx(x,.) : [0,2] ---+ [0,1] given by the formula
bx ' (x, c:) := inf {l-II~(x' +y*)11 : x',y' E Sx', I(x' - y')(x)l:2: c:}
Geometrical property in terms of this modulus and/or this coefficient.
Definition 2.8 V.L. Smulian, 1939. The Banach space (X*, 11·11) is weak' uniformly rotund (W'UR) whenever bx'(x,c:) > ° for all x E Sx and E E (0,2].
Megginson's book [70], on page 466 reads
It should be noted that some sources say that if X is a normed space such that X' satisfies the above definition of weak' uniform rotundity, then it is X instead of X' which is called weak' uniformly rotund.
1 ([70], p. 466) For dual Banach spaces, (WUR)=;.-(W'UR)=;.- (SC).
2.5. Lovagia modulus of weak local uniform rotundity
For x E Sx and x E Sx', the modulus of convexity of X at x with r-espect to x·, is the function 8x(x,x',.): [0,2]---+ [0,1]
8x (x,x',c:) := inf ({1} U {1 -11~(x + y)11 : y E Sx, Ix'(x - y)1 :2: c:}) .
Geometrical property in terms of this modulus and/or this coefficient.
Definition 2.9 Lovaglia, 1955.The Banach space (X, II ·11) is weakly locally uniformly rotund (WLUR) whenever 8x(x,x*, c:) > ° for all x' E Sx and c: E (0,2].
Separation of this property.
1 (LUR)=;.- (WLUR). (See [70]) The converse is not true: The space (£2, 11·llw) given in [86] is (WLUR) but not (LUR).
2 (WUR)=;.- (WLUR)=;.- (SC). None of the implications of this statement is reversible. M.A. Smith's example (h II . ilL) is (WLUR) but not (WUR).
Example 2.10 M.A. Smith, [86]. For all x in £2 define
140
and
IlxiiA := (lIxll~ + 11(~x2' ~X3' •.. )II~)! . Then (i2 , II . IIA) is a reflexive (SC) not (WLUR) Banach space.
2.6. Lindenstrauss Modulus of Smoothness, [67]
It is the function P x : [0,00) ---+ lR given by
px(t) := sup {~(lIx + tyII + Ilx - tyII) - 1 : x, y E EX} .
Related coefficient. . px(t)
po(X):= hm --. t--->O+ t
Geometrical properties in terms of this modulus and/or this coefficient.
1 Definition 2.11 The Banach (X, 11·11) is uniformly smooth (US) whenever po(X) =0.
2 Po(X) < ~ =? X is superreflexive and has (UNS). (See [46] p.70-71 for detailed references about this result).
Facts about this modulus and/or this coefficient.
1 For each positive t, max{O, t - 1} ~ px(t) ~ t.
(See, for instance, [70] for a proof).
2 Alternative formula for Px:
px(t) = sup {~(llx + yll + Ilx - yll) - 1 : x E Ex, Ilyll ~ t} . 3 For a Hilbert space H PH(t) = VI + t 2 - 1 for every t ~ O. px(t) ~ PH(t) for
t ~ O.
4 Let (11, J.L) be a measure space such that J.L takes at least two different values. For 1 < p ~ 2 then
and for p > 2, then
1
( (1 +t)P + 11- tIP)" PLP(O,I')(t) = 2 - 1
for every t ~ O. (See [67]). Clearly this covers the case of the spaces i p .
5 For every Banach space (X, II . II) , the modulus of smoothness of (X, II . II) is a convex continuous function.
Moduli and constants 141
6 Lindenstrauss formulae (see [67]).
px.(t) = sup {~et - 8x (e) : 0 ::; 10 ::; 2} .
px(t) = sup{~et- 8x.(e): 0::; 10::; 2}.
Thus, po(X*) = !eo(X) and po(X) = !eo(X*). Consequently, (X, II . II) is (UC) if and only if its dual space is (US) and (X, II . II) is (US) if and only if its dual space is (UC).
7 po(X) = 0 if and only if for each positive T/ there exits a positive 10 (depending on T/) such that, if x E Sx, Y E X and IIx - YII ::; 10, then
IIx + YII ~ Ilxll + Ilyll - T/llx - yll· (See [22], p. 147).
2.7. Directional modulus of rotundity
For a given non zero z EX, the modulus of convexity of X in the direction of z, is the function 8x( --t z,·) : [0,2] -+ [0,1]
8x (--t z,e) := inf {1-11~(x +y)11 : x,y E Sx, IIx -yll ~ 10,3'>" E R s.t. x - Y =.>..z}.
Related coefficient.
eo,z(X):= sup{e: 8x(--t z,e) = O}.
Geometrical properties in terms of this modulus and/or this coefficient.
1 Definition 2.12 The Banach space (X, II . II) is uniformly convex in every direction (UCED) whenever 8x( --t Z, e) > 0 for all z E X\{O} and 10 E (0,2].
2 ([45]) If eo,z(X) < 1 for all z EX, z =f. 0, then (X, II . II) has normal structure.
3 ([23]' [93]) (UCED)=HWNS).
Separation of (UeED).
1 (WUR)* (UCED)* (SC). The converse implications are not true.
The above example (£2, II· IIA) is a (UCED) Banach space which is not (WUR). On the other hand, (£2, II . IlL) is (SC) but not (UCED).
Facts about this modulus (See Prus' chapter in this book) ..
1 For a Hilbert space H, 8H(--t z,e) = 1 - V1- (~)2, for every z E SH and 10 E [0,2].
2 8 x (--t Z, .) need not be a convex function.
3 Every separable Banach space admits an equivalent norm with respect to which it is (UCED).
142
2.8. Gurarii modulus of convexity, [49]
It is defined by the formula
!3x(E):= inf {1- inf Iltx+ (1- tM : x,y E Sx,lIx -yll = E}. tEIO,lj
Facts about this modulus (See Sanchez and Ullan [81]).
1 For any E E [0,2], OX(E) :S (3X(E) :S 2bx(E).
2 (X, II . II) is (SC) if and only if (3x (2) = 1.
3 (X, II . II) is uniformly nonsquare if and only if there exists E E (0,2) such that (3X(E) > 0.
4 (X, 11·11) is uniformly convex if and only (3X(E) > ° for all E E (0,2].
5 (3X(E) is a continuous function at [0,2). Continuity may fail at E = 2. Nevertheless, if (X, II . II) is a uniformly convex Banach space then (3x is continuous at E = 2.
6 If (X, II . II) is an inner product space, (3X(E) = bx(E).
7 If the inequality (3X(E) 2 1- VI - ~ holds in a normed space (X, 11·11) for every
E E (0,2] then (X, II . II) is an inner product space.
S (3X(E) = in£{l- inftEIO,ljlltx+ (1- tM: x,y E Ex, Ilx -yll = E}.
9 (3x is nondecreasing on [0,2].
10 (3X(E) = in£{ 1 - inftEIO,ljlltx + (1 - t)yll : x, y E Ex,llx - yll 2 E} .
11 For all r E [0,1]' (3x(rE) :S r(3x(E).
12 The function E ...... f3xf?) is non-decreasing in (0,2].
13 (3x is strictly increasing in [EO(X), 2].
14 If (X, II . II) is strictly convex and (3x is continuous at E = 2 then (X, II . II) is uniformly Ilonsquare.
15 For p 2 1 and (X, II . II) is the Banach space flp or £P then for all E E [0,2],
(3 x (E) :S 1 - (1 - (~) p) ~ .
16 For p 2 2 and for all E E [0,2],
2.9. Milman Modulus of Convexity
It is the function dx : [0,00) --+ [0,00) given by (see [71])
dX(E) := inf {max{llx + Eyll, Ilx - EYII : X,y E Sx,} - I}.
Moduli and constants 143
Main features of this modulus.
1 Figiel's formula [37] : For every c E [0,2)
6x(c) _ dx ( c ) 1- 6x(c) - 2(1 - 6x(c)) .
2 This formula shows that (X, II ,11) is (UC) if and only if dx(c) > 0 for all c > O.
2.10. Geremia-Sullivan Modulus of k-rotundity
For Xl, X2, ... ,Xk+l E X the value
is called k-volume of the set co{ Xl, ... ,Xk+1}'
The function
(k) ._ . {1. ) } 6x (c) .- mf 1- k + 111xl + ... + Xk+111 . X!, ... ,Xk+l E Ex, V(Xl, ... ,Xk+1 2: c
is called the modulus of k-rotundity of (X, 11·11). Clearly this function is defined on the interval [0, f.Lk(X)) where
Related coefficient. Characteristic of k-convexity of (X, II . II)
Geometrical properties in terms of this modulus and/or this coefficient.
1 Definition 2.13 Sullivan, (88j.(See also Geremia-Sullivan, [44]). The Banach space (X, II . II) is k-uniformly rotund (k-UR) whenever
6~)(c) > 0 for c E (O,f.Lk(X)).
(The case k = 1 corresponds to Clarkson's uniform convexity.)
2 ([15]) If c~k)(X) < 1 then (X, II· II) has (UNS).
Separation of (k-UR). In [62] D. Kutzarova gave the following
Example 2.14 Let X be the £1 direct sum of the Banach spaces Y = ]Rl and Z = £2·
Then X is 2-uniformly rotund. But co(X) = 2. Of course, X is not uniformly convex.
144
Facts about this modulus and/or this coefficient (See Prus' chapter in this book).
1 In the definition of 81 (e), '2 e' can be replaced by '= e'.
2 The function 81 ( .) is nondecreasing.
3 The function 81(-) is continuous on [0, JL!;(X)).
4 The function 81(-) is lipschitzian on each interval (a, b)where 0 < a < b < JLk(X).
5 The function e f-+ 67c~€'} is nondecreasing.
6 (UC)=> (kUR) => ((k + I)UR). In fact,
8x(c) = 8~)(e) ~ 8t;)(c) ~ 8t;+l)(c).
for all positive integer k and all suitable c.
7 If X is infinite dimensional, as well as the Hilbert space H, 1
k I: ( k 2/1:)" 8x (c) ~ 8H = 1 - 1 - (k + 1)1+1/I:e .
8 ([15]) Let kEN. If c~I:)(X) < 2k then X is superreflexive.
2.11. Milman k-dimensional modulus of convexity
Notation: £1: is the collection of all (k + I)-dimensional subspaces of X.
In the same paper of Milman there seems to have been defined the following modulus of k-dimensional convexity. (Here we are following the survey of Prus [79]).
,6.~)(e):= inf inf sup {lix + cYII- 1 : y E BE}. xESx EEe,
Geometrical property in terms of this modulus and/or this coefficient.
Definition 2.15 The Banach space (X, II· II) is k-uniformly convex (k-UC for short)
whenever ,6.~)(e) > 0 for e > o.
Geremia and Sullivan for k = 2 and P.K. Lin ([65]) in the general case have shown that k-uniform convexity and k-uniform rotundity are equivalent properties.
Facts about this modulus (See Prus' chapter in this book).
1 0 = ,6.~)(o) ~ ,6.~)(e) for every to 2 o.
2 The function e f-+ ~)Z:(€) is nondecreasing in (0,00).
3 ,6.~)(t) ~ ,6.~+l)(t) for every t 2 o. Consequently (k - UC) => ((k + 1) - UC).
4 ([37]) ,6.1 ( e ) _ 8x(e)
x 2(1 - 8x (e)) - 1 - 8x(e)"
Moduli and constants 145
2.12. Milman k-dimensional moduli of smoothness
Notation: £", is the collection of all k-dimensional subspaces of X.
For each x E Sx and t ;::: 0, the k-dimensional modulus of smoothness at x is given by
!3x(k, x, t) := sup {inf{ Ilx - tyjI + Ilx + tyjI - 1 : y ESE}} . EEl:. 2
Milman's k-dimensional modulus of smoothness is given by
/lj;)(t) := sup{,6x(k,x, t) : x E Sx}.
Geometrical property in terms of this modulus and/or this coefficient.
Definition 2.16 A Banach space (X, II . III is called k-uniformly smooth, (k-US) for short, if
,6(k) (t) lim _x __ =0. t~O+ t
Properties of this modulus (See Prus' chapter in this book).
1 ° = ,6~)(0) ::; ,6~)(E) for every E;::: 0.
2 The function E ...... f3j;; (e:) is nondecreasing in (0, 00 ) .
3 ,6~+l)(t) <:::: ,6~)(t) for every t ;::: 0. Consequently (k - US) => «k + 1) - US).
2.13. Partington modulus of "UKK-ness", [74]
It was defined by J.R. Partington in the proof of Theorem 1 of [74]. It is the function Px : [0, K(X)) -t [0,1] given by
Px(c) := inf{I-llxll : 3 Xn E Bx (n = 1, ... ), Xn '::"x, Ilxm - xnll ;::: 10, (m =f- n)}.
Here K(X) = K(Bx) is the so called Kottman constant of the space (X, 11·11) ,that is,
K(Bx) := sup {{inf Ilxm - xnll : m =f- n} : Xn E Bx (n = 1, ... )}.
Main features of this modulus.
1 (See [32], or [13]). For E E [0, 2~)
Pep (E) = 1 - (1- 10;) ~ .
2.14. Goebel-Sekowski modulus of noncompact convexity, [48]
Let us suppose that dim(X) = 00. The modulus of noncom pact convexity of a Banach space (X, II . II) is the function ll.x : [0,2] --+ [0,1] defined by
ll.X(E) := inf {1- dist(O, A) : A c Bx, A =f- 0, A = co(A), a:(A) ;:::E}.
Here a:(A) := inf{ r > ° : A C Bl U ... U Bk, diam(Bi) < r} is the Kuratowski measure of noncompactness of the set A eX.
146
Related coefficient. Characteristic of noncompact convexity of (X, II . II) .
C:l(X):= sup{c: E [0,2] : ~x(c:) == a}.
Geometrical property in terms of this modulus and/or this coefficient.
1 Definition 2.17 A space for which cl(X) = a is said to be ~-uniformly convex.
2 A Banach space is (NUC) if dim(X) < 00 or dim(X) = 00 and ~x(c:) > a for every c: E (0,2].
Recall that a Banach space is said to be nearly uniformly convex (NUC) if for any c: > a there exists 8> a such that for any sequence (xn) in Bx with sep((xn)) > c: one has that
dist(O,co({Xn })) < 1- 8.
Here sep((xn )) := inf{llxn - xmll : m =I- n}.
Separation of this property.
Example 2.18 Let f'{ (~) be the space ]Rn endowed with the norm II . 111, (II . 1100)' We consider the Day spaces
Dl := (Ii~ x Iii x ... x f'{ x ... )l2 Doo:= (Ii;' x Ii~ x ... x ~ x ... )l2'
Then one has (see [48]) that for all c: E [0,2]
~D,(c:) == ~Doo(C:) = 8H (c:) = 1- (1- ~)! Thus Dl and Doo are ~-uniformly convex. However, the spaces Dl , Doo are not uniformly convex, nor even superrefiexive. Thus 8D,(c:) = 8Doo(C:) == a even under equivalent renormings.
The spaces Dl and Doo do not have uniform normal structure although they have normal structure.
Facts about this modulus and/or this coefficient.
1 ~x(c:) ~ 8x(c:) for every c: E [0,2].
2 See Kirk, 1988 ([59]) for a proof. For any positive integer k, and a ::; t < c: ::; 2,
3 ([13]) With AxO being the modulus of noncompact convexity,
Px(c:) ~ ~x (K:x )'
4 For 1 < p < 00,
Moduli and constants
One can see the proof in the joint work of Goebel and Sekowski [48J.
5 If 2 ::; p < 00 and c E [0,2J
~ep(c) = ~£p([O,l])(c) = 8£p([0,1])(c).
147
See the paper of BanaS [12J for a proof. However, for 1 < p < 2 and c E (0,2],
~ep(c) > ~e2(c) = 8H (c) > 8ep (c).
6 ~e,(c) == O. Note that limpll ~ep(c) = ~ =F ~e,(c).
7 (See [79]) Whenever X is the Cp direct sum of a sequence (Xn) of finite dimensional Banach spaces
~x = ~ep'
8 The exact values of ~£p([O,l]) are not known if 1 < p < 2, although they are different from those of 6.ep ' For more details see the work of S. Prus (1994) [78J.
9 For x E Cp , (1 ::; p < 00), we denote by x+ and x- the vectors whose i component are respectively
x+(i) := max{x(i),O} = x(i) ~lx(i)l, x-(i) := max{ -x(i),O} = -X(i); Ix(i)l.
For any q E [1,00), and for x E Cp we denote ,
IIXllp,q := (lIx+ll$ + Ilx-II$) q
Ilxllp,oo := max{llx+llp, IIx-llp}'
It is easy to check that all these norms are equivalent to the usual norm in Cp .
The Banach spaces Cp,q = (Cp, II· IIp,q) where introduced by Bynum.
The modulus of noncompact convexity of the space Cp ,l is ,
6.ep " (c) = 1- (1- (~r) p
for any c E [0,2J. (See [83]).
10 ([48]) If c1 (X) < 1 then (X, II . III is reflexive. (Then, (NUC) Banach spaces are refl exi ve ) .
11 ([48]) If c1(X) < 1 then (X, II· III has normal structure.
12 ([83], [46]) Stability of condition c1 (X) < 1.
Let X be a Banach space and let Xl := (X, II . 111) and X 2 := (X, II . 112) where II . III and II . 112 are two equivalent norms on X satisfying for a, (3 > 0,
allxll1 ::; IIxl12 ::; (3llxl11
for all x EX. If k := ~ then
6.X2 (c) 2 1 - k (1 - ~Xl (~)) .
Consequently, if c1 (Xl) < 1 and
148
2.15. Banas modulus of noncompact convexity, [11]
Let us suppose that dim (X) = 00. The modulus of noncompact convexity with respect to the Hausdorff measure of noncompactness of a Banach space (X, 11·11) is the function ~x.x : [0,1] --> [0,1] defined by
~x.x(c) := inf {I - dist(O, A) : A c Bx, A # 0, closed and convex, X(A) ~ c} .
Here X(A) := inf{r > 0 : A C B1 U ... U Bk, B; balls ofradii smaller than r} is the Hausdorff measure of noncompactness of the set A c X.
Related coefficient.
cx(X) := sup{c ~ 0: ~x.x(c) = OJ.
is called Characteristic oEnoncompact convexity of (X, 11·11) associated to the Hausdorff measure of noncompactness.
Geometrical properties in terms of this modulus and/or this coefficient.
1 A space for which cx(X) = 0 is said to be ~x-uniformly convex.
2 ([11]) (X, II ·11) is (NUC) if and only if cx(X) = O.
3 ([11]) If cx(X) < ! then (X, II ·11) (is reflexive and) has normal structure.
4 ([40]) Banach spaces with nonstrict Opial condition and with cx(X) < 1 have weakly normal structure.
Recall that (X, II . II) satisfies the nonstrict Opial condition provided that if a sequence in X is weakly convergent to x E X then
liminf IIxn - xii::; liminf Ilxn - yll n n
for every y EX.
Facts about this modulus and/or this coefficient.
1 ([11]) ~x(c) ::; ~x.x(c) ::; ~x(2c) for every c E [0,1].
2 ([11]) The function ~x.x(-) is continuous on the interval [0,1).
3 ([11]) If X is a reflexive Banach space then
(a) ~x.x is a subhomogeneous function, that is ~x.x(kc) ::; k~x.x(c) for any k,c E [0,1].
(b) For any c E [0, 1], ~x.x(c) ::; c.
(c) The function ~x.x(-) is strictly increasing on the interval [cx(X),l]. The function c t-> ~x.x(c)c is nondecreasing on (0,1] and
~X.X(c1 + c2) ~ ~x.x(cI) + ~X.X(c2) whenever 0::; C1 + C2 ::; l.
4 ([6], p. 89) If cx(X) < 1 then X is reflexive.
5 ([12])
Moduli and constants 149
6 ([11]) Stability of condition EX(X) < ~. Let X be a Banach space with EX(X) < !. Let B > 1 be such that
1 -~ = Ax (2~ ) . (which exists in view ofthe continuity ofthe function Ax). If Y is another Banach space with d(X, Y) < B, then Ax(Y) < ~.
7 ([31]) Let (X, II '11) be a Banach space, C a nonempty weakly compact subset of X and T : C --+ C and asymptotically regular mapping. Let
h:=SUP{t21:~Ax,x(D 21}. If lim infn ITnl < h, then T has a fixed point. (Here ITI denotes the (exact) Lipschitz constant of T on C).
2.16. Dominguez-Lopez modulus of noncom pact convexity, [32]
Let us suppose that dim(X) = 00. The modulus of noncompact convexity with respect to the Istratescu meaS1Lre (or separation measure) of noncompactness of a Banach space (X, 11·11) is the function AX,f3 : [O,,6(Bx)] ---> [0,1] defined by
AX,f3(E) := inf {I - dist(O, A) : A c Bx, A =1= 0, closed and convex, ,6(A) 2 E}.
Here ,6(A) := sup{r > 0 : A has an infinite r separation}, where a r separation of A is a nonempty subset SeA such that Ilx - yll 2 r for all x,y E S, x =1= y.
Related coefficient.
Ef3(X) := SUp{E 2 0 : AX,f3(E) = O}.
is called characteristic of noncompact convexity of (X, 11·11) associated to the separation measure of noncompactness.
Geometrical properties in terms of this modulus and/or this coefficient.
1 A space for which Ef3(X) = 0 is said to be Af3-uniformly convex.
2 ([6]) (X, 11·11) is (NUC) if and only if Ef3(X) = O.
3 ([32]) If Ef3(X) < 1 then (X, II . II) has normal structure.
4 ([40]) If Ax,f3(I) =1= 0 then (X, 11·11) has weak normal structure.
Facts about this modulus and/or this coefficient.
1 ([6], p.86) .sX(E) ::; AX(E) ::; AX,f3(E) ::; AX,X(E) and consequently
EO(X) 2 q(X) 2 Ef3(X) 2 EX(X).
2 ([6], p. 90. ) If Ef3(X) < 1 then X is reflexive.
150
3 ([32]) For 1 < p < 00,
( c:P) ~ ~ip,.B(C:) = 1 - 1 - 2" .
4 ([6], p. 96)
~Dl'.B(C:) = ~D~,.B(C:) = 1- (1- c:;)! So the spaces Dl and Doc are (NUC) but fail to be k-UC for any k.
5 ([6], Remark 1.12, p. 93) For reflexive spaces, Partington's modulus is identical to the modulus of noncompact convexity associated to {3.
6 ([31]) Let (X, II . II) be a Banach space, C a nonempty weakly compact subset of X and T : C ~ C and asymptotically regular mapping. If
1 limninf ITnl < 1 _ ~x,.B(1
then T has a fixed point. (Here ITI denotes the (exact) Lipschitz constant of T on C).
2.17. Opial's modulus
It was defined by S. Prus in [77]. See also (Lin-Tan-Xu, [66]).
It is the function rx : [0,00) ----> lR given by
rx(c) := inf{liminf Ilx + xnll - 1 : Ilxll ~ c, Xn ~O, liminf Ilxnll ~ I}. n
Geometrical properties in terms of this modulus and/or this coefficient.
Definition 2.19 A Banach space (X, 11·11) is said to satisfy the uniform Opial property if for any c > 0 there exists an r > 0 such that
1 + r:S liminf IIx + xnll n-oc
for each x E X with Ilxll ~ c and each sequence (xn) in X such that Xn ~ 0 and liminf Ilxnll ~ 1.
1 The space (X, 11·11) satisfies the uniform Opial property if and only if rx(c) > 0 for all c > O.
2 ([89] and [20]) rx(c) ~ 0 for all c ~ 0 if and only if (X, II . II) has the nonstrict Opia! property.
3 ([90]) If rx(c) > 0 for some c E (0,1), then (X, 11·11) has weakly normal structure.
Main features of this modulus.
1 For all c 2 0, c - 1 ::; rx{c) ::; c.
Moduli and constants
In particular, rx(c) > 0 for all c> 1. (See [66]).
2 The function rx is continuous on [0,(0).
3 The function c f-+ 1 + rx(c) is nondecreasing on (0, (0). In fact we have c
whenever 0 < ct :S C2. (See [66])
4ForI<p<00 1
rep (c) = (1 + c)p - 1
(c 2': 0). (See [66]).
5 If 1 < p < 00 and 1 :S q < 00 then for all c 2': 0,
rep.q(c) = min {(I + d')~ - 1, (1 + cq)t - I}. (See [89]).
151
6 ([61] See also [58] for a generalization) Let X be a Banach space with rx(I) > 0 and with the nonstrict Opia! property, e a nonempty weakly compact subset of X and T = {Tt : t E G} an asymptotically regular semigroup with
O'(T) = k < 1 + rx(I).
Then there exists z E e such that Tt (z) = zfor all t E G. Here O'( T) : = lim inf ITt I and k := SUPt ITtl, where ITt I denotes the exact Lipschitz constant of Tt in e.
7 ([66]) Suppose e is a weakly compact convex subset of X. If (X, II· II) satisfies the uniform Opial property and T : e -t e is a mapping of asymptotically nonexpansive type ; that is, for every x E e,
lim sup [sup{IITn(x) - Tn(y)II-lix - yll : y E e}] :S O. n
If TN is continuous for some integer N 2': 1, then T has a fixed point.
8 ([31]) Let (X, 11·11) be a Banach space, e a nonempty weakly compact subset of X and T : e -t e and asymptotically regular mapping. If liminfn ITnl < 1 + rx(I) then T has a fixed point.
2.18. Six moduli for the property (/3) of Rolewicz
In a Banach space (X, 11·11) the drop D(x,Bx) defined by an element x E X\Bx is the set co({x} UBx), and we write Rx := D(x, Bx)\Bx.
Recall that (X, II '11) is said to have the property ((3) iffor each c > 0 there exists {j > 0 such that
1 < Ilxll < 1 + {j => JL(Rx) < c
where JL is any of the three measures of noncompactness a, (3, S. (S(A) stands for the separation measure of the set A, that is SeA) := sup{c: > 0 : 3(xn ) E A with sep(xn) > c:}.)
152
In [4] the authors defined three moduli as follows
Rx : [0,2]-. [0,1]
Rx(c) := 1- sup {inf {~llx + xnll : n EN} : {xn} C Bx,x E Bx,a({xn}) 2: c},
R'x : [0,2]-. [0,1]
R'x(c) := 1- sup {inf {~lIx + Xnll : n E N} : {xn} C Bx,x E Bx ,/3({xn}) 2: c},
R'Jc : [0, a] -. [0,1]
R'Jc(c) := 1 - sup {inf {~IIX + xnll : n E N} : {xn} C Bx,x E Bx,sep({xn}) 2: c}
where a is a real number in the interval [1,2) depending on (X, II . II) .
Moreover, the same authors in [5] defined
PX,I': [O,J1(Bx» --> [0,(0) by
PX'I'(c) := inf{lIxlI-1 : x E X, Ilxll > 1, J1(Rx ) 2: c}.
Related coefficients.
Ro(X) := sup{c 2: 0: Rx(c) = O}.
RfJ(X) := sup{c 2: 0 : R'x(c) = O}.
Rf;(X) := sup{c 2: 0 : R'Jc(c) = O}. PO,I'(X) := sup{c 2: 0 : PX,p.(c) = O}.
Geometric properties in terms of these moduli or constants.
1 (X, II . II) has the property (/3) of Rolewicz if and only if either Ro(X) 0, RfJ(X) = 0, or Rg(X) = O.
2 PO,p.(X) = 0 if and only if (X, II . II) has property (/3) of Rolewicz.
It is easy to prove that (UC) implies property (/3) and property (/3) implies NUC. Thus, this property lies between uniform and near uniform convexity.
3 If any of the coefficients Ro(X), Rti(X), Rg(X) is less than one, then both X and X· are reflexive and have normal structure.
4 If PO,p.(X) < !, then the spaces X and X· are reflexive and have normal structure.
Main features of these moduli.
1 8x G) ::; R'x(c) ::; Rx(c) ::; R'Jc(c) ::; ~x,s(c).
2 cs(X) ::; Rg(X) ::; Ro(X) ::; RfJ(X) ::; 2co(X).
Moduli and constants
3 If 1 < p < 00, then
4 If 1 < p < 00, then
5 Rt(E) = R~=(E) = RC=(E) = 0 for all E E [0,2].
6 R~, (E) = R~, (E) = Re, (E) = 0 for all E E [0,2].
E 7 o:s: PX,,,(E):S: (B) . It follows that Px,,, is continuous at O.
f1 x - E
I
8 If 1 < p < 00 and 0 :s: E < 2 P, then
( 2 )1- 1 PCp S(E) = -- P-1.
, 2 - EP
9 If 1 < p < 00 and 0 :s: E < 2, then
10 Pc,,,(E) = Pco,,,(E) = PC",,(E) = Pe=,,,(E) = O. for all E E [0,2].
2.19. Dominguez modulus of (NUS), [27]
It is the function given by
153
rx(t) '.-_ sup {l'nf { IIx1 + tXnl1 +2 IIx1 - tXnl1 } ( ) b . . B } "'-"--_-"-_.::....."--_..:..:..c. : n > 1 : Xn aSlC sequence m x .
Geometrical properties in terms of this modulus and/or this coefficient.
1 If X is (US) then limt ...... orx(t)/t = O.
2 (X, 11·11) is (NUS) if and only if X is reflexive and limt ...... o rx(t)/t = o.
Main features of these modulus.
1 px(t) 2: rx(t) for every t E [0,2].
I
2 Forl<p<oo,rcp (t)=(l+tP)p.
Other properties of this modulus are listed below, among the ones of WCS(X).
154
2.20. The Modulus of squareness, a universal one
It was defined in the joint paper by Przeslawski and Yost [80]. For more information about it see the joint paper by Benitez, Przeslawski, Yost [14] which is the source of all this information. Given a normed space (X, 1/.1/) , one observes that for any x,y E X with IIYII < 1 < I/xl/, there is a unique z = z(x, y) E Sx n [x, y], (where [x, y] is the line segment joining x and y). We put
( ) ._l/x-z(x,y)1/ w x, y.- I/xl/ _ 1
and define Xx : [0,1) --+ [1,00) by
Xx(3):= sup{w(x,y) : Ilyi/ ::; f3 < 1 < I/xll}·
Geometrical properties in terms of this modulus and/or this coefficient.
1 (X,I/· II) is uniformly convex if and only if
lim (1 - (3)xx(f3) = O. /3--+1-
2 (X, 1/ . II) is uniformly smooth if and only if X~(O) = O.
3
Xx' (f3) = () . -1 1 Xx 73
1
4 If Xx(f3) < 1/1 - f3 for some f3, then (X, 1/ . 1/) has uniform normal structure.
5 If a normed space (X, II . II) is uniformly nonsquare then for each f3 E (0,1),
1+f3 Xx(f3) < X1(f3) := 1- f3'
Here XP is Xx where X is lp.
Main features of this modulus.
1 Xx(f3) = SUp{XM(f3) : M C X, dim(M) = 2}.
2 Xx ( .) is a strictly increasing and convex function.
3 xx(f3) = X2(f3):= ~ whenever (X, 1/.1/) is an inner product space. y1-/32
4 For any normed space (X, 1/.1/) containing £1(2), Xx(f3) = X1(f3).
5 For all normed space (X, 1/ .1/) , Xx(f3)::; X1(f3).
6 Unless (X, 1/.1/) contains arbitrarily close copies of £1(2), xx(f3) < n(f3) every-where on (0,1).
7 Almost everywhere on (0,1) we have that X~(f3) < X~ (f3).
8 Unless (X, II .1/) is an inner product space, Xx(f3) > X2(f3) everywhere on (0,1).
9 Whenever 0 ::; f3 <, < 1, Xx(r) - xx(f3) ::; X1(r) - X1(f3).
Moduli and constants 155
10 Fix 8,,6 E (0,1). If Xx (,6) > (1-8,6)X1(,6), then(X, 1I'II)contains atwo-dimensional subspace whose Banach-Mazur distance from £1 (2) is less than 1_(11+/3)6'
11 If X and Yare two isomorphic Banach spaces whose Banach-Mazur distance is less than 1 + 282 for some 8 E [0,1], then for all ,6 E (0,1)
2(8 + 82) Ixx(,6) - Xy(,6)I::; (1 _ ,6)2 .
12 Let px the modulus of smoothness of a Banach space (X, II . II) . Then, for all ,6 E (0, 1)
1 < Xx(,6) -1 < _2_. - px(,6) - 1 - ,6
and X'x(O) = p'x(O).
2.21. Three further moduli for the Nearly Uniform Convexity
They were defined by J.M. Ayerbe and S. Francisco ([7]) as follows.
DX,JL: [O,p,(Bx)) --; [0,00) is the function given by
DX,JL(C:):= inf{llxll-1 : x E X\Bx,ii(Rx ) 2:: E}
where, for a bounded subset A of X, j:i(A) := sup{p,(C) : C = co(C),C c A} and p, E {a,x,,6}.
Related coefficient. The coefficients of noncompact convexity of (X, II . II) corre-sponding to this modulus are the numbers
DO,JL(X) := SUp{E 2:: 0: DX,JL(E) = O}.
Geometrical properties in terms of this modulus and/or this coefficient.
1 The Banach space (X, II· II) is (NUC) if and only if DO,JL(X) = O.
2 If DO,/3(X) < 1/2, then the space (X, II . II) is reflexive and has normal structure.
Main features of this modulus (See [7]).
1 DX,JL is nondecreasing in [O,p,(Bx)), and DX,JL(O) = o.
2 For every .0 E [0, ~/L(Bx)) we have 0::; Dx,JL(C:)::; (B c:) . /L x - .0
3 DX,JL is continuous at c: = O.
2.0 4 For every .0 E [O,/L(Bx)) we have DX,JL(E) < /L(Bx)'
5 C:/3(X) ::; 2Do,/3(X) ::; 4E/3(X),
6 Let H be a separable and infinite dimensional Hilbert space. Then
2E2 DII,a(E) = 4 _ c:2 .0 E [0,2),
156
2.22. A further modulus for Uniform Convexity
It was defined by S. Francisco ([36]) as follows.
Dx : [O,oe) --+ [O,oe) is the function given by
Dx(e) := inf{lIxlI-l : x E X\Bx, diam(Rx ) ::::: e}.
Related coefficient. The coefficient of convexity of (X, II . II) corresponding to this modulus is the number
Do(X) := SUp{e ::::: ° : Dx(e) = OJ.
Geometrical properties in terms of this modulus and/or this coefficient.
1 The Banach space (X, II . II) is (UC) if and only if Do(X) = 0.
2 If Do(X) < 1/2, then the space (X, II . II) is reflexive and has normal structure.
Main features of this modulus (See [36] and [8]) ..
1 Dx is nondecreasing on [O,oe), and Dx(O) = 0.
2 For every e E [0,1] we have that ° ::::: Dx(e) ::::: 2=-". Hence, if e E (0,1), then Dx{e) < e.
3 Dx is continuous at e = 0.
4 eo{X) ::::: 2Do{X) ::::: 8eo{X).
5 Let H be a separable Hilbert space. Then
{ _2 __ 1 e E [0, v'3]
DH{e)= r 1 +e2 -1 e E [v'3,oe).
6 If (X, 11·11) is a uniformly convex Banach space, then Dx is strictly increasing on [O,oe).
2.23. Modulus of uniform "nonoctahedralness"
It was defined by A. Jimenez-Melado in [52]. Denote by 8{X) the supremum of the set of numbers e E [0,2] for which there exist points xl, X2, X3 in Bx with
min{lIxi - xjll: i # j} ::::: e.
Define the function 8x : [0, 8{X)) --> [0,1] by
8x {e) = inf {1- ~lIxl +X2 +X311: Xi E Bx, and min{lIxi -xjll: i # j}::::: e} .
Moduli and constants 157
Related coefficient with this modulus.
EO(X) := SUp{ c E [0, 8(X)): 8x (c) = O}.
Geometrical properties in terms of this modulus and/or this coefficient.
1 ([41]) EO(X) < 1 =? X has (UNS).
Main features of this modulus (See [52]).
1 For any c E [O,8(X)).
2 EO(X) 0/ co(X)
In some cases, the inequality is strict.
3 Consider the classical real sequence space £2 endowed with its usual euclidean norm II . II· Let I . I be a norm on £2 such that
for some b ;::: 1 and let X = (h 1·1). Then EO(X) < 2 for b < If· 4 Now consider the space E(3 := (£2,1,1(3), where Ixl,a = max{llxll,,6llxll oo }. Since
Ilxll ::; Ixl,a ::; ,6l1xll2 for all x E £2 then Eo(E,a) < 2 for ,6 < If. On the other
hand, for ,6 ;::: v'2 we have co(E(3) = 2.
5 If (X, 11·11) is a Banach space with the (WORTH) property such that EO(X) < 2 then (X, 11·11) has the (WFPP).
Recall that (X, II '11) has the (WORTH) property if
lim IlIxn - xll-Ilxn + xIII = 0 n
for all x E X and for all weakly null sequences (xn) in X.
6 Suppose that there exist {j > 0 and c E (0,2) such that for all points x, y, z E Sx with II(x + y + z)/311 > 1 - {j we have
Ily - zll dist(x, [y, z]) < c.
Then EO(X) < 2. (Here [y, z] is the affine span of {y, z}.)
3. List of coefficients
3.1. Jung constant
It is the oldest of this list. It was defined by Jung in a work published in 1901 [55]. We will follow the Amir's paper [2] to list some important facts about it.
For a bounded subset A of X, and a subset Y of X, we denote by ry(A) the relative Chebyshev radius of A with respect to Y, that is,
ry(A) := inf {sup{llx - yll : x E A} : y E Y}.
158
The Jung constant of (X, II ·11) is
J(X) := sup{2rx(A) : A c (X), diam(A) = I}.
Besides the Jung constant J one may study also the "self-Jung" constant
Js(X) := sup{2rA(A) : A C X, diam(A) = I}.
Main features of these coefficients.
1 1:S J(X) :S 2.
2 J(~) = flfr. 3 If dim(X) = n, J(X) = 1 if and only if X = ~.
4 J(1!2) = V2. 5 ([9])
J(E ) = {f3V2 1:S f3 :S V2 (3 2 V2<f3<=
6 J(X) = 1 if and only if X = C(T) for a stonian T.
7 If the compact Hausdorff space T is not extremely disconnected, then for every finite-codimensional subspace E of C(T) we have J(E) = 2.
8 J(X) :S Js(X).
9 If (X, II . II) is a dual space,
J(X) = sup{2rx(K) : K finite, K C X,diam(K) = I}.
10 If (X, II . II) is a reflexive Banach space, then
Js(X) = sup{2rco(K)(K) : K finite, K C X,diam(K) = I}.
11 If (X, II . II) is a nonreflexive Banach space, then Js(X) = 2.
12 For every infinite dimensional space (X, II . II) , Js(X) 2: V2. 13 For every n-dimensional space (X, II . II) , Js(X) :S n2.;\.
14 For every n and every c > 0 we have
Js(X):S 2 max {1- 1 ~ c ,1- or)(c)}. n.c
3.2. Bynum's coefficient of (uniformly) normal structure
Notation: Let A be a nonempty bounded subset of X.
r(A) := inf {sup{lla - yll : yEA}} aEA
is called the Chebyshev self-radius of the set A.
Bynum in 1980 [17] defined the following coefficient of (uniform) normal structure:
N(X) := inf {di:0~) : C bounded, convex, C C X,diam(C) > o}.
Moduli and constants 159
Geometric properties in terms of this constant.
Definition 3.1 A Banach space (X, II . II) with N(X) > 1 is said to have uniform normal structure.
Separation of this property. The Banach space (X, II . II) obtained by the R2 direct sum of the sequence of Banach spaces (Rn) is (UCED), and hence it has normal structure. Nevertheless N(X) = 1.
Main features of this coefficient.
1 ([10], [69]) If N(X) > 1 then X is reflexive.
2 ([18]) If C is a nonempty weakly compact convex subset of X and T : C ---> C is uniformly k-lipschitzian with k < VN(X) then T has a fixed point in C.
3 Bynum's lower bound of N(X) ([17]). If X is a reflexive Banach space,
1 N(X) ;::: 1 - 8x (I)'
4 ([2])Amir's lower bound of N(X)
N(X) ;::: sup { (max { 1 - n~n (1- c), 1- 8(k) (c) }) -1 : 0 ::; c ::; I} . 5 ([56]) If liminft ..... o px(t)/t < 1/2 then N(X) > 1.
6 Prus' lower bound for N(X) ([79]).
N(X);::: (inf{l+px(t)-i:tE [o,~]}rl
7 ([17]) For a Hilbert space H, N(H) = v'2. 8 ([18]) For 1::; j3 < v'2, N(E{3) = v'2/j3. (For j3 ;::: v'2, N(E{3) = 1).
9 ([17]) For X, Y isomorphic Banach spaces, N(X) :S d(X, Y)N(Y), where d(X, Y) is the Banach-Mazur distance between the Banach spaces X, Y.
10 ([75], [25]) For 1 < p < 00,
1 1_1 N(Rp) = N(D'([O, 1]) = min{2p,2 p}.
11 ([69]) If dim(X) = 00 then N(X) :S v'2. 12 W.L. Bynum defined in [17] the coefficient BS(X) as the supremum of the set
of all numbers M with the property that for each bounded sequence (xn) with asymptotic diameter A, there is some y in the closed convex hull of the (range of the) sequence such that MlimsuPn IIxn - yll ::; A.
Here diama ((xn)):= lim (sup IIxi - Xjll : i,j ;::: n)
n ..... oo
160
is called the asymptotic diameter of the sequence (xn), and
ra ((xn)) := inf {lim sup Ilxn - xII: x E co{xn: n = I, ... }}
is called the asymptotic radius of the sequence (xn ).
An equivalent definition is
BS(X) := inf {diama((xn)) : (xn) is a nonconvergent bounded sequence in x} . ra((xn))
It was shown in [63] that for all Banach space X, N(X) = BS(X).
13 ([18] Suppose that, C is a weakly compact convex subset of X and T : C -+ C is k-uniformly Lipschitzian on C with k < )N(X), then T has a fixed point.
3.3. Bynum's weakly convergent sequence coefficient
It was defined by L.B. Bynum in [17] as follows
WCS(X) is the supremum of the set of all numbers M with the property that for each weakly convergent sequence (xn) with asymptotic diameter A, there is some y in the closed convex hull of the (range of the) sequence such that M lim supn Ilxn - yll ::; A.
This is probably one of the Banach space constants most widely studied, although with considerable confusion because there are many equivalent definitions. We will follow in this summary the illuminating work by Sims and Smith [85]. See also the Ph.D. dissertation of M.A. Smyth [87].
An equivalent definition is (see [63])
WCS(X) := inf { di:~(;:~))) : (xn) is a weakly (not strongly) convergent seq in X} .
Geometrical properties in terms of this constant.
1 Definition 3.2 We quote [85], p. 500.
Some authors have said that a space X has weak uniform normal structure if WCS(X) > l. We shall say that X satisfies Bynum's condition if this inequality holds.
2 Recall that a Banach space has the generalized Gossez-Lami Dow property (GGLD) if for every weakly null sequence (xn) such that lim Ilxnll = 1 we have that D[(xn)] > 1, where
D[(xn)] := lim;up (lim,;;up Ilxm - xnll) .
This property was defined by A. Jimenez-Melado in [51]. In the same work there was defined the following coefficient in order to obtain stability results for (GGLD) in terms of the Banach-Mazur distance:
f3(X) := inf{D[(xn)] : Xn ~O, Ilxnll -+ I}.
Obviously (X, 11·11) has property (GGLD) if f3(X) > 1. Moreover (GGLD) implies (WNS).
Moduli and constants 161
Separation of this property.
Example 3.3 The space Co equivalently renormed by
II (xn) II := II (xn) 1100 + L I~:I, n
was considered by A. Jimenez-Melado [51J. It enjoys Opial condition, and hence (WNS), but lacks (GGLD).
Main features of this coefficient.
1 ([85])
wes(x) =· f{diama«xn)). ~o -'.,.o} III Ta«Xn)). Xn , Xn T' .
'diama ' can be replaced with 'diam' in the above equality.
2 ([85J and the references therein) The following constant are equal.
(1) WeS(X).
(2) inf{diama«xn)): xn~O, IIxnll--> I}.
(3) (1(X).
(4) infb«xn)) : Xn ~O, Ilxnll --> I}.
(5) inf{o:«xn)): xn~O, IIxnll--> I}.
3 ([17]) For a reflexive Banach space (X, II ·11) ,
1 :S N(X) :S BS(X) :S WCS(X) :S 2.
4 ([3]) If X is a reflexive infinite dimensional space such that the duality mapping is continuous, then
1 :S N(X) :S J(~) :S WeS(X) :S 2.
5 ([32])
6 ([17]) Let X, Y be isomorphic Banach spaces. Then,
WCS(X) :S d(X, Y)WeS(Y).
(See also [6J, p. 119)
7 ([51]) If WCS(X) = (1(X) > 1 and d(X, Y) < (1(X) then Y has property GGLD. 1
8 ([17]) For p ~ 1, WeS(flp) = 2p.
9 For a Hilbert space H, WeSCH) = ..;2.
10 ([9])
{fl
WCS(E{3) = t 1 :S {1..;2 ..;2 < {1 < 00.
162
11 ([6]) WCS(co) = 1.
12 For any Banach space we have
WCS(X) :::: 1 + rx(l).
(See [66]). This inequality may be strict, as in the following example.
Example 3.4 ([61]) Let X = f!2 EEl ~ equipped with the norm
II(x, r)1I := IIxll2 + max {Irl - ~lIxIl2' ° } where II . 112 denotes the euclidean f!2 norm. The space X has the nonstrict Opial property, rx(l) = ~ and WCS(X) = y2.
1 1
13 ([34]) WCS(f!p,q) = min{2', 2q}.
14 ([32]) If (X, II· II) is a non-Schur Banach space
. 1 WCS(X):::: hm A ()
e~l- 1 - uX,{3 c:
15 ([27]) Let X be a reflexive Banach space. Then
tc:WCS(X*) tc: sup - L"lx,O'(c:) :::; rx·(t):::; sup WCS(X) - L"lx,O'(c:)
O::;e::;a 4 O::;e::;a
for every t > 0, where a = <7(Bx). If, in addition, X* satisfies the Opial condition then
tc: sup - - L"lx,a(C:) :::; rx·(t).
O::;e::;a 2
16 ([27]) Let X be a reflexive Banach space. Then
. rx.(t) 4. rx.(t) WCS(X)hm-- <c:oa(X):::; WCS(X) hm--.
t~O t - , * t~O t
17 ([27]) Let X be a reflexive Banach space satisfying the Opial condition. Then
tc: WCS(X)WCS(X*) sup - L"lx,{3(C:) :::; rx.(t):::; sup tc: - L"lx,{3(C:)
O::;e::;a 4 O::;e::;a
for every t > 0, where a = <7(B x). If, in addition, X* satisfies the Opial condition then
tc:WCS(X) sup - L"lx,{3(C:):::; rx·(t).
O::;E::;a 2
18 ([26]) Let (Xn, I· In) be a sequence of reflexive Banach spaces. Let 1 < p < 00.
Let X be the f!p-direct sum of (Xn, I· In). Then,
1
WeS(X) = inf{WCS(Xi), 2. : i EN}.
19 ([26]) Let (Xl, I. 11), .. · ,(Xn , I· In) be a finite sequence of reflexive Banach spaces. Let Z be ~n with a monotone norm. Then,
wes ((Xl EB ... EB Xn)Z) = min{WeS(X;) : i = 1, ... ,n}.
Moduli and constants 163
20 ([75], [25]) Let (O,~, fL) be a a-finite measure space. Let 1 :S p < +00 and assume that LP(O) is infinite dimensional. Then
WCS(.D'(O)) = N(.D'(O)) = min{2~, 21-~} either p :::: 2 or J.L is not purely atomic.
21 ([33], [6]) Suppose that WCS(X) > 1, C is a weakly compact convex subset of X and T : C ...... C is asymptotically regular on C. If
liminflTnl < JWCS(X),
then T has a fixed point.
3.4. Gao-Lau coefficient
This was defined in an attempt to simplify Shafer's notion of girth and perimeter of the unit ball of a normed space [82].
G(X):= sup{IIx+yll 1\ IIx -YII: x,y E Sx}·
Geometrical properties in terms of this constant.
1 ([42]) G(X) < 2 if and only if X is uniformly nonsquare.
2 ([43]) If G(X) < ~ then (X, II . II) has uniform normal structure.
Main features of this coefficient.
1 ([42]) For any Banach space (X, II . II) , y'2 :S G(X) :S 2.
2 ([42]) G(fip) = G(LP) = max{2~, 21-~}. 3 ([43]) G(X) < c if and only if dx(c) > 1 - ~.
4 ([76]) N(X) :S G(X) + 1 - ((G(X) + 1)2 - 4)~.
5 ([43]) For any isomorphism T from X to fip or LP, 1 < p < 00,
G(X) :S IITilIIT-1II max {2~, 21-~}.
G ([43]) For any Banach space (X, II . II) and for any nontrivial ultrafilter U on N,
G(Xu) = G(X).
7 ([43]) G(X) = sup {c :::: 0: ox(c) :S 1 - c/2}.
8 ([42]) If X and Yare Banach spaces and T : X ...... Y is a (bicontinuous) isomorphism, then
(IITilIIT-1II)-1:s ~i~~ :~ :S II TIl IIT-l II·
9 ([43]) If G(Y) < 3/2 and
~(X, Y) < In (2G(:) + 2) , then X has uniform normal structure.
Here ~(X,Y):= inf {In(IITIlIIT-111): T E I(X,y)}.
164
3.5. A measure of the degree of WORTHwileness, [84]
w(X) := supp: >.liminf IIxn + xII S liminf IIxn - xII, Xn ~o x EX}.
Geometrical properties in terms of this coefficient.
1 The Banach space (X, II . II) has property (k) if
w(X) > max {~c, 1 - 8x(c)} for some positive E.
Recall that a Banach space (X, II . II) has property (k) if there exists k E [0,1) such that whenever Xn ~ 0, Ilxnll -> 1 and lim infn Ilxn - xii S 1 we have Ilxll s k. One has that property (k)=>(WNS). Property (k) is equivalent to r(l) > 0. (See [20])
Main features of this coefficient.
1 1 k(Lp([O, 1])) = (1- 2-P)p.
2 (See [53]) A Banach space (X, II . II) has the (FPP) whenever
co(X) _1_ 1 4 + 2w(X) < .
3.6. A coefficient related with (NUS) property
It was defined by J. Garcia-Falset in [38]
R(X) := sup {liminfllxn +xlI : X,Xn E Ex, n = 1, ... , Xn ~O}. n~oo
Geometrical properties in terms of this coefficient.
Let (X, II ·11) a Banach space. The following conditions are equivalent:
(a) X is (WNUS).
(b) X is reflexive and R(X) < 2.
Recall that (X, II . II) is (WNUS) if for some c > 0 there exists /-L > 0 such that if 0 < t < /-L and (xn) is a basic sequence in Bx then there exists k > 1 so that II Xl + xnll s 1 + t:t.
Main features of this coefficient. (See [38] and [39]).
1 1:S; R(X) S 2.
2 If X is finite dimensional, R(X) = l.
3 Let X be a weakly orthogonal Banach Lattice. Then R(X) S a(X) where a(X) is the Riesz angle of X.
Moduli and constants 165
1
5 For 1 <p<oo,R(l:'p))=2v.
6 R(c) = 2.
7 Let (X, II . II) be a Banach space with the property (WORTH). Then R(X) ::; G(X).
1 8 R(l:'p,oo) ::; 2v.
9 ([20]) If R(X) = 1 then the following are equivalent. (X, II . II) is a Schur space; (X, II ·11) has (WNS); (X, II ·11) has the (KK) property; Co f+ X.
Recall that a Banach space is said to have the (KK) property if
IIxn - xII --> 0 whenever xn:!!'...x and Ilxnll -+ IIxlI·
10 If R(X) < 2 then X has the weak Banach-Saks property.
11 eo(X) < 1 =? R(X) < 2.
12 R(X) < 2 =? (X, II ·11) has (WFPP).
13 A Banach space (X, II . II) has the (WFPP) if there exists a isomorphic Banach space Y such that d(X, Y)R(Y) < 2.
14 If :[;? < 2 then R(X) < 2.
3.7. Dominguez generalization, [28]
Given a nonnegative real number a, one defines
R( a, X) := sup{lim inf II x + Xn II },
where the supremum is taken over all x E X with IIxll ::; a and all weakly null sequences (xn) in the unit ball of X such that
D(xn) = lim:up (lim~up IIxn - xmll) ::; 1.
Moreover we put
{ 1+a } M(X) := sup R(a,X): a::::: 0 .
Main features of this coefficient. (See [28])
1 If R(a, X) < 1 + a then (X, II . II) has the (WFPP).
2 Let X, Y be isomorphic Banach spaces. Then
R(a, Y) ::; d(X, Y)R(a,X)
for every nonnegative number a.
3 If M(X) > 1, then (X, II . II) has the (WFPP).
166
4 If M(X) > 1 and Y is a Banach space which is isomorphic to X and d(X, Y) < M(X), then Y has the (WFPP).
5 R(O, X) = WC1(Xl'
6 M(X) ~ WCS(X).
7 M(f2,oo) = v'2 > WCS(f2,oo) = 1.
8 Let X be a reflexive Banach space and denote
rex) := inf { 1 + res) - ~ : s E [0,1]}.
Then
(a) For every a ~ 0, R(a, X) ::; 1 + ar(X).
(b) M(X) ~ q\-l'
9 Let X be a reflexive Banach space and denote
{ sWCS(X) } r/(X) := inf 1 + f(s) - 2 : s E [0,1] .
Then
(a) M(X) ~ r'(x),
(b) M(X) > 1 iff/CO) < WC~(X).
(c) M(X) > 1 if f/(O) < ~.
10 ([6]) Let X be a Banach space and denote
Then
!3x := inf { 1 + i3}(s) - 2sk : s E [0,1]} .
1 +2k M(X) ~ 1 + 2ki3x .
11 Let X be a reflexive Banach space. If c E (0,1) satisfies I'X' (c) > 0 then
R(a,X)::; max { 1 +ac,a+ 1 +1'~.(c)}· In particular, if I'x.(l) > 0 then R(a, X) < 1 + a and M(X) > 1.
12 For 1 < p < 00,
( 1); 1 ~ R(a,fp) = aP + 2 =}- M(fp) = (1 + 2P - 1 ) P
1 E.::! 13 R(a,fp,oo) = (1 +aP)" =}- M(fp,oo) = 2 P •
14 For a Hilbert space X, M(X) = v'3.
Moduli and constants
15 ([30])
M(E(3) = { ~ (: + V",-') 1 + v'2
16 ([30]) For 1 ::; p < 00 and 1 ::; q < 00,
17 R(a,co) = a ~ M(co) = 2.
(3::; ft
ft<(3<V2 V2::; (3.
p"5.q
q <po
167
18 (See [54]) If we define, for B ~ 1 the number Gx(B) := {c ~ 0 : rx(c) ::; B-1} and 1·1 is an equivalent renorming of the Banach space (X, 11·11) such that for all XEX
and
{ l+a } B < sup ("- () ): a ~ 0 , R BGX B ,X
then (X, I ·1) has the (WFPP).
3.8. Lifschitz and Dominguez-Xu coefficients for uniformly lipschitzian mappings
In 1975, Lifschitz ([64]) introduced the coefficient K(M) for a metric space (M, d) as follows
K(M) := sup{(3 > 0: :la > 1 s.t. 'ix, y E M and 'ir> 0, d(x, y) > r
and B[x,(3r] nB[y,ar] C B[z,r] for some z EM}.
He proved that if (M, d) is a complete bounded metric space and T is a k-uniformly Lipschitzian selfmapping of M with k < K(M), then T has a fixed point. For a Banach space (X, 11·11) ,it is often denoted by KO(X) := inf{K(M)} where the infimum is taken over all the closed convex and bounded subsets of X.
The following definitions are inspired on this Lifschib:'s coefficient, and were given in [33] by T. Dominguez-Benavides and H.K. Xu.
Let M be a bounded convex subset of X. A number b 2': 0 is said to have property (P) with respect to M if there exists some a > 1 such that for all x, y E M and r > 0 with Ilx - yll 2': r and each weakly convergent sequence (xn) in M for which
lim sup Ilxn - xii::; ar, and
lim sup Ilxn - yll ::; br
there exists some z E M such that lim inf Ilxn - zll ::; r. Then one defines
K(M) := sup{b > 0 : b has property (P) w.r.t. M}.
K(X) := inf{K(M) : M as above}.
168
Main features of these coefficients.
1 ([6], p. 145) Let (X, II . II) a Banach space and h a solution of the equation
Then h <::: KO(X). Furthermore c:o(X) < 1 if and only if KO(X) > 1.
2 ([9])
{ [
l+~p-l ] ~ KO(IJ') :::: 1 + (1+Tp
P)p-i p> 2
/P P<::: 2
where Tp E (0,1) is the unique solution of the equation
3 ([6], p. 145, [9])
(p - 2)TP- 1 + (p - 1)TP- 2 = 1.
1 <::: KO(X) <::: N(X) <::: WCS(X) <::: 2 1 <::: KO(X) <::: N(X) <::: V2
1 <::: KO(X) <::: N(X) <::: J(:x.) <::: 2
K(X) <:::~.
4 ([64]) KO(X) :::: 1-6~(1)'
5 ([64]) Ko(H) = V2 whenever H is an infinite dimensional Hilbert space.
6 In [33] one can see easier equivalent definitions of K(X) when (X, 11·11) is a Banach space satisfying the uniform Opial condition.
7 ([33]) If (X, II . II) is a Banach space satisfying the uniform Opial condition then K(X) = 1 + rx(l).
1
8 ([33]) For 1 < p < 00, K(£p) = 2-.
9 ([33]) Let (X, 11·11) a Banach space with the uniform Opial condition and M be a bounded convex subset of X. Then K(M) :::: h where h is the unique solution of the equation
10 ([33]) Let (X, II . II) a Banach space with the uniform Opial condition and M be a bounded convex subset of X. Then K(M) :::: ho where
{ ( WCS(X)) 1 } ho := sup t > 1 : Ll'x t + t > 1 .
11 ([33], [6]) If (X, II . II) is a Banach space satisfying the Opial condition, or X is reflexive then
K(X) <::: WCS(X).
Example 3.5 Let X be £2 renormed by
Moduli and constants 169
where 2 < p < 00. Then QX)::; 2~ < WCS(X) = v'2.
12 ([6], Theorem IX.2.3) If (X, II . II) is a Banach space satisfying the uniform Opial condition, then
13 ([33]) Suppose that C is a weakly compact convex subset of the Banach space (X, II . II) and T : C --t C is a uniformly Lipschitzian mapping. If
liminf ITnl < ii,(C), n
then T has a fixed point in C. (Here ITnl denotes the exact Lipschitz constant of Tn).
14 ([29]) Let (X, II . II) a Banach space, C a closed convex bounded subset of X and T : C --t C a k-uniformly lipschitzian mapping. If
k 1 + VI + 4N(X)("o(X) - 1) < 2
then T has a fixed point.
15 ([29])
"o(X) ::; 1 + VI + 4N(;)("o(X) - 1) ::; N(X).
16 ([92]) If f3 2: ~v'5, then "o(EfJ ) = 1.
17 ([29]) If 1 ::; f3::; ~v'5, then
18 ([29]) Let (X, 11·11) be a reflexive Banach space, C a bounded closed convex subset of X and T : C --t C an asymptotically regular mapping.If
I· . f ITnl 1 + VI + 4WCS(X) (ii,(X) - 1) ImlIl < 2
then T has a fixed point.
3.9. Coefficients for asymptotic normal structure, [16)
They were defined in [16] in order to get a quantitative description of the asymptotic normal structure, playing a similar role to that Bynum's coefficients for normal structure.
For a subsequence (XnJ of a bounded sequence (Xn) in X, we will denote by T'a ((xnJ) the asymptotic radius of this subsequence with respect to the set co( {xn : n EN}), i.e.
170
Moreover the bounded sequence (xn) in X is asymptotically regular (a.r.) whenever Xn - Xn+l --+ O. Now we define
AN(X) := sup{k: k.inf(xni ) ra ((xn,)) ~ diama((xn)) for each a.r. bounded sequence (xn)} .
If we add in this definition the condition that the sequence (xn) is such that the set cot {xn : n E N}) is weakly compact, then we get the asymptotic normal structure coefficient with respect to the weak topology
w - AN(X) := sup{k: k.inf(xni) ra ((Xn,)) ~ diama((xn)) for each a.r. bounded sequence (xn) with cot {Xn : n E N}) weakly compact}.
Geometrical properties in terms of these constants.
1 Definition 3.6 If AN(X) > 1 we say that (X, II . II) has uniform asymptotic normal structure, (UAN).
2 Definition 3.7 If w - AN(X) > 1 we say that (X, 11·11) has uniform asymptotic normal structure with respect to the weak topology, (w-U AN) .
Main features of these coefficients.
1 1 ~ AN(X) :s: w - AN(X).
2 1 :s: WCS(X) :s: w - AN(X).
3 In the definitions of w - AN(X) we can replace diama((xn)) by diam((xn )).
4 If a Banach space (X, 11·11) has AN(X) > 1 then it is reflexive.
5 w - AN(X) > 1 =} (WFPP).
6 Let (X, II ,11) and (Y, I· I) be isomorphic Banach spaces. Then we have
AN(X) :s: d(X, Y)AN(Y),
w - AN(X) :s: d(X, Y).w - AN(Y).
7 AN(X) = 00 if and only if (X, II . II) is finite dimensional.
8 w - AN(X) = 00 if and only if (X, II· II) is a Schur space.
1
9 AN(fp)=2p.
10 Example 3.8 For (3,p > 1 let us consider fp endowed with the norm
Ilxllf3 := max {llxllcx), ~lIxllp } .
We denote this space by X~. If we take the space Z := X~ Xfl equipped with the
f I -norm then this space is nonreflexive. Thus, AN(Z) = 1. But w-AN(Z) = J2.
REFERENCES 171
3.10. Coefficient for Semi-Opial property, [16]
It was defined in [16]. Recall that a Banach space is said to have the semi-Opial (weak semi-Opial) property, (SO) (( w-SO)) for short, if for each bounded nonconstant asymptotically regular sequence (xn) in X (with weakly compact closed convex hull), there exists a subsequence (xnJ weakly convergent to x, such that
liminf IIx - xnJ < diam({xn}). , Now we define the semi-Opial coefficient with respect to the weak topology as follows
w - SOC (X) := sup{k: k.infexn,),xni C".yra (y, (xnJ) :S diama((xn)) for each a.r. bounded sequence (xn) with co({xn: n EN}) weakly compact}.
Geometrical properties in terms of this constant.
1 Definition 3.9 If w-SOC(X) > 1 we say that (X, 11·11) has uniform semi-Opial property, with respect to the weak topology, (w-USO).
Main features of this coefficient.
1 1:S WCS(X) :S w - SOC(X) :S w - AN(X).
2 In the definitions of w - SOC(X) we can replace diama((xn)) by diam((xn )).
3 If a Banach space (X, II . II) has the nonstrict Opial property, then
w - SOC(X) = w - AN(X).
4 The space £P([O, 1]), 1 < p < 00, p i= 2, has (w-USO) property but does not satisfy the nonstrict Opial condition. inxxcondition,nonstrict Opial
5 Let (X, II . II) and (Y, I . I) be isomorphic Banach spaces. Then w - SOC(X) :S d(X, Y).w - SOC(Y).
6 w - SOC(X) = 00 if and only if (X, II . II) is a Schur space.
7 Suppose that X = WEB Z where W is a closed subspace of X, Z is a Schur space, and the projection onto W has norm 1. Then we have
w - SOC(X) = w - SOC(W).
8 Let (X, 11·11) , (Y, 1·1) be Banach spaces. If (X, 11·11) is (w-USO) and WCS(Y) > 1 then (X x Y)ep (1 :S p < 00), is also w-USO.
1 9 w - SOC(Cp ) = 2p.
10
w - SOC(X%) = max [l,min (2~' ~)] . 11 If we take the space Z := X0 x Cl equipped with the C1-norm then this space is
nonreflexive. Thus, AN(Z) = 1. But w - SOC(Z) = V2.
172
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Chapter 6
ULTRA-METHODS IN METRIC FIXED POINT THEORY
M. A. Khamsi
Department of Mathematical Sciences and Computer Science
The University of Texas at El Paso
El Paso, Texas 79968, USA
aminel11mcs.sci.kuniv.edu.ku
B. Sims
Mathematics, School of Mathematical and Physical Sciences
The University of Newcastle
NSW 2308, Australia
1. Introduction
Over the last two decades ultrapower techniques have become major tools for the development and understanding of metric fixed point theory. In this short chapter we develop the Banach space ultrapower and initiate its use in studying the weak fixed point property for nonexpansive mappings. For a more extensive and detailed treatment than is given here the reader is referred to [1] and [21].
2. Ultrapowers of Banach spaces
Throughout the chapter I will denote an index set, usually the natural numbers N, for most situations in metric fixed point theory.
Definition 2.1 A filter on I is a nonempty family of subsets F <:;; 21 satisfying
(i) F is closed under taking supersets. That is, A E F and A <;;; B <:;; I ==* B E F.
(ii) F is closed under finite intersections: A, B E F ==* An B E F.
Examples.
(1) The power set of I, 2I , defines a filter.
(2) The Fhkhet filter {A <:;; I : I\A is finite}
177
WA. Kirk and B. Sims (eds.), Handbook of Metric Fixed Point Theory, 177-199. © 2001 Kluwer Academic Publishers.
178
(3) For io E I, Fio := {A ~ I : io E A}. Filters of the form Fio for some io E I are termed trivial, or non-free filters.
(4) If (1, j) is a lattice, then the family of supersets of sets of the form Mio = {i : it io}, for io E I, is a filter. To see this, note that Mio nMjo = MioVjo.
A filter F is proper if it is not equal to 21, the power set of I. Equivalent conditions are: 0 rt F, or F has the finite intersection property; that is, all finite intersections of filter elements are nonempty.
Throughout this chapter, we will take filter to mean proper filter.
Definition 2.2 An ultrafilter U on I is a filter on I which is maximal with respect to ordering of filters on I by inclusion: that is, if U ~ F and F is a filter on I, then F = U. Zorn's lemma ensures that every filter has an extension to an ultrafilter.
Lemma 2.3 A filter U C 21 is an ultrafilter on I if and only if for every A ~ I precisely one of A or I\A is in U.
Proof. (=» We show that if I \ Art U, then A E U. If I \ Art U, then I \ A has no subset which is an element of U; hence every element of U meets A. The family B = {A n U : U E U} therefore has the finite intersection property and so its supersets form a filter FB. But U ~ FB, because U ;2 UnA E FB, and so by the maximality FB = U. Also, A = A n I E FB and so A E U.
(¢=) Note: the condition automatically ensures U is proper because lEU and so 0= I \ I rt U. Now, let F be a filter on I with U ~ F, we show F = U. Assume not, then there exists A E F with A rt U. However, we then have I \ A E U ~ F. So both A and I \ A belong to F which implies that 0 = An (I \ A) E F, contradicting F being proper. •
As a consequence of this lemma: For an ultrafilter U on I if Al U A2 U ... U An E U then at least one of the sets AI, A2, ... , An is in U, and an ultrafilter is nontrivial (free) if and only if it contains no finite subsets.
It will henceforth be a standing assumption that all the filters and ultrafilters with which we deal are nontrivial.
We say U is countably complete if it is closed under countable intersections. Ultrafilters which are not count ably complete are particularly useful for some purposes. It is readily seen that an ultrafilter U is count ably incomplete if and only if there exist elements Ao, AI, ... , An, . .. in U with
00
I = Ao :::l Al :::l A2 :::l •.. :::l An::J ... and n An = 0. n=O
We shall see that this structure allows us to readily extend inductive and diagonal type arguments into ultrapowers. Every ultrafilter U over N is necessarily countably incomplete (consider the countable family of nested sets An := {n, n + 1, n + 2, ... } E U).
One of the most exiting result about ultrafilters deal with compactness. Before we state this result, we will need to link ultrafilters with the concept of convergence in topological spaces.
Ultra-methods 179
Definition 2.4 For a Hausdorff topological space (0, T), an ultrafilter U on I, and (Xi)iEI ~ 0 we say
l\Txi (== T -1\TXi) = Xo
if for every neighbourhood N of Xo we have {i E I: Xi EN} E U.
Limits along U are unique and if U is on Nand (Xn) is a bounded sequence in lR then
liminfxn ::; limxn::; lim sup X n. nUn
Moreover, if C is a closed subset of 0 and (Xi)iEI ~ C, then limu Xi belongs to C whenever it exists.
Remark 2.5 Let X be a metric space. If U is an ultrafilter and limu Xn = X, with (xn) C X, then there exists a subsequence of (xn) which converges to x.
The next theorem is fundamental since it characterizes compactness by use of ultrafilters.
Theorem 2.6 Let K be a Hausdorff topological space. K is compact if and only if limu Xi exists for all (Xi)iEI C K and any ultrafilter U over I.
When the space in question is a linear topological vector space, convergence over an ultrafilter has similar behaviour to traditional convergence. In particular, we have:
Proposition 2.7 Let X be a linear topological vector space, and U an ultrafilter over an index set I.
(i) Suppose that (Xi)iEI and (Yi)iEI are two subsets of X such that limu Xi and limu Yi exist. Then
for any scalar a E R
(ii) If X is a Banach lattice and (Xi)iEI is a subset of positive elements of X, i.e. Xi ~ 0, then limu Xi is also positive.
Now we are ready to define the ultrapower of a Banach space. Let X be a Banach space and U an ultrafilter over an index set I. We can form the substitution space
Then,
.e",,(X) := {(Xi)iEI: II(xi)II",,:= sup II Xi II < oo}. iEI
Nu(X) := {(Xi)iEI E £",,(X) : lim II Xi II = O} u
is a closed linear subspace of .e",,(X).
Definition 2.8 The Banach space ultrapower of X over U is defined to be the Banach space quotient
(X)u := .eoo(X)/Nu(X),
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with elements denoted by [xilu, where (Xi) is a representative of the equivalence class. The quotient norm is canonically given by
II[XiJull = lim IIxill· u
Remark 2.9 The mapping J: X ~ (X)u defined by
J(x) := [xJ := [xilu, where Xi = X, for all i E I
is an isometric embedding of X into (X)u. Using the map J, one may identify X with J(X) seen as a subspace of (X)u. When it is clear we will omit mention of the map J and simply regard X as a subspace of (X)u.
In what follows, we describe some of the fundamental results related to ultrapowers. We will not be exhaustive and leave it to the interested reader to pursue the subject further by consulting [21], for example.
Proposition 2.10 Let (X)u be an ultrapower of a Banach space X. Then for any c: > 0 and any finite dimensional subspace M of (X)u, there exists a subspace N of X with the same dimension and a linear map T : Xo -> Yo such that
(1 - c:)llxll ::; IIT(x)11 ::; (1 + c:)lIxll
holds for all X E Xo (Such a map is referred to as a c:-isometry).
Proof. Let x{l), x(2), ... ,x(n) be a unit basis for M and choose representatives
(X;k)) of x(k) such that IHk) II ::; 2, (k = 1,2, ... ,n).
Consider the vector space Mi = (x(k)\n
, I k=1
and define a linear map T; : M -t Mi by its action on the basis;
Then, IITil1 ::; 2K, where
For any X = 2.::;=1 Ak x(k) E M, we have
IIxll = IltAkx(k)11 = IltAk (x~k))ull
=1~lltAkx(k)11 = lim IITiXl1 u
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Thus,
Ix = {i E I: IIITiXIl-lIxlll::; ~lIxll} EU.
Now, let 8 be a positive number (to be chosen later) and let y(1), y(2), ... , y(m) be a finite 8-net in the unit sphere of M and set
m
10 = n Iy(k), k=l
then 10 =1= 0 and for any i E 10 and x E M with IIxll = 1 we have
IIITiXIl-lIxlll ::; k=5~~,m (1ITi(X - y(kl ) II + Ilx - y(klll + IIITi(y(kl ) 11-lly(kllli)
::; (2K + 1)8 + ~.
The conclusion now follows by taking 8 = ( c: )' N := Mi and T := Ti . • 2 2K +1
Any Banach space which enjoys a similar property as the one described above for (X)u is called finitely representable in X. Therefore, any ultrapower of X is finitely representable in X. Note that we may avoid using the map T by introducing the so-called Banach-Mazur distance between normed spaces.
Definition 2.11 Let X and Y be Banach spaces. The Banach-Mazur distance between X and Y is
d(X, Y) = inf{IITIIIIT-11I :: where T is an isomorphism from X onto Y} .
When X and Yare not isomorphic we simply set d(X, Y) = 00.
Therefore, X is finitely representable in Y if and only if for any c: > 0 and any finite dimensional subspace Xo of X, there exists a subspace Yo of Y with the same dimension such that d(Xo, Yo) < 1+c:. It is a stunningly useful fact that an ultrapower of a Banach space X can capture isometrically all the spaces finitely represented in X. Indeed, we have
Theorem 2.12 Let Y be a separable Banach space which is finitely represented in X. Then there is an isometric embedding ofY into the ultrapower (X)u for each countably incomplete ultrafilter U.
Proof. Let U be a count ably incomplete ultrafilter on Ij that is, there is a countable chain It ;2 h ;2 ... with In E U and
00
By the separability of Y we can find a linearly independent sequence (x(n))~=l such that Y = ({x(n)}~=l). Since Y is finitely represented in X, for each N in N, there exists a liN-isometry
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Now define J: Y -7 (X)u by its action on the x(m),
J(x(m)) = (xi(m)) ,
where
{ 0 if i E I \ 1m,
xi(m) = Tn (x(m) ) if i E 1m, where n ~ m and Tn(x(m)) is the unique number such that i E In \ In+!.
Note that since n::'=l In = 0, Xi is defined for each i E I. To see that J is an isometry observe that; for
we have
K
X = LoAkx(mk) (such x are dense in Y) k=l
IIJxl1 = liE oAk (Xi (mk))U II
=1~IIEoAkXi(mk)11 = Ilxll·
To see this, given c: > 0, choose N > (1/c:, maxk mk), then we have x E XN and for i E IN E U,
III E oAk xi(mk)II-IIXIlI = III E oAk Tnx(mk)II-IIXIII (for some n ~ N)
= IIITnXII-llxlll ::; c:114
• Example 2.13 Ultrapowers of a Hilbert space It is known that a Banach space X is a Hilbert space if and only if
for all x,Y E X. Let (X)u be an ultrapower of X and let [(Xi)] and [(Yi)] be two elements in (X)u, then we have
and
II [(Xi)] - [(y;)] 112 = II [(Xi - Yi)]11 2 = l~ IIxi - Yil1 2 .
Since
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and using the Hilbert structure of X, we get
Whence,
which implies (X)u is a Hilbert space.
This example, though easy to prove, is extremely rich in many ways. Indeed, what the reader should learn from it is that the ultrapower catches any finitely determined property satisfied by the Banach space. Maybe one of the most useful instances of this concerns lattice structure. If X is a Banach lattice, then any ultrapower (X)u is also a Banach lattice when the order is defined by taking x E (X)u to be positive if and only if one can find a representative, (Xi), of x all of whose elements are positive in X. In this case, (X)u enjoys most of the important lattice properties satisfied by X.
Also from the above example, we see that a nonreflexive Banach space can not be finitely represented in a Hilbert space. In other words, only reflexive Banach spaces may be finitely represented in a Hilbert space. This leads to the concept of a super-property.
Definition 2.14 Let P be a property defined on a Banach space X. We say that X has the property "super-P" if every Banach space that is finitely representable in X has P.
The following result is an immediate consequence of Proposition 2.10 and Theorem 2.12.
Theorem 2.15 IfP is a separably determined Banach space property that is inherited by subspaces, then a Banach space X has super-P if and only if some ultrapower of X over a countably incomplete ultrafilter has P (and hence every ultrapower of X has P).
Remark 2.16 Reflexivity satisfies the requirements of the above theorem. Thus, a Banach space X is superreflexive if and only if some (and hence every) count ably incomplete ultrapower of X is reflexive.
We also note that Theorem 2.12 remains valid if we replace 'every countably incomplete ultrafilter' by 'there exists an ultrafilter', without the assumption that the property be separably determined. Thus, we always have:
HP is a Banach space property that is inherited by subspaces, then a Banach space X has super-P if and only every ultrapower of X has P.
In particular Hilbert spaces are superreflexive. One may think that these are the only examples of superreflexive Banach spaces. In the following example, we show that this is far from the case, indeed the family of superreflexive Banach spaces is quite a rich one.
Example 2.17 Let X be a Banach space. For any e > 0, define
8x (e) = inf {1- ~llx+YII: X,y E X and IIxll::; 1, Ilyll::; I}
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The function 8x (e:) is called the modulus of uniform convexity of X. The characteristic of uniform convexity of X is defined by
co(X) = sup{c : 8x(c) = o} .
A Banach space X is uniformly convex if 8x (c) > 0 for any c > O. A space is said to be uniformly nonsquare, or in quadrate if and only if co(X) < 2. We next discuss the link between these concepts and ultrapowers.
Let (X)u be an ultrapower of X. Then, for any e: > 0, we have
Consequently, we also have e:o(X) = e:o((X)u). In particular, a Banach space is uniformly convex (uniformly nonsquare) if and only if some, and hence every, ultrapower is uniformly convex (uniformly nonsquare). It is also worth mentioning that an ultrapower is uniformly convex if and only if it is strictly convex. Since uniformly nonsquare spaces are reflexive, we deduce that uniformly nonsquare Banach spaces are also superreflexive. In fact, Enflo [8] (see also Pisier [18]) has shown that X is superreflexive if and only if there exists an equivalent norm (on X) which is uniformly convex. More on this may be found in, for example, [3], or [21].
Next we discuss ultraproducts of maps. Let X and Y be two Banach spaces and let (X)u and (Y)u be their associated ultrapowers with respect to a given ultrafilter U on I. Let T; : D c X -> Y he a family of maps indexed by I. Consider
[D]u := {x E (X)u : there exists a representative (£4) of x with £4 ED} .
Define [(Ti)]U : [D]u ---t (Y)u by
[(T;)]u([(di)]U) = [(Ti(£4))]u.
We of course have to ensure that the Ti satisfy suitable conditions for [(T;)]u to be well defined. When this is the case we have, in particular, the following.
Proposition 2.18 Using the above notations, we have
(i) [D]u is convex if D is convex;
(ii) [D]u is closed if D is closed;
(iii) [D]u is bounded if D is bounded;
(iv) [(Ti)]U is Lipschitzian provided the T; are Lipschitzian mappings whose Lipschitz constants Ai are uniformly bounded, in which case the Lipschitz constant of [(Ti))U is equal to limu Ai;
(v) [(T;)]u is a bounded linear operator provided the Ti are linear operators which are uniformly bounded; that is, sUPiEI IITili < 00, and then II [(Ti))ull = limu IITill·
Proof. Most of these results follow directly from the relevant definitions. Consequently, we restrict ourselves to proving (ii) in the case of particular interest when U is an ultrafilter over N. Thus, let N = Ao ~ Al ~ A2 ~ ... ~ An ~ ... , be a nested sequence of sets with each An E U, and nn>1 An = 0.
Ultra-methods 185
Suppose [ttlu, [t;]u, ... is a sequence of points in [Dlu, with each tf E D, which converges to [xi1u E (X)u. By passing to a subsequence if necessary we may without loss of generality assume that
II[ti1- [xdll
For each mEN let
1 lim Iiti - xiii < U m
and put Bo := Nand t? := 0, then
From this it follows that for each i E N there is a unique m such that i E Bm \Brn+1.
Define Yi := ti, for this m, in particular then Yi E D.
Now, given any mEN, for each i E Bm there is a unique p 2:: m with i E B p\Bp+1 .
Thus,
and so
2 2 IIYi - xiii = Iltf - xiii < - :::; -, p m
{i EN: IIYi - xiii < ~} "2 Bm E U.
We therefore have that U -lim IIYi - xiII = 0, which yields the desired conclusion that ~~E~u· •
For the above theorem, recall that a map T : D c X --> Y is said to be a Lipschitz mapping with Lipschitz constant).. if
IIT(X) - T(Y)II :::; )..llx - yll
for all x,Y E D.
The above results are useful for studying the dual space of an ultrapower. Indeed, let (xi) be a uniformly bounded family of linear funetionals defined on X (i.e. elements of the dual space X*). Then from the above results, we can generate a bounded linear functional [(xil1u. This linear functional belongs to the dual of the ultrapower; that
is, ((X)u) *. One may then ask whether this construction yields all the elements of
( (X)u ) *. An answer is provided by the following theorem.
Theorem 2.19 Let X be a Banach space. Then
((X)u) * = (X*)u
if and only if X is superrefiexive.
More on this and similar results may be found in [211.
We will close this section with an important example.
186
Example 2.20 In this example, we discuss ultrapowers of the Lp-spaces, 1 ~ p < 00.
Let (0, E, /L) be a a-additive measure space and for 1 ~ p < 00, let Lp(/L) denote the real space Lp(O, E, /L). We will show that if U is an ultrafilter on I then there exists a measure space (n, f;, ji,) with (Lp(/L))u lattice isometric to Lp(ji,).
First note that under a 'component-wise' definition of order, (Lp(/L))u is a Banach lattice. Thus, by the classical theorem of Bohnenblust and Nakano, see for example [14], it is sufficient to prove that the norm in (Lp(/L))u is p-additive; that is, whenever x /\ Y = 0 we have that
To this end, let (Xi)U and (Yi)U be elements of (Lp(/Li))U such that (Xi)U /\ (Yi)U = o. Let Zi = Xi /\ Yi, then (Xi - Zi) /\ (Yi - Zi) = 0 and so
On the other hand, we have 0 = (Xi /\ Yi)U = (Zi)U, so limu IIZili (Xi)i7(Xi - Zi), (Yi)i7(Yi - Zi) and (Xi + Yi)i7(Xi + Yi - 2Zi) and so
as required.
0, but then,
This argument does not provide us with any information on the structure of the measure space (n, f;,ji), for information on this and related questions see, for example, [21].
3. Fixed Point Theory
Definition 3.1 A Banach space X is said to have the weak fixed point property (w-fpp) if for every nonempty weakly compact convex subset C of X and every nonexpansive mapping T : C -+ C the fixed point set of T, Fix(T), is nonempty. Recall that x E Fix(T) if and only ifT(x) = x.
To establish the w-fpp for a Banach space X we work toward a contradiction. Thus, assume that X fails to have the w-fpp then there exists a weakly compact convex subset C of X and a nonexpansive mapping T : C -+ C with Fix(T) = 0.
Let F denote the family of all nonempty closed convex subsets K of C that are invariant under T (that is, T(K) C K). Clearly, F is not empty, since C E F. The weakcompactness of C ensures that F satisfies the assumptions for Zorn's lemma. Therefore F has minimal elements.
Definition 3.2 A convex set K is said to be a minimal invariant set for T if K is a minimal element of F.
Clearly any set K which is a minimal invariant set for T contains more than one point; that is,
diam(K) = sup{lIx - yll : X,Y E K} > 0,
otherwise T would have a fixed point.
We proceed to investigate the properties of minimal invariant sets.
Ult:ra-methods 187
Proposition 3.3 Let K be a minimal invariant set for T. Then
The next result gives an interesting property satisfied by minimal invariant sets.
Lemma 3.4 Let K be a minimal set for T, and let a : K --> lR+ be a lower semicontinuous convex function such that
a(T(x)) ~ a(x) , for all x E K.
Then a is a constant function.
Taking a(x) := sup{!lx-y!l : y E K} and using proposition 3.3 to replace the supremum over K with a supremum over T(K) we see that the above lemma applies and readily yields:
Proposition 3.5 Any minimal invariant set K for T is a diametral set; that is, diam(K) > 0 and
sup{llx - y!l : y E K} = diam(K)
for all x E K.
Spaces which contain no weakly compact convex diametral sets are said to have weak normal structure, clearly such spaces have the w-fpp.
The property of normal structure (the absence of diametral closed bounded convex subsets) was introduced by W. A. Kirk in 1965 when he showed that reflexive spaces with the property had the fixed point property. It was quickly realized that this result subsumed most of the then known existence results for fixed points of nonexpansive mappings by F. Browder, D. Gohde, M. Edelstein and others. The main thrust of metric fixed point theory during the late 1960's and throughout the 1970's was the quest for natural, and easily verified, conditions on a Banach space that are sufficient for weak normal structure coupled with an exploration of other consequences of normal structure and related properties such as asymptotic normal structure. Details of this work, together with relevant references, may be found in the chapter entitled the Classical theory of nonexpansive mappings.
Initially, it was unknown whether all reflexive spaces necessarily had normal structure, or if weak normal structure and the weak fixed point property were equivalent. Then, in 1975 and 1976, the two questions were settled in the negative by R. C. James and L. Karlovitz respectively.
Example 3.6 For f3 > 1 let X~ denote the Hilbert space 12 equipped with the equivalent norm
James observed that these spaces are all superreflexive, but that X2 fails to have normal structure. Indeed, it is quite easy to verify that X~ fails to have normal structure for f3 2: "j2. On the other hand, Karlovitz showed that X..j2 has the fixed point property for nonexpansive mappings. Subsequently, this family of spaces has been the subject of considerable investigation. For example, in 1981 Baillon and Schoneberg [2) observed
188
that, for {3 < 2, XfJ has asymptotic normal structure; a geometric property which they showed implies the fixed point property. For larger values of {3 the situation remained unclear, though Baillon managed to give some highly technical demonstrations of the fixed point property for certain values of {3, until finally, in 1984, it was shown [4] that XfJ has the fixed point property for all values of {3 (see also [15]).
Normal structure precludes the presence of diametral sets and as such only involves the mapping T in so far as minimal invariant sets of fixed point free nonexpansive maps provide instances of such diametral sets. To establish the weak fixed point property in the absence of weak normal structure requires properties of minimal invariant sets that involve the mapping T in a more explicit way. One such property was used by Karlovitz to establish the fixed point property for the space Xy'2' The property was independently discovered by K. Goebel and the result has subsequently become known as the Goebel-Karlovitz lemma. Before presenting it we need some more facts about nonexpansive mappings.
Let K be a nonempty, bounded, closed, convex subset of a Banach space X, and T : K -> K be nonexpansive. Fix c E (0,1) and Xa E K, and consider the map TE : K -> K defined by
Te(x) = cXa + (1 - c)T(x)
for all x E K. TE is clearly a contraction mapping. Hence it has a unique fixed point XE E K; that is, Te(xe) = XE' We have
IIT(xE) - Xell ~ cdiam(K) .
In other words, we have
inf{IIT(x)-xlI: XEK}=O.
Definition 3.7 A sequence (xn) satisfying limn~oo IIxn - TXnll = 0, is called an approximate fixed point sequence.
The above construction shows that a nonexpansive self mapping of a closed bounded convex set always has an approximate fixed point sequence.
The Goebel-Karlovitz lemma is the following
Lemma 3.8 Let C be a weakly compact convex set and let K be a minimal invariant set for T : C -> C. Then for any approximate fixed point sequence (Xn) of T in K, we have
lim IIx - xnll = diam(K) n~oo
for all x E K; that is, (xn) is a diameterizing sequence for K.
The proof is an easy consequence of lemma 3.4 with a(x) := limsuPn IIx - xnll·
In the first instance, one might think that the presence of diameterizing sequences in minimal invariant sets of fixed point free nonexpansive mappings would provide a lever for establishing the w-fpp in the absence of normal structure. Unfortunately this is not the case. A simple construction shows that if a space contains a diametral set D then it also contains a diametral set with a diameterizing sequence. Indeed, one can construct within D a sequence (xn) with dist(Xn+l,COnV{Xl,X2,'" ,Xn}) -> diam(D). Such a
Ultra-methods 189
sequence is diameterizing for its closed convex hull which is therefore a diametral subset of D with the same diameter as D. To proceed in the absence of weak normal structure, the mapping T must be brought back into play, via the Goebel-Karlovitz lemma, and the fact that the diameterizing sequence is an approximate fixed point sequence for T exploited. Such arguments are necessarily both delicate and subtle. It was B. Maurey [17J who, in a brilliant series ofresults (see section 4), first demonstrated the usefulness of ultrapowers as a setting for such arguments. His methods brought a new dimension to metric fixed point theory and, together with Alspach's seminal example showing that LdO,IJ fails the w-fpp, began what might be described as the 'non-classical theory'.
We now turn to the basic constructions such methods employ.
Let C be a nonempty bounded convex subset of a Banach space X and T : C -+ C a nonexpansive mapping with no fixed point. Let U be an ultrafilter on the set of natural numbers. In (X)u we may define
o := {[xnJU: Xn E C, for all n EN}.
Then, 0 is a convex subset, with diam(O) = diam(C), containing an isometric copy, .1(C), of C and on which T : 0 --> 0 defined by
T([xnJu) = [T(xn)Ju,
where the representative (xn) is chosen to be a sequence of points from C, is a well defined nonexpansive mapping [proposition 2.18 (iv)J which leaves .1(C) invariant. We now list a number of basic results for 0 and T constructed as above. From proposition 2.18 (ii) we have the following.
Proposition 3.9 The set 0 in (X)u is closed. Hence, when X is a superrefiexive space 0 is weakly-compact.
The next proposition follows directly from the definitions.
Proposition 3.10 If (xn) is an approximate fixed point sequence for T, then [xnJu is a fixed point ofT. Consequently, T always has fixed points in O.
Conversely, from a fixed point (indeed an approximate fixed point sequence) for T in o we can readily extract an approximate fixed point sequence for T.
If C is a weakly compact minimal invariant set for T, so that the Goebel-Karlovitz lemma applies, then in the above proposition we also have II [xnJu - .1xlI = diam(C), for all x E C. Since, in this case we can always assume without loss of generality that diam C = 1 and that (xn) converges weakly to 0 (so, 0 E C), we may suppose that lI[xnJull = dist([xnJu,.1C) = 1.
The following is a significant observation of B. Maurey [17J.
Lemma 3.11 Given any two fixed points ii = [anJu and b = [bnJu ofT in 0 there is a fixed point e with
llii - ell = lie - bll = ~lIii - bll·
Proof. We may assume that A := llii - bll := limu Ilan - bnll > O. For each mEN let
Am:= {n ~ m: lIan - bnll::; A+ ~2 and Ilan -Tanll, Ilbn -Tbnll::; ~2}'
190
then Am E U, N =: Ao :::l Al :::l A2 :::l •.. :::l An :::l ... and nn?:l An = 0.
For each n E N let Cn := {c E C: Ilan - cll, Ilbn - cll ::; >../2 +~} where m is the unique element of N for which n E Am \Am+I . Then Cn is bounded, closed, convex and nonempty since
1 A 1 A 1 lIan - 1/2 (an + bn)1I = 2 II an - bnll ::; 2 + m2 ::; 2 + ;;;;,
and similarly, Ilbn - 1/2 (an + bn)11 ::; >../2 + 11m, so
~ (an + bn) E Cn·
Now, define a strict contraction, Tn on Cn by,
1 1 Tnz:= (1- - )Tz + -(an + bn ).
m 2m
To see that Cn is Tn-invariant let z E Cn, then
Ilan - Tnzll = Ilan - ( (1 - ~)Tz + 2~ (an + bn )) II
::; (1 - ~) lIan - Tzil + _1 lIan - bnll m 2m
::; (1- ~)lIan - Tan II + (1- ~)IITan - Tzil + 2~llan - bnll
::; (1- ~)llan - Tanll + (1- ~)llan - zll + 2~lIan - bnll
::; (1 - ~) ~ + (1 _ ~) (~ + ~) + _1 (A + ~) m m 2 m 2 m 2m m 2
11 All 111 = - - - + - + - - -A - - + -A + -m 2 m 3 2 m 2m m 2 2m m 3
A 1 = 2+;;;;"
and similarly, Ilbn - Tnzll ::; A/2 + 11m.
Thus, Tn has a unique fixed point, Cn E Cn. That is,
1 1 Cn = Tnen = (1 - - )Ten + -(an + bn).
m 2m
and we have,
It therefore follows from the above construction that for each mEN the set of n for which IITen - cnll ::; (11m) (>../2 + 11m) contains Am and so is in U. Consequently, for e := [enju we have
lie - Tell := lim IiCn - Tenll = 0 u
Ultra-methods 191
and so e is a fixed point of T. Similarly, from Ilan -enll, Ilbn-enll :::; >../2+1/m for all n E Am \Am+1 , and consequently for all n ~ Am, we have lIa - ell and lib - ell are less than or equal to >../2. Since >.. = lIa - bll, the triangle inequality then ensures that lIa - ell = lib - ell = Iia - bll/2 and the result is established. •
This Lemma states that the fixed point set of T is metrically convex. An appeal to Menger's theorem then ensures the existence of a continuous path of fixed points joining any two fixed points of T and lying within the metric segment between them.
Remark 3.12 When C is weakly-compact and a minimal invariant set for T it is always possible to find two such fixed points a and b of T with lIa - bll = diamC. To see this, we may without loss of generality suppose that diam C = 1 and that we have an approximate fixed point sequence (xn) for T, with (xn) weakly convergent to O. Applying the Goebel-Karlovitz lemma we may extract a subsequence (xnJ such that IIxni - xni+111 --+ diam C. Taking a := [xn2J and b := [Xn2i_1] yields two fixed points ofT with
The following generalization of the Goebel-Karlovitz lemma, due to P. K. Lin [15] has proved basic for establishing the fixed point property using ultrapower methods.
Lemma 3.13 Suppose C is a weakly-compact minimal invariant set for T. If (an) is an approximate fixed point sequence for T in C then
lim lIan - J"xil = diam(C), for all x E C. n
Proof. Suppose this were not the case. Without loss of generality we may take diam(C) = diam(C) = 1, and by passing to a subsequence if necessary assume that Ilan - Tan II < l/n, for all n.
Then there are co > 0, Xo E C, and no E N with
Ilan - .Jxoll < 1- eo, for all n > no· Let an = [a~]u, with a~ E C, and define
An:= {m: lIa;-:' - xoll < 1- eo/2}, and
Bn:= {m: lIa;-:' - Ta;-:'II < 2/n}.
Then An and Bn are in U.
Put mo = 0 and for n E N inductively choose ffln E AnnBnn{mn-l+1, mn-l+2, ... } E U. Then the sequence (a~n) is such that
lIa;-:'n - Ta;-:'J < 2/n.
That is, (a~J is an approximate fixed point sequence for T in C. But,
lIa;-:'n - xoll < 1 - eo/2, an observation which is difficult to reconcile with the fact that (a~n) is, by the GoebelKarlovitz lemma, diameterizing for C. •
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Remark 3.14 If W is any nonempty closed convex and T-invariant subset of C, then, by the standard construction using Banach's contraction mapping principle, W contains an approximate fixed point sequence for T. SO, by the above lemma, for every x E C we have sup{lIw - .lxll : w E W} = diamC. In particular, if we have 'normalized' so that diam C = 1 and 0 E C, then
sup Ilwll = 1. wEW
This leads to an important strategy for establishing the fixed point property in a class of spaces. Namely, try to construct a nonempty closed convex and T-invariant subset W of C in such a way that the hypotheses on the spaces preclude the existence of elements in W with norms arbitrarily close to one. Thereby contradicting the above lemma and hence denying the existence of a fixed point free nonexpansive self mapping of a nonempty weakly compact convex subset in the space.
Indeed, we know of only one proof establishing the fixed point property for a class of spaces via ultraproduct methods that does not use this approach, and that is S. Prus' proof [19, 20l (also see [12]) that uniformly non-creasey spaces have the fixed point property.
We illustrate the strategy outlined in the above remark with just one example, due to Garcia-Falset [9], others may be found scattered throughout this Handbook. See also the Notes and remarks section for references to the literature.
Let U be a given ultra filter over N and for each Banach space X define a coefficient R(X) by,
R(X) := sup{l~ IIx + xnll : Ilxll :::: 1; Ilxnll :::: 1, for all nand (xn) -t 0 weakly}.
Equivalently, R(X) is the 'smallest' number such that
lim IIx + xnll :::: R(X)llxll V (lim Ilxnll), u u
for all x E X and all weak null sequences (xn).
In general 1 :::: R(X) :::: 2 and R(CO) = 1, while R(Lil = 2, .
Proposition 3.15 If X is a Banach space with R(X) < 2, then X has the weak fixed point property.
Proof. Suppose X fails the weak-fixed point property. Then there exists a weaklycompact convex set C with diam(C) = 1 which is a minimal invariant set for some nonexpansive mapping T. Further we may assume that C contains a weakly-null approximate fixed point sequence (an) for T. Let C and T be defined as above and define
W := {[wnlu: Wn E C, for n E N; II [wnlu - [anlll :::: 1/2 and D[wnl :::: 1/2},
where D[wnl := limu,m limu,n Ilwm - wnll. Then, W is readily seen to be aT-invariant, closed, convex, nonempty (as (1/2)[an l E W) subset of C. Thus, by the above remark
sup{llwll : wE W} = 1.
Ultra-methods 193
On the other hand, let W = [wn]u be any element of W, where without loss of generality Wn E C, for all n E N, and let Wo be the weak-limit with respect to U of (wn ). Then,
Ilwll = lim Ilwnll u = lim II(wn - wo) + wall u < R(X)(lim Ilwn - wall) V Ilwoll, - u
by definition of R(X), as (wn - wo) converges weakly to 0, hence
Ilwll :S: R(X) lim lim Ilwn - wmll V IIwn - anll, U,nU,m
by lower semi-continuity of the norm, since
Hence
Ilwll :S: R(X) x (~V~) = R(X)/2 < 1, a contradiction which establishes the result. • Our choice of the above result to illustrate the strategy in the previous remark is based on its utility; the parameter involved is readily evaluated for many spaces and the criteria is satisfied in a large class of spaces.
Since nearly uniformly smooth (NUS) Banach spaces are readily seen to have R(X) < 2 (see [9]), the result answers in the affirmative the long standing question of whether or not NUS spaces have the weak fixed point property.
In a weakly orthogonal Banach lattice R(X) is less than or equal to the Riesz angle a(X) introduced in [4], thus, this proposition generalizes results of [4], [22] and [23].
The above argument is typical of those for many of the more recent 'non-classical' results in metric fixed point theory, starting with Maurey's 1982 proof of the weak fixed point property for Co , for which it provides an alternative proof. Note that, since a numeric contradiction is arrived at, by carefully analyzing the proof, the gap (here between R(X)/2 and 1) can be exploited to establish the weak fixed point property for spaces whose Banach-Mazur distance from a space satisfying the assumptions is not too great. This is the basis for many of the results given in the chapter entitled Stability of the fixed point property for non expansive mappings, where the reader can find many more existence results, in the morc general guise of stability results, together with other applications of the methods outlined here.
4. Maurey's fundamental theorems
Maurey's results were deep and particularly significant coming as they did just after Alspach demonstrated the failure of the weak fixed point property in L1 [0,1]. As we have already remarked, his results set the stage for the second major revolution in metric fixed point theory. We will not give the details of the proofs for many of his results, and the interested reader is referred to [17], [7] and [1].
Maurey began by establishing the w-fpp for the space Co.
194
Theorem 4.1 The space Co has the weak fixed point property.
This result had eluded proof for many years. From a geometric point of view the space Co is a bad space, exhibiting many of the features found in 100 , Previously, only partial results related to the fixed point property for special domains in Co were known and the arguments employed were often extremely intricate and tedious. We will not give Maurey's original proof, as the result is a special case of proposition 3.15 above. However, his proof was both elegant and open to generalization. It exploited the lattice structure of Co induced from the canonical basis. Others (see, for example, [4], [22], [23]' [15J and the Notes and Remarks section below) quickly refined and generalized these ideas to a large class of Banach lattices.
Perhaps the most important result of Maurey is the following.
Theorem 4.2 Any reflexive subspace X of L1 [0,1 J has the fixed point property; that is, any nonexpansive self mapping of a nonempty bounded closed and convex subset of X has a fixed point.
The ideas behind the original proof of this result have been generalized [7, IJ to obtain the following.
Theorem 4.3 Let X be a Banach lattice with a uniformly monotone norm and assume that II is not finitely representable in X. Then X has the fixed point property.
Recall that a Banach lattice X has a uniformly monotone norm if for all c > 0 there exists 8 > 0 such that IIxll ~ Ilyll + 8 whenever x ~ y ~ 0 and Ilx - yll ~ c, with Ilyll = 1.
In his investigation of the fixed point property, Maurey discovered many fundamental results which led to new insights and a better understanding of the property. For example, in his proof of the above theorem, Maurey used lemma 3.11 and the lattice structure of L1 [0, IJ to show that the ultrapower (X)u of X would contain isometric copies of 1'1, for all n, if X failed to have the fixed point property. Since reflexive subspaces of LdO, IJ are superreflexive, this gave the desired contradiction. Following the appearance of his result there have been many attempts to identify a geometric property enjoyed by the reflexive subspaces of LdO, IJ which would imply the fixed point property. So far such attempts have been in vain.
In the years prior to the appearance of Alspach's example, the w-fpp had been established for many of the classical Banach spaces and it was commonly conjectured that all Banach spaces enjoyed the weak fixed point property. His example therefore came as a surprise to many, and helped redefine the direction of investigation. It cast doubt on the likelihood of positive answers to three of the most basic open questions, which we list in decreasing order of strength:
(1) Do all reflexive Banach spaces have the fixed point property? [And conversely; does having the fixed point property imply reflexivity of the space?J
(2) Do all superreflexive Banach spaces have the fixed point property?
(3) Does the Hilbert space £2 have the fixed point property in all equivalent norms?
To which we would add.
Ultra-methods 195
(4) Does Co have the weak fixed point property in all equivalent norms? [If on no other ground than in its natural norm the space is about as bad as it can get.J
Maurey's results, in particular theorem 4.2, offset Alspach's finding and point in the direcL:m of an affirmative to (2) and hence (3). Lin's recent stability result for £2 (see section 5 and [16]) also lends support to (3). The recent progress described in the chapter entitled Renormings of £1 and Co and fixed point properties may be seen as support for the converse of (1). Further support for this is provided by a result of van Dulst and Pach [6J which shows that the 'super fixed point property' implies superreflexivity. Maurey was unsuccessful in his attempts to settle (2), however, in the course of his investigations he discovered the following tantalizing result, the proof of which again relies on constructions in an ultrapower of the space, [7J and [1 J.
Theorem 4.4 Let X be a superrefiexive Banach space and let K be a bounded nonempty closed convex subset of X. Then any isometry T ; K --t K has a fixed point.
In other words, superreflexive Banach spaces have the fixed point property for isometries.
Before we close this section, it is worth mentioning that Maurey [17J also proved that the Hardy space HI has the fixed point property.
5. Lin's results
We will not attempt to give a detailed list of the results obtained in the two decades following Maurey's discoveries, many of which may be found in the chapter entitled Stability of the fixed point property for nonexpansive mappings. However, some of the most important contributions were due to P-K. Lin [15]' and we discuss two of these.
Theorem 5.1 Let X be a Banach space with a 1-unconditional basis, then X has the weak fixed point property.
Proof. Assume that there exist a weakly compact convex nonempty subset C of X and T ; C --t C a nonexpansive map with no fixed point. Let K be a minimal set for T. Let (xn ) be an approximate fixed point sequence in K. Without loss of generality, we may assume that (xn) converges weakly to 0 E K and diam(K) = 1. Passing to a subsequence, one can construct a sequence of natural projections (Pn), associated to the Schauder basis of X, such that
PnoPm=O ifn#m,
lim IlPn(xn) II = 0 for any x E X, and n~oo
Using the Goebel-Karlovitz lemma, one may assume that
lim IIXn+1 - xnll = 1. n~oo
Let (X)u be an ultrafilter of X. Let k and f be associated to K and T. Set
196
Both X and iI are fixed points for T. Consider the projections (on (X)u)
F = [(Pn )] and Q = [(Pn+1)]'
Hence F(x) = Xj Q(f}) = ilj F(f}) = Q(x)F(x) = 0,
for any x E X. Since the constant of unconditionality of the basis is 1, we have
F 0 Q = OJ III - FII ::; Ij III - QII ::; Ij IIF + QII ::; 1 where I is the identity operator of (X)u. Now set
W = {WE k : Ilw - xII ::; ~, for some x E KjllW - xII ::; ~ and IIw - ilil ::; ~} . Since
II~ + ilil = IIF(x) + Q(f})11 ::; IIF(x) - Q(f})11 = IIx - ilil = 1, we ~a':,e x + U E W, so W)s not empty. It is easy to check that W is T-invariantj that is, T(W) C W. Let w E Wand x E K such that IIw - xII ::; 1/2. Then,
2w = (F + Q)(w) + (I - F)(w) + (I - Q)(w)
= (F + Q)(w - x) + (I - F)(w - x) + (I - Q)(w - iI),
so
211wll ::; II(F + Q)(w - x)11 + 11(1 - F)(w - x)11 + 11(1 - Q)(w - iI)11 1 1 1 3 <-+-+-=_. - 2 2 2 2
which implies Ilwll ::; 3/4. This contradicts remark 3.14. • This result was quickly extended, see for example [22], [23], [13] and [5]. In fact more was proved. For example, by exploiting the gap between 3/4 and 1, Lin obtained the the conclusion for any Banach space X with an unconditional basis provided that the constant of unconditionality ,\ is less than (J33 - 3)/2. This conclusion brings to the surface the problem of whether or not the above result is valid for all Banach spaces with an unconditional basis. This problem is still open and clearly related to the stability problem: Does there exists a Banach space for which the above conclusion is true for any equivalent norm? In the particular case of Hilbert space, Lin [16] improved on all previously known results by establishing the following stability bound.
Theorem 5.2 Let (H, 11·11) be a real Hilbert space. Let 1·1 be an equivalent norm such that
Ilxll ::; Ixl ::; .Bllxll for all x E H·
Then (H, I· I) has the fixed point property provided
f3 < J 5 + 2.,ff3 ~ 2.07.
The proof uses all of the ingredients developed in this chapter and may be found in the chapter entitled Stability of the fixed point property for nonexpansive mappings.
Ultra-methods 197
6. Notes and Remarks
The results developed in this chapter have been used to establish a variety of results in metric fixed point theory, in particular the weak fixed point property for a large variety of Banach spaces.
The notion of an ultrafilter dates back to work of Tarski in 1930 and that of an ultrapower to Skolem in 1934. A rich theory for ultrapowers and ultraproducts (of sets and models) has been built up by a succession of logicians: Los, Frayne, Morel, Scott, Tarski, Hanf, Chang, Keisler, Robinson, Luxemburg and Shelah to mention only a few.
Banach space ultrapowers and ultraproducts were formally introduced by DacunhaCastelle and Krivine in 1972 and were subsequently developed and applied by Stern, Heinrich and many others. They are now a significant and widely used tool for probing the geometry and structure of Banach spaces. They have been particularly important in the study of local theory, superproperties, operator ideals and the isomorphic classification of Banach spaces.
Ultrapower methods were first introduced into metric fixed point theory by B. Maurey [17] in 1982 when he used this technique to provide a positive resolution to the long standing question of whether or not CO had the weakly-fixed point property. He took the W of remark 3.14 to be the metric midpoint set for two fixed points of T constructed as in remark 3.12. This was generalized in [4] to obtain the weak fixed point property for Banach lattices with a Riesz angle
a(X) := sup{lIlxl V Iylll : x, y E Ex} < 2
and for which
liminfliminf IIlxnl/\ IXmll1 = 0, whenever (xn) converges weakly to O. m n
Lattices with this last property were referred to as weak orthogonal Banach lattices. A stronger variant of weak orthogonality, namely:
liminf IIlxnl/\ Ixlll = 0, whenever (xn) converges weakly to 0 and x EX, n
was shown to imply the weakly-fixed point property by Sims [22, 23]. The proof employed the W first defined by Lin in 1983 and used to establish the weak fixed point property for Banach spaces with a I-unconditional basis [15]. The set W used in these proofs consisted of those points in Maurey's W whose distance from .:fC is less than or equal to a half, where in addition the points ii and b were chosen to be 'orthogonal' to one another so that llii + hll = llii - hll = 1. A class of spaces in which such a choice is always possible was considered in [22]. Such spaces were said to have property WORTH. It remains an open question whether or not all spaces with WORTH have the weakly-fixed point property.
Several more 'geometric' variants of these conditions have been introduced. For instance Jimenez-Melado and Llorens-Fuster [11] considered the property of orthogonal convexity, gave examples of orthogonally convex spaces, and showed that it entails the weak fixed point property. A Banach space X is orthogonally convex if for every weak-null sequence (xn) with
D(xn):=limsuplimsupllxm-xnll > 0 m n
there exists f3 > 1/2 such that
lim sup lim sup sup{lIzll : liz - xmll, liz - xnll ::::; f3l1xm - xnll} < D(xn). m n
198
The characteristic of a sequence, D(wn), first introduced in the above context and used in the proof of proposition 3.15, has played a crucial role in many of the more recent results. For example, in the proof that spaces with Kalton's property (M) have the weak fixed point property [1Ol where it was found necessary to employ a W similar to that used in the proof of proposition 3.15, but defined by the asymmetric constraints Il[wn]u - [anlll ::; 1/2 - 10 and D[wnl ::; 1/2 + 10, with 10 > o. Recall that X has property (M) if weak null types are constant on spheres about O. That is, limu Ilx - xnll = limu lIy - xnll whenever Ilxll = Ilyll and (xn) weakly converges to O. Starting with the proof of the Goebel-Karlovitz lemma, weak null types are seen to play an essential role in many aspects of metric fixed point t.heory. Indeed, understanding the behaviour of weak null types in a space is often the key to its fixed point properties.
Lin used a W defined by a combination of all the constraints discussed above to establish what is currently the best known bound for the stability of the fpp in £2 discussed in section 5.
For many of the results discussed in this chapter, and in many other applications, a Banach space ultrapower (X)u over N can be replaced by the space
£oo(X)/co(X),
where the quotient norm is canonically given by II [xnlll = lim sUPn Ilxnll, see for example: [4, 9, 10]. However, calculations in this space usually entail an infestation of subsequence taking. In many instances it is possible to avoid the use of these larger ambient spaces altogether; for example, see [7] where an ultrapower free proof of Maurey's result on the reflexive subspaces of £1 may be found. However, such proof often obscure the essential argument in a veritable plague of epsilons and deltas. None-the-less, the disadvantages and advantages are largely cosmetic and it is up to the individual to choose which approach is most to their taste.
References
[I] Aksoy, A. G. and Khamsi, M. A., Nonstandard methods in fixed point theory, Springer-Verlag, 1990.
[2] Baillon, J. B. and Schoneberg, R., Asymptotic normal structure and fixed points of nonexpansive mappings, Proc. Amer. Math. Soc., 81 (1981),257-264.
[3] Benyamini, Yoav and Lindenstrauss, Joram, Geometric Nonlinear Functional Analysis, Vol. 1, Amer. Math. Soc., Colloquium Publications 48, Providence Rhode Island, 2000.
[4] Borwein, J. M. and Sims, B., Nonexpansive mappings on Banach lattices and related topics, Houston J. Math., 10 (1984), 339-356.
[5] Dalby, T., Facets of the fixed point theory for nonexpansive mappings, Ph. D. dissertation, Univ. of Newcastle, Australia, 1997.
[6] van Dulst, D. and Pach, A. J., On flatness and some ergodic super-properties of Banach spaces, Indagationes Mathematical, 43 (1981), 153-164.
[7] Elton, J., Lin, P. K., Odell, E. and Szarek, S., Remarks on the fixed point problem for nonexpansive maps, Contemporary Math. 18 (1983), 87-120.
[8] Enflo, P., Banach spaces which can be given an equivalent uniformly convex norm, Israel J. Math. 13 (1973), 281-288.
[9] Garcia-Falset, J., The fixed point property in B spaces with NUS property, J. Math. Ann. Appl., 215 (1997), 532-542.
[10] Garcia-Falset, J. and Sims, B., Property (M) and the weak fixed point property, Proc. Amer. Math. Soc., 125 (1997), 2891-2896.
[ll] Jimenez-Melado, A. and Llorens-Fuster, E., A sufficient condition for the fixed point property, Nonlinear Anal., 20 (1993), 849-853.
REFERENCES 199
[12] Khamsi, Mohamed A. and Kirk, William A., An introduction to metric spaces and fixed point theory, John Wiley, 2000.
[13] Khamsi, M.A. and Thrpin, Ph., Fixed points of nonexpansive mappings in Banach lattices, Prec. Amer. Math. Soc., 105 (1989), 102-110.
[14] Lacey, H. Elton, The isometric theory of classical Banach spaces, Springer-Verlag, 1974.
[15] Lin, P. K., Unconditional bases and fixed points of nonexpansive mappings, Pacific J. Mat.h. 116 (1985), 69-76.
[16] Lin, P.K., Stability of the fixed point property of Hilbert spaces, Prec. Amer. Mat.h. Soc., 127 (1999), 3573-358l.
[17] Maurey, B., Points fixes des contractions de certains faiblement compacts de £1, Seminaire d'Analyse Fonctionnelle, Expose No. VIII (1980), 18.
[18] Pisier, G., Martingales with values in uniformly convex spaces, Israel J. Mat.h. 20 (1975), 326-350.
[19J Prus, Stanislaw, Multidimensional uniform convexity and uniform smoothness of Banach spaces, in: Recent. Advances on met.ric fixed point. t.heory, Editor: T. Dominguez-Benavides, Universidad de Sevilla, Serie: Ciencias, 48 (1986), 111-136.
[20] Prus, S., Banach spaces which are uniformly noncreasy, Nonlinear Anal., 30 (1987), 2317-2324.
[21] Sims, B., Ultra-techniques in Banach Space theory, Queen's Papers in Pure and Applied Mathematics, No. 60, Kingston, Canada (1982).
[22] Sims, B., Orthogonality and fixed points of nonexpansive maps, Prec. Centre Math. Anal., Austral. Nat. Uni. 20 (1988), 178-186.
[23] Sims, B., Geometric condition sufficient for the weak and weak' fixed point property, Proceedings of the Second International Conference on Fixed Point Theory and Applications, Ed. K. K. Tan, World Scientific Publishers (1992), 278-290.
Chapter 7
STABILITY OF THE FIXED POINT PROPERTY FOR NONEXPANSIVE MAPPINGS
Jesus Garcia-Falset
Department d'Analisi Matematica
Facultat de Matematiques
Universitat de Valencia
Doctor Moliner 50, 46100 Bur/assot, Spain
Antonio Jimenez-Melado
Department de Matematiq'ues
Universitat de Malaga
campus de Teatinos/290'l1
Malaga, Spain
Enrique Llorens-Fuster
Department d'Analisi Maternatica
Facultat de M atematiques
Universitat de Valencia
Doctor Moliner 50, 46100 Bur/assot, Spain
Enrique.Llorens. uV.es
1. Introduction
In 1971 Zidler [Zi 71] showed that every separable Banach space (X, II . II) admits an equivalent renorming, (X, II . 110), which is uniformly convex in every direction (UeED), and consequently it has weak normal structure and so the weak fixed point property (WFPP) [D-J-S 71],
Later, in 1981, Alspach [A 82] showed that the separable Banach space (L1[0, 1], 11·111) lacks the WFPP. As a consequence of both Zidler's and Alspach's results it was known that, in general, the WFPP is not preserved under topological isomorphisms.
Following on from this observation, came the well known, yet still open, problem in metric fixed point theory (see [E-L-O-S 83]): To find a nontrivial class of isomorphic Banach spaces such that each one of its members has the WFPP. A trivial example of
201
WA Kirk and B. Sims (eds,), Handbook of Metric Fixed Point Theory, 201-238. © 2001 Kluwer Academic Publishers,
202
such a class is the family of all Banach spaces isomorphic to £1 or to any other Schur space.
On the positive side of this problem there are some well known results that show that the WFPP is inherited from Banach spaces with nice geometric properties. This suggests that we may concentrate our attention on what is nowadays known as the stability problem, that is: Given a Banach space (X, II . II) with the WFPP, does this property extend to isomorphic spaces close enough to (X, II . II) in the Banach Mazur sense. Of course, the question we would like to answer is how far from X is the WFPP preserved. However, with the techniques currently available this is over ambitious and we must content ourselves with lower estimates on how far from X the space may be.
Notice that we may try to answer the above question, without appealing to equivalent renormings, by showing that the existence of a fixed point for the mapping T is guaranteed by some property of T inherited by those mappings that are nonexpansive with respect to an equivalent norm, for instance to be uniformly lipschitzian. From the early 1970's some fixed point results for this type of mappings were known. So we may start by considering uniformly lipschitzian mappings. To be more precise, we have the following elementary result.
Theorem 1.1 Let (X, II . II) be a Banach space such that every weakly compact convex subset of X has the fixed point property for uniformly k-lipschitzian mappings. If d(X, Y) < k then Y has the WFPP.
Proof. Since d(X, Y) < k there exists a bicontinuous isomorphism U : Y -+ X such that 11U1111U-1 11 < k. Suppose that C is a weakly compact convex subset of Y and T: C -+ C is II· lIy-nonexpansive. Then the mapping f : U(C) -+ U(C) given by
f(U(x)) = U(T(x))
is well defined and it is straightforward to see that for all positive integer n and all x E C,
Thus, for all x, y E C,
Ilfn(u(x)) - fn(U(y))lIx :::; 1IUIIIITn(x) - Tn(y)lIy
:::; 1lUllllx - ylly
:::; 1IUIIIIU- 1111IU(x) - U(y)lIx
for any positive integer n. This shows that f is uniformly >.-lipschitzian on the weakly compact convex subset UlC) with>. = IIUIIIIU- 1 11 < k. Consequently there exists U(x) E U(C) such that T(U(x)) = U(x). That is there exists x E C such that U(T(x)) = U(x), which implies that T(x) = x. •
This theorem shows the close relationship between fixed point theorems for uniformly lipschitzian mappings and bounds of stability for the WFPP. In fact, any fixed point theorem for uniformly k-lipschitzian mappings on weakly compact convex sets, yields k as a "radius of stability" for the WFPP.
In [G-K-T 74] Goebel, Kirk and Thele showed that weakly compact convex subsets of any Banach space (X, II . II) with characteristic of convexity co(X) < 1, have the FPP for uniformly k-lipschitzian mappings where k is a constant depending on the modulus of convexity of (X, II . II) . More precisely,
Stability of the F P P for nonexpansive maps 203
Corollary 1.2 Let (X, II . II) be a Banach space with EO(X) < 1. Let k > 1 be the unique solution of the equation
k (1- ox(k- l )) = 1. (1.1)
If d(X, Y) < k, then Y has the WFPP.
For a Hilbert space the solution of equation (1.1) is k = .../5/2.
Theorem 1.1 and the corresponding Corollary allow us to say that the WFPP is stable in spaces with EO(X) < 1, in the sense that it is retained when the norm is changed slightly.
In this same sense, some sufficient conditions for the WFPP are also stable. A typical instance of such a condition would be EO(X) < 1.
Theorem 1.3 [D-T 83J Let (X, 11·11) a Banach space with EO(X) < 1. If k is the unique solution of the equation (1.1), and d(X, Y) < k, then EO(Y) < 1.
Proof. There exists a bicontinuous isomorphism U : X --> Y such that 11U1111U-lll < k. Without loss of generality we may suppose that II U-lll = 1. Choose Yl, Y2 E By such that
IlUII IIYl - Y211y ~ k and take Xl := U-l(Yl), X2 := U- l (Y2). It is immediate that II xIII x ::; 1 and IIx211x ::; 1, and
II~II ::; IIYl - Y211y = IIUU-lYl - UU- lY2I1y ::; IIUllllxl - x2l1x. Thus, -Ie ::; IIXl - x211x, and by the definition of ox,
Therefore
This implies that
Oy e~lI) ~ 1 - 11U11 (1 - Ox G) ) > O. Thus EO(Y) ::; llifll < 1 and the proof is complete. • Although both Corollary 1.2 and Theorem 1.3 yield the same stability bound for the WFPP, we can see that they are results of different kind and that the second one is stronger.
The aim of this chapter is to give the best known stability bounds for the WFPP. Nevertheless, as the above results illustrates, one can find in the literature two main types of stability theorems, depending on one or other of the following schemes.
A) If (X, 11·11) has property (P) and (Y, 1·1) is a Banach space isomorphic to X, then (Y, I . I) has (P) whenever d(X, Y) < k(X).
B) If (X, 11·11) has property (P) and (Y, 1·1) is a Banach space isomorphic to X, then (Y, I· I) has the WFPP whenever d(X, Y) < k(X).
Here (P) is a geometric property that implies the WFPP.
204
2. Stability of normal structure
Although weak normal structure (WNS) is the most fruitful sufficient condition for the WFPP, it is not a property that is easy to check. It is perhaps for this reason that many sufficient conditions for WNS have been studied. We examine in this section the stability of some of them.
2.1. Behaviour of Bynum's coefficients [By 80]
Bynum's paper [By 80], p. 432 reads: When any of the normal structure coefficients is greater than one, this condition is contagious.
The following definitions are stated according to Bynum's paper [By 80].
Definition 2.1 Let (xn) be a bounded sequence in the Banach space (X, II . III . The real number
diama ((xn»:= lim (sup Ilxi - xjll : i,j ~ n) n->oo
is called the asymptotic diameter of the sequence (xn). In the same way, the real number
ra((xn» := inf{limsup Ily - xnll : y E co((xn))}
is called the asymptotic radius of the sequence (xn).
The normal structure coefficient of X is defined by
N(X) := inf {di;0~) : C bounded, convex, C C X, diam(C) > o} and we say that X has uniform normal structure (UNS) if N(X) > 1.
Definition 2.2 The weakly convergent sequence coefficient of X, denoted by WCS(X) is defined by
WCS(X) := sup {M: V'(xn) E WCS(X) 3y E co{xn} such that
M lim sup Ilxn - yll ::; diama ((xn» }. n
Here WCS(X) is the set of all the weakly convergent sequences in X.
Bynum's theorems below seem to be the oldest results on stability of normal structure.
The normal structure coefficient of X: N(X).
Theorem 2.3 For X, Y isomorphic Banach spaces, N(X) ::; d(X, Y)N(Y).
Corollary 2.4 If N(X) > 1 and d(X, Y) < N(X) then Y has UNS.
Proof. From the above theorem
N(X) < N(Y). d(X,Y) -
Thus, N(Y) > 1 and Y has uniform normal structure. •
Stability of the FPP for nonexpansive maps 205
The weakly convergent sequence coefficient of X: WCS(X). Bynum's weakly convergent sequence coefficient, WCS(X), is perhaps one of most widely studied in metric fixed point theory. In particular one can find in the literature many different equivalent definitions for it. Among others, we have the following.
Definition 2.5 For each bounded sequence (xn) in the Banach space (X, II . II) we define the following constants.
The diameter of (xn ):
The sepamtion of (xn) :
sep«xn)) := inf{llxm - xnll : m # n}.
We will use the following notation
It is easy to see that, for each bounded sequence (xn) in X,
We recall the following combinatorial result.
Ramsey's Theorem. Suppose n E Nand C is an infinite set. If Pn(C) is the collection of all the subsets of C with n elements, and
Pn(C) = AUB,
then there exists an infinite subset D of C such that Pn(D) C A or Pn(D) C B.
Lemma 2.6 Let C a bounded infinite set and r :::: O. Then there exist a sequence (Yn) with infinite mnge in C so that either diam«Yn)) ::; r or sep«Yn)) :::: r.
Proof. We have
where A:= {{x,y} E P2(C): d(x,y)::; r}
and B:= {{x,y} E P2(C) : d(x,y) :::: r}.
Using Ramsey's theorem we obtain an infinite subset D of C such that P2(D) C A or P2(D) C B. We can define in D an infinite sequence (Yn). If P2(D) c A then diam«Yn)) ::; T. If P2(D) c B then sep«Yn)) :::: r. •
Because of the indefiniteness of Y in the expression which defines WCS(X) it is convenient to get easier equivalent formulae for this coefficient. Among others, we have the following.
206
Lemma 2.7 Let (X, II . II) be a non-Schur Banach space. The following constants are equal.
a) WCS(X).
b) A (X):= inf {di;~a~~»)) : (xn) is a weakly (non strongly) convergent seq. in X}.
c) (3(X) :=inf{D[(xn)] :xn~O, IIxnll ~ 1}.
d) B(X) := sup{M > 0: V(xn), with Xn ~O, Mlimsup IIxnll ::; diama((xn))}.
e) C(X) := inf{diama((xn)) : Xn ~O, Ilxnll ~ I}.
Proof. Let
Mo E {M: V(xn) E WeS(X) :3y E co{xn} s.t. Mlim:up Ilxn - yll ::; diama ((Xn))}
where WeS(X) is the set of all the weakly convergent sequences in X. Let WeS'(X) be the set of all the weakly convergent but not norm convergent sequences in X.
In particular, for each (xn) E WeS'(X) there exists y E co{xn} such that
... diama ((xn)) diama ((xn)) 1V10 < < .
- limsuPn Ilxn - yll - ra((xn))
Thus, taking the infimum on the right hand side of the above inequality,
Then it follows that, WCS(X) ::; A(X).
On the other hand, fix). > 1. Then for all (xn) EWeS', ra((xn))). > ra((xn)) and hence there exists y E col {xn}) such that
).ra((Xn)) > lim sup Ilxn - yll 2: ra((Xn)). n
This gives 1 1
).ra((xn)) < limsuPn Ilxn - yll'
Hence
which yields
A~X) lim:up IIxn - yll ::; diama ((xn)) .
In summary, for each (xn) E WeS'(X) there exists y E co((xn)) for which the above inequality holds. Of course, if (xn) is a convergent sequence, then there exists y = limn Xn for which this inequality also holds. By the definition of WCS(X) this gives
AC:) ::; WCS(X).
Letting A ~ 1+ we have A(X) ::; WCS(X). We thus get A(X) = WCS(X).
Stability of the F P P for nonexpansive maps 207
Let Mo be such that for all weakly null sequence (xn ),
Mo lim sup Ilxnll ::; diama((xn)).
In particular, if (xn) is weakly null and IIxnll ----> 1, Mo ::; diama((xn)). Therefore Mo ::; C(X), and hence B(X) ::; C(X).
On the other hand, let (xn) be a weakly null sequence. If p := lim sup IIxnll > 0 then for a subsequence (xnk ) of (xn)
1 w 1 -xnk ~O and II-xnkll----> 1. p p
Then
That is, C(X)p = C(X) lim sup IIxnll ::; diama((xn))
11.
which holds trivially if p = O. This yields C(X) ::; B(X) and thus we have shown that C(X) = B(X).
Since for each bounded sequence (xn) in X, D[(xn)] ::; diama((xn)), it is immediate that f3(X) ::; C(X).
Suppose, for a contradiction, that f3(X) < C(X). Then there exists a sequence (xn) with xn~O and IIxnll-> 1, such that D[(xn)] < C(X)::; diama((xn ))::; diam((xn )).
Let r be such that D[(xn)] < r < C(X). As the set {xn : n E N} is infinite (since Xn ~ 0 and IIxnll -> 1), then we can apply Lemma 2.6 to this set. Hence there exists a subsequence (Yn) of (xn) such that either diam((Yn)) ::; r or sep((Yn)) 2: r. But Yn ~ 0 and IIYnll ----> 1. Hence r < C(X) ::; diama((Yn)) ::; diam((Yn)), and then diam((Yn)) ::; r which is absurd. Thus we have
which is absurd again. Hence we have that f3(X) = C(X).
Finally we will show that WCS(X) = B(X). Let M be a constant such that for all weakly null sequences (xn), M lim sup Ilxnll ::; diama((xn)). Let (Yn) be a sequence which converges weakly to Y EX. Thus (Yn - y) is weakly null and hence
Mlimsup llYn - yll ::; diama((Yn - y)) = diama((Yn)).
As Y E cO({Yn : n EN}, M::; WCS(X). Thus, B(X) ::; WCS(X).
On the other hand, let
Mo E { M : \f(xn) E WCS(X) 3y E co{xn} S.t. Mlim;up Ilxn - yll ::; diama ((xn)) } .
Let (Yn) be a weakly null sequence in X, and define
Ak := cO{Yn : n 2: k}.
An easy application of Mazur's theorem (see [A-D-L 97]) gives
00 n Ak = {O}. k=!
208
The function q, : X -> R defined by q,(z) := limsuPn liz - Ynll is norm-continuous and convex. Hence it is weak sequentially lower semicontinuous. Since each Ale is weakly compact, for each positive integer k there exists Zle E Ale such that
q,(ZIe) = inf{ q,(z) : z E Ale}.
As {Zle : kEN} c cO{Yn : n E N} and cO{Yn : n E N} is weakly compact, (ZIe) admits a subsequence (Znp) such that Znp:!'!.z E cO{Yn : n EN}. Bearing in mind that the sequence of sets (Ale) is decreasing, as well as Lemma 1.9. of [A-D-L 97),
As 0 E Ale for each positive integer k, and the sequence (q,(ZIe)) is nondecreasing because Ale J Ak+1, there exists
lif q,(ZIe) ::; q,(0) ::; lif q,(Zk)·
For each positive integer k let us consider the sequence (Yn+k)n. By definition of Mo, there exists Y E cO{Yn+k : n ~ I} such that
Mo lim sup Ily - Yn+kll ::; diama((Yn+k)n). n
Thus,
Moq,(zle)::; Molimsup Ily - Yn+kll ::; diama((Yn+k)n) = diama((Yn)). n
Letting k -> 00,
That is Mo lim sup IIYnll ::; diama((Yn)).
n
It follows that WCS(X) lim sup IIYnll ::; diama((Yn)),
n
which yields WCS(X) ::; B(X), and the proof is complete. • The coefficient (3(X) was defined by Jimenez-Melado ((J 92]) twelve years after Bynum's paper was published. In addition, he defined a sufficient condition for weak normal structure as follows.
Definition 2.8 A Banach space (X, 11·11) has the generalized Gossez-Lami Dozo property (GGLD) whenever D[(xn)] > 1 for every weakly null sequence (xn) such that lim IIxnll = 1.
Of course, (3(X) > 1 =? X has the GGLD property.
One of the most important results about stability of weak normal structure is the following one.
Theorem 2.9 [By 80] For X, Y isomorphic Banach spaces,
WCS(X) ::; d(X, Y) WCS(Y)
Stability of the FPP for nonexpansive maps 209
or equivalently f3(X) :::; d(X, Y) f3(Y).
Proof. Let U : Y --> X be a bicontinuous isomorphism. Given any weakly null sequence (Yn) in Y with IIYnll --> 1, there exists a subsequence (Ynk) of (Yn) such that
As
letting n --> 00 we have
In particular A > O.
The sequence GU(Yn'»)k>O is weakly null in X and "!:U(Yn~)" --> 1. Thus, by definition of f3(X) , we have: -
f3(X) :::; D [GU(Ynk ») k~O] = lim:up [lim:up II~U(Ynp) - ~U(Ynq)ll]
:::; lim:up [lim:up II~U(Ynp) - ~U(Yn)ll]
:::; lim~up [lim:up II~U(Ym) - ~U(Yn)ll] :::; 1I~lIlim~up [lim:up IIYm - Ynll]
= ll.!!llD[(Yn)J A
:::; 11U1l11U-1IID[(Yn)J.
As the inequality f3(X) :::; 11U1111U-1IID[(Yn)J holds for each weakly null sequence (Yn) in Y with IIYnl1 --> 1, it is immediate that
f3(X) :::; 11U1l11U-11lf3(Y). Finally, as the above inequality holds for each bicontinuous isomorphism U : Y --> X,
f3(X) :::; d(X, Y)f3(Y)
which completes the proof. • Corollary 2.10 [By 80J, [J 92J If WCS(X) > 1 and d(X, Y) < WCS(X) then Y has the GGLD property, and hence WNS and the WFPP.
Proof. By the above theorem we have that
WCS(X) < WCS(Y) d(X,Y) - .
Thus, WCS(Y) > 1 and Y has weak normal structure. •
210
We present now a well known computation of weakly convergent sequence coefficient for X = ip that yields stability results.
1 Corollary 2.11 Let 1 < p < 00. flY is a Banach space such that d(ip, Y) < 2" then Y has the GGLD property and hence NS and the FPP.
Proof. Let (xn ) be a weakly null sequence in i p. Given x E ip it is well know that
lim sup IIxn - xll~ = IIxll~ + lim sup IIxnll~·
Thus, for each positive integer m,
lim sup IIxn - xmll~ = IIxmll~ + lim sup IIxnll~. n n
If, in particular, IIxn ll p -t 1, then
lim,:up (limnsup IIxn - xmll~)) = 2,
1 1
that is, D[(xn)] = 2". Hence WCS(ip) = f3(ip) = 2;;. • The coefficient WCS(X) has been computed for many other Banach spaces. The reader is referred to the book [A-D-L 97] for a comprehensive study. We list some of these computations.
1 For x E ip, 1 :S p < 00, we denote by x+ and x- the vectors whose i component are respectively
x+(i) .- max{x(i) O} - x(i) + Ix(i)1 x-(i).= max{-x(i) O} = -x(i) + Ix(i)1 .- , - 2 ,. , 2'
For any q E [1, 00) and for x E ip we denote
1
IIxllp,q := (lIx+lI~ + IIx-II~)-IIxllp,oo := max{llx+llp, IIx-lIp}'
It is easy to check that all these norms are equivalent to the usual norm in i p .
The Banach spaces ip,q := (ip, 11·llp,q) were introduced by Bynum. In [D-L-X 96] the authors showed that for p E (1,00) and q E [1,00),
1 1
WCS(ip,q) = min{2",2-}.
2 For f3 > 1 the Banach space E{3, introduced by James, is i2 renormed accordingly to Ixl{3 = max{lIxI12,f3l1xlloo}. It was shown in [A-X 93] that
WCS(E ) = {Yj 1:S f3v'2 {3 1 v'2 < f3 < 00.
3 Let J be the real James space which consists of all real sequences x = (xn ) for which limxn = 0 and IIxll < 00, where
IIxll := sup {[(xPl - xP2 )2 + ... + (Xp=_l - xp=)2 + (xp= - XPl)2]!}
Stability of the FPP for nonexpansive maps
and the supremum is taken over all choices of m and Pl < P2 < ... < Pm.
Other equivalent norms on J are
and
211
In [J 92J Jimenez-Melado showed that WCS(J) = VI. Later on, Dominguez
and Xu [D-X 95] showed that WCS((J, 11·112)) = WCS((J, 11·111)) = viz. We will write Ji for the space (J, 11·lli) (i = 1,2).
4 Let (D, I:, p,) be a a-finite measure space, 1 ::; P < 00, and assume that IJ'(D) is infinite dimensionaL Then
N(IJ'(rl)) = min {2l-~, 2~ } .
Moreover WCS(IJ'(D)) = N(IJ'(D)) if either P ::::: 2 or p, is not purely atomic. (See [A-D-L 97]).
As a direct consequence of the computations of WCS(X) we have the following stability result.
Corollary 2.12 a) Let 1 < P < 00. IfY is a Banach space such that
d ( £p,q, Y) < min { 2 ~ , 2 ~ }
then Y has the GGLD property and hence NS and the FPP.
b) Let 1 < f3 < viz. If Y is a Banach space such that d(E{3, Y) < vIz/ f3 then Y has the GGLD property and hence NS and the FPP.
c) If d(Y, J) < J372 then Y has property GGLD and hence WNS and the WFPP.
d) If d(Y, IJ'(D)) < min {2l-~, 2~} then Y has property NS and hence the FPP.
2.2. Further remarks about the stability of WNS
Next we list some stability theorems for some other conditions which are sufficient for normal structure.
Stability of condition cl(X) < 1 [S 86], [G-K 90] .
Theorem 2.13 Let X be a Banach space and let Xl := (X, II· lid and X2 := (X,II·12) where II . 111 and 11·112 are equivalent norms on X where there exists a, f3 > 0 such that,
212
for all x EX. If k := ~ then
Consequently, if Cl (Xl) < 1 and
Stability of condition ex(X) < i [Ba 87] .
Theorem 2.14 Let X be a Banach space with cx(X) < ~. Let B > 1 satisfy
1 - ~ = ~x (2~) . (which exists in view of continuity of the function ~x) If Y is another Banach space with d(X, Y) < B then ~x(Y) < 1/2.
Bernal-Sullivan convexity property.
Theorem 2.15 Let (H, 11·11) be a Hilbert space, and let 1·1 be a norm on H such that, for all x E H,
1 I3lxl ::; Ilxll ::; Ixi
for some f3 with 1 ::; f3 <.J2. Given e > 0, there exists 6> ° and M, a positive integer, such that for m ~ M, if xl, ... ,Xm E B(HJI) and
1~(Xl + ... +xn)r > 1- 6,
then D(I·I,Xl, ... ,xm ) < e. Here
D(I·I,Xl, ... ,Xk+l) = IXk - Xk_ll·dist (Xk-b [Xk,Xk+l]) .... . dist (Xl, [X2, ... ,Xk+l])'
where [Xi, ... ,Xj] is the affine span of the vectors {Xi, ... ,Xj}.
Corollary 2.16 Let (H, 11·11) be a Hilbert space, and let 1·1 be a norm on H such that, for all X E H,
1 I3 lxl ::; IIxll ::; Ixi
for some f3 with 1 ::; f3 <.J2. Then (H, I .1) has normal structure.
3. Stability for weakly orthogonal Banach lattices
In 1980-81 Maurey [M 80] showed that the classical Banach space Co enjoys the WFPP. In order to generalize Maurey's ideas to a larger class of Banach lattices, Borwein and Sims [B-S 84] introduced the notion of a weakly orthogonal Banach lattice as well as the concept of Riesz angle.
Stability of the FPP for nonexpansive maps 213
Definition 3.1 A subset C of a Banach lattice X will be called weakly orthogonal if
lim inf lim inf Illxn - xol!\ IXm - xolll = 0 n m
whenever (xn ) is a sequence in C, weakly convergent to Xo E C. A Banach lattice X is said to be weakly orthogonal whenever all its weakly compact subsets are weakly orthogonal.
It is not hard to show that Co, c, ip (1 5 P < 00) are weakly orthogonal while 100 and non-atomic lJ' spaces are not.
Definition 3.2 The Riesz angle of a Banach lattice X is defined by
a(X) := sup{lIlxl V Iylll : x,y E Ex}.
The fundamental inequality involving the Riesz angle is:
Proposition 3.3 Let (X, II . II) a Banach lattice and consider x, y, z E X. Then
Ilzll 5 a(X) (11x - zll V liz - yll) + Illxl !\ Iylll·
Proof. By the definition of Riesz angle it is clear that
III x-z II y-z III Illx-zlvly-zlll a(X) 2: Ilx _ zll V lIy - zll V IIx - zll V lIy - zll = Ilx - zll V lIy - zll .
Therefore,
Illx - zl V Iy - zlll 5 a(X)(llx - zll V lIy - zll)· (3.1)
On the other hand, since
Izl 5 (Ix - zl V Iy - zl + Ixl) !\ (Ix - zl V Iy - zl + IYI)
we have
Izl 5 Ilx - zl V Iy - zl + Ixl!\ IYII· (3.2)
Since X is Banach lattice from (3.2) we obtain that
IIzlls Illx - zl V Iy - zlll + Illxl!\ Iylll· (3.3)
Finally by (3.1) and (3.2) we conclude
IIzll 5 a(X)(lIx - zll V liz - yll) + Illxl!\ Iylll·
• By using these notions they obtained the following stability result.
Theorem 3.4 [B-S 84] If X is a weakly orthogonal Banach lattice such that
d(Y, X)a(X) < 2
then Y has the WFPP.
214
Proof. By standard procedures we can suppose, to get a contradiction, that T is a nonexpansive mapping on a minimal invariant weakly compact subset C of X such that diam(C) = 1, and we can select a weakly null approximately fixed point sequence {xn}.
Since d(Y, X)a(X) < 2 we can pick an isomorphism U of Y onto X with
Since X is weakly orthogonal, we can find subsequences {an} and {Yn} of {Xn} such that
lim IllUanl II IUYnll1 = O. n~oo
Since any approximate fixed point sequence is diameterizing we can also assume that
lim Ilan - Ynll = 1-n~oo
On the other hand, it is well known that given {an} and {Yn} we can consider another approximate fixed point sequence {zn} satisfying
1 lim Ilan - znll = lim Ilzn - Ynll = -2'
n---+oo n---+oo
Now by using the above proposition, we have
lim sup IIUznl1 S a(X) limsup(IIUan - U znll V IIUYn - Uznll) +limsup(IIIUanl II IUYnlll)· n n n
Therefore,
which is a contradiction. • As a consequence of the above theorem it is clear that every weakly orthogonal Banach lattice (X, II . II) with Riesz angle a(X) < 2 has the WFPP. However, this result does not work when the Riesz angle is equal to 2. To avoid this difficulty in 1988 Sims lSi 88] introduced a strong version of weakly orthogonal Banach lattices as follows.
Definition 3.5 Let (X, 11·11) a Banach lattice. We say that X is strong weakly orthog-onal if
lim Illxnl II Ixlll = 0 n~oo
for all x E X, whenever {xn} is a weakly null sequence.
For this class of Banach lattices, it was shown in lSi 88] that
Theorem 3.6 a) If (X, 11·11) 'is a strongly weak orthogonal Banach lattice then (X, II· II) has the WFPP.
b) If (X, II . III is a strongly weak orthogonal Banach lattice then (X, II . III has the non-strict Opial property.
Recall that a Banach space (X, II . II) has the non-strict Opial property if for every weakly null sequence (xn) in X the following inequality holds
liminf Ilxnll S liminf Ilxn - xii
Stability of the F P P for nonexpansive maps 215
for every x E X.
It is not difficult to see that the Banach lattices Co and £p (1 :::; p < 00) are strongly weak orthogonal. Since it is well known that c does not enjoys the non-strict Opial property, then c cannot be a strongly weak orthogonal Banach lattice.
Stability results for the strong weakly orthogonality have been found by Sims lSi 86], Khamsi and Turpin [Kh-T 89J (for other topologies besides the weak topology) and Dalby [Da 97]. The 'radius of stability' in each case was, respectively, V5 ~ 1, 4/3, (v'33 ~ 3)/2. In this context the following observation due to Garcia-Falset [GF 94] is of interest.
Let (X, II . II) a Banach space. In [GF 94] the author defines the coefficient
R(X) := sup{lim inf IIxn + xii}
where the supremum is taken over all x E Ex and all weakly null sequences in Ex.
Proposition 3.7 If (X, 11·11) is a strong weakly orthogonal Banach lattice, then R(X) :::; a(X).
Proof. Let {xn } be a weakly null sequence in Ex and let x E Ex. It is well known that
Then
Therefore, by the definition of the Riesz angle
Illxnl + Ixlll :::; a(X) + Illxnllllxlll·
Since X is a strong weakly orthogonal Banach lattice, we have
liminf Illxnl + Ixlll :::; a(X), n
and hence,
lim inf Ilxn + xii:::; lim inf Illxn + xiii:::; lim inf Illxnl + Ixlll :::; a(X). n n n
This implies that R(X) :::; a(X). • Later, in [GF 97] the author showed that a Banach space (X, II . II) has the WFPP whenever R(X) < 2 and an easy computation allows us to see that if (X, II . II) and (Y, I . I) are isomorphic Banach spaces then R(Y) :::; d(X, Y)R(X). Thus, in order to obtain stability results for strongly weakly orthogonal I3anach lattices (X, II . II) it is clear from the above proposition that computing the coefficient R(X) is better than a computation of the Riesz angle. Moreover, the Banach space £p,OC>l which is £p renormed by Ixl := max{llx+llp, IIx-llp}, does not have asymptotically normal structure [By 80J and it is not a Banach lattice. However, we have
Proposition 3.8 [GF 94] A Banach space (X, II . II) has the WFPP whenever there exists 1 < p < 00 such that
216
1
Proof. It is enough to show that R(f!p,oo) < 2".
Let (xn) be a weakly null sequence in the unit ball of f!p,oo and let x be an element in the unit ball. Then we have
Since (xn) is a weakly null sequence there exists a subsequence (xnk ) of (xn) and a sequence (Pk) of projections of the form Pk = P[ak.b.]' where
and limk~oo ak = 00, such that
(3.4)
and
(3.5)
Moreover, we can assume that x has finite support, that is x = P[a,b]X, Therefore, there exists ko E N such that ak > b for all k 2: ko.
Consequently, for each k 2: ko we have
(3.6)
and
(3.7)
On the other hand, by (3.5) we have
liminf IXn + xl::; liminf IXnk + xl::; liminf IPkXnk + xl n~oo k-----+oo k---+oo
(3.8)
and then by using (3.4), (3.6), (3.7) and (3.8) we obtain
lim inf IXn + xl ::; lim inf !PkXnk + xl n--oo k---+oo
::; max {liminf IlpkX;; + x+11 ,liminf IlpkX;;:k +x-II } ::; 2~. k---+oo k P k-----+oo P
1
Since (xn) and x are arbitrary, we get that R(f!p,oo) ::; 2". • On the other hand, if (X, 11·11) is a weakly orthogonal Banach lattice which is not strong weakly orthogonal, the coefficient R(X) can be bigger than the Riesz angle of (X, 11·11)
This is the case for the classical space (e, 11.11(0)' Indeed,
Proposition 3.9 R(e, 11.11(0) = 2 and a(e, 11.11(0) = 1.
Proof. First, let us see that ate) = 1. Indeed, let x,y E e with Ilxlloo ::; 1 and IIYlloo ::; 1. It is clear that given € > 0 there exists n E N satisfying
Illxl V Ivili oo - c: <:: IXnl V IYnl
Stability of the FPP for nonexpansive maps 217
but, we know that
Therefore,
Illxl V Iyliloo - c s Ilxll oo V IIvlloo.
Consequently a( c) = l.
Second, if we consider the sequence (en) where en = (l5i ,n) and the element e (1,1,1, ... ) it is easy to see that lien + ell 00 = 2 and therefore R(c) = 2. •
As a consequence of both Borwein-Sims's result and the above comment we have the following stability corollaries.
Corollary 3.10 [B-S 84] If X is a Banach space such that
d(X,Cp )) < 21-~
for some 1 < p < 00, then X has the WFPP.
Corollary 3.11 [B-S 84] If X is a Banach space such that
d(X,c(S))) < 2 or d(X,co(S))) < 2
then X has the WFPP.
4. Stability of the property M(X) > 1
Inspired by the proof of [GF 97], Dominguez-Benavides [D 96] in 1996 improved the stability constant 2R(X)-1. In order to do this, he stated the following definitions.
Definition 4.1 ([D 96]) Let (X, II . II) be a Banach space. We define the coefficient M(X) as
SUP{R~:;) :a~o} where for a nonnegative real number a,
R(a,X) := sup {liminf IIx +xnll},
where the supremum is taken over all x E X with Ilxll s a and all weakly null sequences (xn ) in the unit ball of X such that D[(xn )] S 1.
It is shown in [D 96] that a Banach space Y has the WFPP if there exists a Banach space X such that
{ I +a } d(X, Y) < M(X) = sup R(a, X) : a ~ 0 .
It turns out that for 1 < p < 00,
and that for a Hilbert space H, M(H) = v'3. Hence, the result of [D 96] improves all previously known stability results in Hilbert spaces and Cp , 1 < p < 00. See [B-S 84]' [By 80], [Pr 90], [J-L 92]' [Kh 94]' [J-L 93].
218
In this section we improve the stability constant M(X) for a certain class of Banach spaces. To do this, we develop the arguments due to Jimenez and Llorens (J-L 00].
Recall that a Banach space X has the uniform Opial property [Pr 92] if for every c > 0 there exists r = r( c) > 0 such that 1 + r ::; lim inf IIx + Xn II for every x E X with IIxll ? c and every weakly null sequence (xn) in X with lim inf IIxnll ? 1.
To any Banach space X we can associate its Opial's modulus [L-T-X 95] by
rx(c) = inf {liminf IIx + xnll-1}
where c ? 0 and the infimum is taken over all x E X with IIxll ? c and all weakly null sequences (xn) in X with liminfllxnll? 1.
In order to simplify the statement of the following theorem, for B ? 1 we define the number
Cx(B) := sup{c? 0 : rx(c) ::; B-1}.
The inequality (see [X 97]) c - 1 ::; rx(c) ::; c
holds for all c ? 0, so Cx(B) ? B-1.
Theorem 4.2 Let X be a Banach space with norm II . II and denote by rx its Opial's modulus. Let I· I be a norm on X equivalent to II . II and let B be a positive real number such that, for all x EX,
IIxlI::; Ixi ::;Bllxll·
Let Y be the Banach space X with the new norm 1·1 and let C(B) = Cx(B). If B is a solution of the inequality
{ l+a
B < sup R (iC(B),X)
then Y has the weak fixed point property.
To prove the above theorem we need some notation and a preliminary lemma.
Let (X,II ·11) be a real Banach space and let loo(X) denote the space
{(Yn) : Yn E X and {IIYnll} E loo}
with the norm II(Yn)III~(X) := sUPn IIYnll and let co(X) be the closed subspace of loo(X)
{(Yn) : Yn E X and {IIYnll} E eo}.
Set x = loo(X)/eo(X).
Thus for any [Yn] EX,
II [Yn] Ilx = lim sup IIYnll· n~oo
Finally, given a nonexpansive mapping T : K -> K where K is a nonempty, convex weakly compact subset of X we consider
k = {[Yn] EX: Yn E K}.
Stability of the FPP for nonexpansive maps 219
Clearly, k is a closed convex subset of X. Let T : k -+ k be the mapping defined by
Since T is nonexpansive on K, T is a well-defined nonexpansive mapping on k. Moreover, if (:z;,n) is an approximate fixed point sequence (a.f.p.s.) in K, then [xnJ is a fixed point ofT.
Let (xn) be any fixed a.f.p.s. of T. By passing to a weakly convergent subsequence of (xn), and then translating K, we may assume that (xn) converges to 0 weakly. For convenience, we assume that diam(K) = 1 and we denote the fixed point [xnJ by x. For any 0 < t < 1 define, in X,
wl = B(1-t) [xnJ
wl = {[WnJ E k: lim:uplim~up IIwm - wnll s t} wl = {[WnJ E k : 3x E K with II [wnJ - xlix s t} .
Each set is closed, convex and T invariant, so
is closed, convex and T invariant and non-empty as tx E Wt .
Lemma 4.3 [L 99J For 0 < t < 1, let [wnJ be an element in Wt .
(a) limn ..... oo Ilxn - wnll = 1 - t.
(b) There is x E K such that liilln--+oo IIwn - xii = t.
(c) For any x E K, liminfn--+oo IIwn - xii ~ t.
(d) limsupm ..... oo limsupn ..... oo Ilwn - wmll st. Moreover, if (wnk ) converges weakly to W E K, then limn--+oo Ilwn - wil = t and lim sUPk-->oo IIxnk - wnk + wll ~ 1 - t.
Proof of Theorem 4.2. First, let us observe that we can suppose that there exists a > 0 such that
l+a R (~C(B),X) > B.
Suppose, for a contradiction, that Y lacks the WFPP. Then let t = 1/(1 + a) and consider the corresponding set Wt as in Lemma 4.3. From Lin's lemma [L 85], Wi contains elements of norm arbitrarily close to l.
Let [znJ be any element of Wt . Since K is weakly compact, there exists a subsequence (znk) of (zn) which converges weakly to y E K. By Lemma 4.3, (d) we have that
lim IIzn - yll st. n ..... oo
Again from Lemma 4.3 (d) and the relationship between the norms we get that
1 1- t lim IIxnk - znk + yll ~ B- lim sup IXnk - Znk + yl 2: -B .
k--+oo k---+oo
220
Since t < 1, we can consider the sequence (Uk) defined by
B Uk = -1- (-xnk + znk - y).
-t
Then (Uk) is weakly convergent to 0 and limk_oo IIUkll ~ 1. By the definition of rx we have that
liminf IIUk + ~YII ~ 1 + rx (I B IIYII) . k ..... oo 1 - t - t
On the other hand, since Uk + l~tY = (Znk - Xnk)l~t' we have
liminf IIUk + IB yll = 1 B lim IIznk - xnkll ::; IB t lim sup IZnk - xnkl ::; B. k_oo - t - t k-oo - k-oo
Hence, we have B-1 ~ rxC~tIlYII) and then lIyll ::; lstC(B).
Pick any c: > O. Since limk_oo IIznk - yll ::; t, there exists a subsequence (Yk) of (znk) such that IIYk - yll < t + c: for every positive integer k.
On the other hand, we have that
D (Yk - Y) = D (~) < D (~) < _t < 1 t+c: t+c: - t+c: - t+c:
and also that
II -Y-II < I-tC(B) < I-tC(B) =aC(B). t+e: - t+c: B - t B B
i,From the above facts we conclude that
lim II ~ II = lim II Yk - Y + -Y II::; R (a CB(B) ,X) k ..... oo t+c: k ..... oo t+c: t+c:
and, since c: > 0 is arbitrary,
. 1 (C(B) ) hm IIYkll::; --R a-B ,X . k-oo 1 + a
Finally, we get that
. . B (C(B) ) I[znll = hm IZnkl = hm IYkl::; --R a-B ,X < 1 k ..... oo k ..... oo 1 + a
which contradicts Lin's lemma. • Corollary 4.4 Let X be a Hilbert space with norm II . II. Suppose that I . I is a norm on X such that
IIxll ::; Ixi ::; Bllxll for all x E X.
If B < )2 + y'2, then (X, I . I) has the WFPP.
Proof. Since II . II is an Euclidean norm, we have rx(c) = VI + c2 - 1 for all c ~ 0 and C(B) = ~. On the other hand,
R(a,X) = J~ +a2
Stability of the FPP for nonexpansive maps
for all a ~ 0 and
R (a C(B) x) = V~ + a2B2 - 1. B ' 2 B2
By the Theorem, we only have to show that
That is, that there exists a ~ 0 such that
which is equivalent to
F(a):=2(1+a)2~a2 >B2. 1 + 2a
221
The left hand side F(a) in the above inequality takes its maximum value at ao = V2/2. Then a solution of the above inequality is every B > 1 such that
F( ~) >B2
and this happens if B < ,/2 + )2. •
In order to obtain a similar result for Cp when 1 < p < 00 and p =1= 2 we will need to find the maximum of the positive real function
F (a) := 2 (1 + a)P + aP p 1 + 2aP
defined for a ~ O.
It is easy to see that 2 = Fp(O) = lim Fp(a).
a-+oo
On the other hand, for nonnegative a
F;(a) = (2aP2: 1)2 (aP- 1 + (a + 1)P-l - 2 (a(a + l)P-l) .
Since the term (2a;~1)2 is positive, the sign of the derivative F'(a) is the same as the sign of the continuous function
Gp(a) := aP- 1 + (a + l)p-l - 2 (a (a + lW-1 .
We have
G - = - > o. ((l)P:l) 1 p 2 2
We will give the argument only for p > 2. Since for p > 2 we have
222
< v1 + (1 - V2) ( v1 + 1) p-l
< v1 + (1 - V2) ( v1 + 1) == 0,
there exists ap in the open interval
for which F~(ap) == O. Then, using arguments from elementary calculus we conclude that ap is the unique maximum of Fp.
1
Corollary 4.5 Let p > 1 be a real number, and cp := (Fp( ap)) 1'. Suppose that I . I is a norm on ip such that, for all x E i p,
Ilxll :S Ixl :S Bllxll where II . II is the ordinary norm of i p. If B < cp, then (ip, I . Il has the WFPP.
1
Proof. Since (see [X 97]) rcp(c) = (1 + cP)1' - 1 for all c:::: 0, we have
1
GCp(B) == G(B) = (BP -1)1'.
On the other hand, (see [D 96]), for all a 2': 0
1
( G(B) ) _ (1 aP(BP -1)) P R a----rJ' ip - "2 + BP .
By the Theorem, we only have to show that there exists a :::: 0 such that
That is
This is equivalent to
l+a l+a <?J!!l - 1 > B.
R(a B ,ip) (! + ap(BP-l)) l' 2 Bp
( 1 BP-1) (1 +aJP > BP - +aP-- . 2 BP
2 (1 + a)P + aP > BP. 1+2aP
The left hand-side Fp(a) in the above inequality attains its maximum value at some ap > (1/2)1/(p-l). Then a solution of the above inequality is every B > 1 such that
and this happens if B < cpo • Remarks
Stability of the FPP for nonexpansive maps 223
1. The effective computation of the constant cp in the above corollary seems to be hard, specially if p is not a natural number. Nevertheless, it is clear that
The right hand-side in this inequality furnishes bounds of stability in ip that are greater than M(ip), at least for p < 6.
2. Let X be a Banach space with Opial modulus rx, and let B > 1. Since rx(c) :::: c-l for every c:::: 0, if rx(c) :::; c - 1 then c:::; B. Therefore, by the definition of Cx(B) it is clear that Cx(B) :::; B. Hence, for every a :::: 0,
R(aC~(B),X) :::;R(a,X).
Consequently
{ l+a } M(X) :::; sup ( ) : a :::: 0 . R aCx(B) X
B '
Thus, we have that the above theorem is more general than the stability theorem due to Dominguez-Benavides [D 96]. Nevertheless, in [L-T-X 95] the authors proved that rx is continuous on [0,00) and that, for 0 < Cl :::; C2,
If rx satisfies the condition rx(l) = 0, then for c:::: 1, we have
c-1:::; rx(c):::; (c-1)(rx(1) + 1) = c-1.
Therefore, for those Banach spaces X with rx(l) = 0, Cx(B) = B (B:::: 1).
It is obvious that, for such spaces, the stability bound given by the above theorem is M(X).
Spaces verifying rx(l) = 0 are, for example, X = i2 EBlR (see [L-T-X 95], Remark 3.1.), and the Bynum spaces ip,DO (1 < p < 00).
3. It is worthwhile to notice that for all Banach space X, R(O,X) = (WCS(X))-l and therefore M(X) :::: WCS(X). (See [A-D-L 97], and [D-J 98] for more information about the coefficient M(X)).
5. Stability for Hilbert spaces. Lin's Theorem
The stability constant for a Hilbert space was improved by several authors over a short period of time. We present here the last of these improvements, due to P.K. Lin [L 99]. His result gives a lower bound for the stability constant that is slightly greater than 2.07, dispelling the conjecture that for Hilbert spaces the constant was 2.
Theorem 5.1 Let (H, II '11) be a real Hilbert space. Let 1·1 be an equivalent renorming of H such that for all v E H
224
If b < J 5+Z© then (H, I . IJ has the FPP.
Proof. Suppose, for a contradiction, that (H, I . I) does not have the FPP. Then we may suppose that there exists T, a nonexpansive mapping on a minimal invariant weakly compact subset C of H such that 1·1- diam( C) = 1, and we can select a weakly null approximately fixed point sequence (xn).
Let
D := inf {limninf llYn - yll : (Yn) is an a.f.p.s. in C, for T, Yn 3"... Y} .
Notice that by the Goebel-Karlovitz's Lemma, D :::: !. As a first step, let us show that for all t E (0,1), if Wt is defined as in Lemma 4.3 we have
sup { lim Ilwnll : (w:) E Wt} :::: D. n~U
Indeed, suppose that there exists to E (0,1) such that
sup {lim IIwnll : (w:) E WtD} < D. n~U
Then there exists c > ° such that
sup { lim Ilwnll : (w:) E WtD} < D - c. n~U
Since the bounded closed convex set WtD is t invariant, we can select an a.f.p.s. (wm )
for t, with
Wm := (wii')n::y
Fix a positive integer k. Then there exists mk EN such that
Thus the set Ak := {n EN: Ilwii'k - T(w;:'k)11 < D E U. Moreover, since
lim IIW;:'kll < sup {lim IIwnll : (w:) E WtD} < D - c, n--+U n--+U
we have that the set
Bk:= {n EN: Ilw;:'kll < D - c} E U.
As U is a free ultrafilter we know that Ak n Bk is an infinite set, and therefore we can select a strictly increasing sequence (nk) of positive integers such that for any kEN,
and
Ilw~kll < D - c.
Put Zk := W~k. It is clear now that the sequence (Zk) is an a.f.p.s. for T and moreover it satisfies IIZkl1 < D - c for each kEN.
Stability of the FPP for nonexpansive maps 225
Since the set C is weakly compact, we may suppose without loss of generality that Zn ~ Z E C. Then, as (H, II . II) is a Hilbert space, we obtain
liminf Ilzk - zll :::; liminf Ilzkll :::; D - c < D
which is a contradiction.
As a second step, we show that D2 2:: ~.
Given any t with VI - l/b < t < 1, by the definition of D, there exists an a.f.p.s. (Yn) for T in C with Yn ~Y E C, such that
After a translation, if necessary, we may suppose without loss of generality that Y = o.
Let Wt be defined as in Lemma 4.3, with x = (Yn) , we can apply the first step to obtain
sup {lim Ilwnll : (w;3 E Wt} 2:: D. n-'U
Hence, we can select ~jj := (w;3 E Wt such that
lim Ilwnll 2:: D - (1 - t)2. n-'U
As (H, II . II) is a Hilbert space, the ultrapower flu is also a Hilbert space. Let P = x, Qt = tx and Rt = w. Consider the triangle t:,.PQtRt. First note that this is a proper triangle because w does not lie on the line that contains x and tx. Otherwise, since by Lemma 4.3 Iw - xl = 1- t, we would have that w = tx. But then, the above estimates on Ilwll and Ilxll give D - (1- t)2 :::; Ilwll = Iltxll :::; tD + t(l- t)2 and so 1- t2 > D 2:: i which contradicts the choice of t.
Denoting by 'Y the angle at P we obtain
211w - xllllxli cosb) :::; IIxll2 - IIwl1 2 + Ilw - xll2
22 22 (1_t)2 :::; (D + (1 - t) ) - (D - (1 - t) ) + -b-
:::; (4D + 1/b2)(1 - t)2
Thus we may choose t near enough to 1 so that 'Y is arbitrarily close to or larger than ~. Let the perpendicular from P meet the line through Rt and Qt, at F. Let a be the distance, in 11·11, between P and Qt and let h be the distance, again in 11·11, from P to F. For convenience, let d be the distance between P and Qt. Then
and
d = (1- t)D, a:::; 1- t
1-t -b-:::;h.
To se~this last inequality note that, if F = Atx + (1 - A)w,then for A E [O,lJ we have FE W t and, by Lemma 4.3 (a), IP - xl = 1- t. For A > 1 we have again, by Lemma 4.3 (a), that IP - xl 2:: It x - xl - (1 - A)lw - xl = 1 - t. For A < ° we can use similar reasoning to obtain the same conclusion. In any case, IP - xl 2:: 1 - t.
226
Now, the distance from R t to F is va2 - h2 and the distance from F to Qt is Vd2 - h2 ,
thus given E > 0, using the above estimates and the observation about "( we may choose t sufficiently close to 1 so that we have
This leads to
d2 ~ h2 (1 + (a/h~2 -1 + O(E)).
Substituting the value for d and using the bounds on a and h, we obtain
and so 2 1
D ~ b2 _ 1 + O(E).
Since E is arbitrary this establishes the inequality and completes the proof step two.
For the third step, let us see that for any t E [0,1] the following inequality holds
Indeed, consider (wn ) E Wt. Since C is a weakly compact subset of H, we know that there exists wEe such that w = (w -limn~u)wn. Then it is clear that
Now, since (wn ) E Wt by Lemma 4.3 (a), we have
Ilxn - wnll 2 :s: IXn - wnl2 :s: (1- t)2.
Moreover, by Lemma 4.3 (d), it is easy to see that
2 1 2 (l_t)2 Ilxn - Wn + wll ~ [;2lxn - Wn + wi ~ -b-
Therefore,
Now, by using again the facts that (H, II . II) is a Hilbert space and that w = (w -limn~u )wn it is not difficult to see that
lim lim Ilwn - wm l1 2 = 2 lim Ilwn - w11 2 . m--+U n-+U n--+U
Thus, we may apply Lemma 4.3 (d) in the above equality to obtain
Stability of the FPP for non expansive maps 227
Consequently,
Combining Lemma 4.3 together with the inequalities of first, second and third steps, we finally derive
In particular, taking t = ~t~:.:~ this yields
b>J5+vTI. - 2
• We end this section with a question raised in the introduction: it was noted there that every I·I-nonexpansive mapping in a I3anach space X is II· II-uniformly lipschitzian with respect to any norm II . lion X equivalent to I· I. Then, the question of whether any equivalent renorming of a Hilbert space X, with norm II . II, has the FPP would have an affirmative answer if the same were true for the following second question: does any Hilbert space have the fixed point property for uniformly lipschitzian mappings? Some well known examples show that this last question has a negative answer. One of them, due to J.B. Baillon [B 78], is the mapping T : B -4 B defined on the closed unit ball of the Euclidean space £2 by
T(x) = { cos (~llxI12) e + sin (~llxIl2) ~ x 7' 0 e x=O
where (en) is the usual Schauder basis in £2 and S is the right shift operator associated to (en).
It is known that T is a fixed point free uniformly ~-lipschitzian mapping, and this suggest that an strategy for answering in the negative our first question could be to show that T is nonexpansive with respect to some equivalent norm on £2. Unfortunately, the following theorem [G-J-L 97] shows that this is not the case.
Theorem 5.2 The mapping T is not non expansive with respect to any norm on £2 equivalent to the Euclidean norm.
Proof. Suppose that 1·1 is a norm on £2 equivalent to the Euclidean norm, II ·112, such that T is I . I-nonexpansive.
Let a > 0 be a real number such that allxl12 ::; Ixl for all x E h Assume that a is optimal and choose y such that IIyl12 = 1 and a ::; Iyl < aV2. Then we have that
1 1 V2 = IIT(O) - T(y)112 ::; -IT(O) - T(y)1 ::; -Iyl, a a
which is a contradiction. •
228
6. Stability for the T-FPP
A fundamental tool used to prove the results in the above sections has been the GoebelKarlovitz Lemma, which does not work in a general topology T on X.
Nevertheless, as we said in the introduction, the fixed point theorems for uniformly lipschitzian mappings can be used to obtain stability results for nonexpansive mappings. For the case of the weak topology, these results have not showed to be better than those which we have seen in the above sections.
However, for a general topology T it is possible to obtain stability results by using some fixed points results for asymptotically regular and uniformly lipschitzian mappings.
Let (X, II· II) be a Banach space and let C be a closed convex subset of X, a mapping T: C -> C is said to be k-uniformly lipschitzian whenever the inequality
lIT'x - T'yll ~ kllx - yll
holds for every n E N and for every x, y E C.
Recall that T is asymptotically regular on C if
lim IITn+1x - Tnxll = 0 n-oo
for all x E C.
It is well known (see [I 76J,[G-K 90]) that if (X, 11·11) is a Banach space and C is a closed convex and bounded nonempty subset of X and 8 : C -> C is nonexpansive then the mapping T := I~S : C --+ C is asymptotically regular on C.
Definition 6.1 Let M be a bounded convex subset of (X, II· II) .
(a) A number b ~ 0 is said to have the property (Pr ) with respect to M (b E (Pr ))
if there exists a > 1 such that for all x, y E M and r > 0 with IIx - yll ~ r and each r-convergent sequence (n) eM for which limsuPn lI(n - xii ~ ar and limsuPn II(n - yll ~ br, there exists some z E M such that liminfn lI(n - zll ~ r.
(b) Kr(M) := {b > 0: b E (Pr )}.
(c) Kr(X) := inf{Kr(M)} where M is any subset of X as above.
Theorem 6.2 Let (X, 11·11) be a Banach space, T an arbitrary topology on X. Assume that I . I is an equivalent norm on X such that
for every x E X and k < Kr(X). Then (X, I . I) has the T-FPP.
Proof. Let C be a T-sequentially compact norm-bounded convex subset of X, and 8: C --+ C a I· I-nonexpansive mapping. By [176] we know that T := (I + 8)/2 is a I . I-nonexpansive and asymptotically regular mapping on C, therefore it is clear that T is k-uniformly lipschitzian for the norm II . II. Let us define a function r on C as follows
r(x) = inf {r > 0 : 3y E C : lim inf IIx - T'yll ~ r} . n-oo
Stability oj the FPP Jor nonexpansive maps 229
Since k < K,T(C), it js readily seen that there exists positive numbers a, 1-£ E (0,1) such that for every x,y E C with IIx - yll 2: (1- I-£)r and any T-convergent sequence (en) in C for which limsuPn lien - xII :::; (1 - I-£)r and limsuPn lien - yll :::; k(1 - I-£)r, there exists some z E C such that limsuPn lien - zll :::; ar. By the definition of rex), there exist some integer m 2: 1 such that Ilx - Tnm II > r(1 - 1-£) and ayE C such that
liminf Ilx - ynyll :::; (1 + I-£)!r(x). n--->oo
(6.1)
Take a subsequence (Tnky) of (Tny) such that
liminf IITny - xII = lim IITnky - xii-n-+oo k-+oo
Since C is T-sequentially compact, we may assume that (ynky) is T-convergent. Now using the asymptotically regularity of Ton C, we obtain
lim sup lIynky - ynmxll :::; k lim sup Ilynk-nmy - xii k--+oo k-+oo
= k lim sup Ilynky - xii:::; (1 + I-£)kr(x). (6.2) k--->oo
It follows from (6.1) and (6.2) that there exists z = z(x) E C such that
liminf IITnky - zll :::; ar(x). k--->oo
Which implies that rex) :::; ar(x).
Also, we have
liz - xII :::; liminf liz - ynkyll + liminf IIx - ynkyll :::; ar(x) + (1 + I-£)r(x) = Ar(x), k-+oo k-+oo
where A = 1 + a + 1-£. Proceeding in this fashion, we get a sequence (zn) in C (zo = x and Zk = z(zk-d) such that
(6.3)
Therefore, from (6.3) it is not difficult to see that (zn) is a norm-Cauchy sequence and thus strongly-convergent. Let Zoo = limn--->oo Zn. Clearly r(zoo) = 0, which implies that Zoo is a fixed point of T. Indeed, for any f > 0, since r(zoo) = 0, there exists y E C such that
lim IIzoo - ynkyll :::; f. k--->oo
(6.4)
Then,
IIzoo - Tzooll :::; limsup(lI zoo - Tnk+1y ll + IITnk+1y - TzoolI) k--->oo
:::; (1 + k) lim IITnky - Zoo II < (1 + k)f. k--->oo
• The definition of K,T(X) makes its computation difficult. In some classes of Banach spaces we can use an easier definition. In this way, let us recall the following.
Definition 6.3 A Banach space (X, II· II) is said to enjoy the property L(T) if there exists a function {j : [0, +00 [ x [0, +00[-> [0, +oo[ satisfying
230
(1) 6 is continuous
(2) 6(·,s) is nondecreasing
(3) 6(r,·) is nondecreasing
(4) 6(limsuPn IIxnll, lIylj) = limsuPn Ily-xnll for all y E X and every T-null sequence.
Lemma 6.4 Let (X, II . II) a Banach space with the property L(T) where T is a linear topology on X. Then the norm II . II is T-slsc.
Proof. We are going to show that the unit ball Bx is T-sequentially closed which easily implies that the norm is T-slsc. Let (xn) be a sequence in the unit ball which is T-convergent to Xo. Since (xn - xo) is T-null, we obtain from the property L(T)
1:::: limsupllxnll = limsupllxn -Xo +xoll = limsupllxn - 2xoll·
Hence, 112xoll ::; lim sup Ilxn - 2xoll + lim sup Ilxnll ::; 2,
and thus we may conclude that Xo E Bx. • It is not difficult to show that the Banach spaces which satisfy the L(T) property such that 6(1,·) is an increasing function are a subclass of Banach spaces with the T-uniformly Opial condition. That is,
Definition 6.5 A Banach space is said to have the T-uniformly Opial condition if for c> 0 there exists r > 0 such that liminfn IIx + xnll :::: 1 + r for every x E X such that Ilxll :::: c and every T-null sequence (xn) in X with liminfn IIxnll :::: 1.
If (X, 11·11) is a Banach space satisfying the T-uniformly Opial condition and with the norm 11·11 T-slsc, then
KT(M) = sup{b > 0 : b E (PT),}
where a number b:::: 0 is said to have the property (PT ), with respect to M, b E (PT )" if for all Z,y E M and r > 0 with IIx - yll :::: r and each sequence ((n) eM T-convergent to z E M such that limsuPn II(n - yll ::; br, we have that liminfn lI(n - zll ::; r. (See [D-X 95]).
Proposition 6.6 Let (X, II ,11) be a Banach space and consider T a linear topology on X. Assume that X satisfies both property L(T) and the T-uniformly Opial condition. Then KT(X) :::: 6(1,1). If, in addition, there exists a T-null sequence that is not normconvergent, then KT(X) = 6(1,1).
Proof. By above Lemma the norm of X is T-slsc, so we may use the equivalent definition of KT(M). Without loss of generality we can assume that r = 1 and z = 0 in such definition of KT(M). Suppose that b < 6(1,1). If ((n) is T-null sequence, lIyll :::: 1 and lim sUPn II(n + yll ::; b we have lim sUPn lI(nll ::; 1, since otherwise we arrive to the contradiction
b:::: lim;up lI(n - yll = 6 (lim;up II (n II , lIyll) :::: 6(1,1).
Thus KT(X) 2: 6(1,1).
Stability of the FPP for nonexpansive maps 231
To prove the second assertion, let (x71) be a T-null normalized sequence and M = co((x71 ) U {a}). Consider y = Xl and (71 = X 71 , n 2 2. Then Ilyll = 1, limsuP71 11(71 -yll = 0(1,1) and liminf71 II(nll = 1. Thus KT(X) ::.; 0(1,1). •
Example 6.7 Let (O,~, iLl be a positive IT-finite measure space. For every 1 ::.; p < 00,
consider the Banach space Lp(O) with the usual norm. Now, consider Lp(O) with the topology of the convergence locally in measure (elm). If (In) is a elm-null sequence in Lp(O), 1 ::.; p < 00, and fis a function in Lp(O), then
lim sup Ilfn - fll~ = Ilfll~ + lim sup Ilfnll~· 71 n
Thus, Lp(O) endowed with the elm topology satisfies the property L(elm) with o(r, s) =
(rP + sP)l/p . In particular, if 0 = N with the cardinal measure and p > 1 we obtain £p with the weak topology. For £1, the elm-topology coincides with the weak* topology on the norm-bounded subset of £1 = c5.
As a consequence of the above results we have:
Corollary 6.8 If Y is a Banach space isomorphic to Lp(O), 1 ::.; p < 00, such that d(Y, Lp(O)) < 21/ p then Y has the clm-FPP.
In particular, if Y is a Banach space isomorphic to £1 and d(Y, £t) < 2, then Y has the weak*-FPP. On the other hand, it must be noted that 2 is the best possible stability bound for £1 with respect to the weak*-topology and thus for L1 (D) with respect to the elm topology. Indeed,
Example 6.9 [Li 85J Consider the elassical Banach space £1 endowed with the norm
Now we define the set,
It is easy to see that K is convex and compact with respect to the weak*-topology of £1. On the other hand, if we define the mapping T : K --> K as
T((xn)) = (1- I:xn,Xl,X2, .•• ,Xn , ... ) .
71=1
Clearly T is I . I-nonexpansive and fixed point free on K.
Finally, an easy computation shows that d((£l, 11·111), (£1, 1·1)) = 2.
7. Further remarks
7.1. Stability for uniformly convex spaces
Bynum's paper [By 80] also contains the following theorem of stability:
232
Theorem 7.1 If (X, II . II) zs a uniformly convex Banach space and if (Y, I . I) is a Banach space such that
d(X, Y) ::; WCS(X),
then Y has the FPP.
Notc that the condition d(X, Y) < WCS(X) yields weak normal structure and hence the WFPP for Y, without the assumption of uniform convexity for (X, II . II) . On the other hand there are spaces Y with d(h Y) = WCS(£2) = V2 and without normal structure. So it is clear that this theorem improves Theorem 2.9. It is also clear from Lin's result that the above theorem is not sharp concerning the stability of the FPP, at least for Hilbert spaces.
7.2. Stability of asymptotic normal structure,[B-K-R 98]
In order to get a quantitative description of asymptotic normal structure, we may mimic the definition of the Bynums's coefficients to obtain the following.
Definition 7.2 Recall that the bounded sequence (Xn) in X is asymptotically regular (a.r.) whenever Xn - Xn+! ---> O.
We define
AN(X) := sup {k :k· inf ra ((xn.)) ::; diama((xn)) (xn.J
for each a.r. bounded sequence (Xn)}.
If in the above definition we assumc the additional condition of weak compactness of the set co( {xn : n E I'll} ), then we get the asymptotic normal structure coefficient with respect to the weak topology
w - AN(X) := sup {k :k. inf ra ((xn.)) ::; diama((xn)) (x ni )
for each a.r. bounded sequence (xn)
with co( {xn : n E I'll}) weakly compact}.
Definition 7.3 If AN(X) > 1 we say that (X, II . II) has uniform asymptotic normal structure (UAN).
Definition 7.4 If w- AN(X) > 1 we say that (X, 11·11) has nniform asymptotic normal structure with respect to the weak topology (w-U AN) .
We list some well known properties of these coefficients
1 1::; AN(X) ::; w - AN(X).
2 1::; WCS(X) ::; w - AN(X).
3 If a Banach space (X, II . II) has AN(X) > 1 then it is reflexive.
4 w-AN(X»l*WFPP.
1
5 AN(Cp ) = 21'.
Stability of the FPP for nonexpansive maps 233
6 Example 7.5 For {3,p > 1 let us consider ip endowed with the norm
Ilxli/3 := max {llxlloo, ~ Ilxlip } .
We denote this space by X~. If we take the space Z := X~ X il equipped with the
iI-norm then this space is nonreflexive. Thus, AN(Z) = 1. But w-AN(Z) = J2.
Theorem 7.6 Let (X, II ,11) and (Y, 1·1) be isomorphic Banach spaces. Then we have
AN(X) :S d(X, Y)AN(Y),
w - AN(X) :S d(X, Y).w - AN(Y).
Corollary 7.7 Let (X, II . II) and (Y, I . I) be isomorphic Banach spaces.
a) Ifd(X, Y) < AN(X) then AN(Y) > 1, and hence (Y, 1·1) has UAN.
b) Ifd(X, Y) < w-AN(X) then w-AN(Y) > 1, and hence (Y, 1·1) has the WFPP.
7.3. Stability of Semi-Opial property, [B-K-R 98]
Semi-Opial property was defined in [B-K-R 98]. Recall that a Banach space is said to have the semi-Opial (weak semi-Opial) property, (SO) ((w-SO», if for each bounded nonconstant asymptotically regular sequence (xn) in X (with weakly compact closed convex hull), there exists a subsequence (xnJ weakly convergent to x, such that
liminfllx - xn.ll < diam({xn}). ,
Definition 7.8 The semi-Opial coefficient (with respect to the weak topology) is defined by
w - SOC(X) := sup {k :k. inf w ra (y, (xnJ) :S diama((xn» (Xni ) 'Xni -" Y
for each a.r. bounded sequence (xn)
with co( {xn : n E N}) weakly compact}.
Definition 7.9 If w - SOC(X) > 1 we say that (X, 11·11) has the uniform semi-Opial property,with respect to the weak topology (w-USO).
The following are well known properties.
1 1:S WCS(X) :S w - SOC (X) :S w - AN(X).
2 If a Banach space (X, II . II) has N(X) > 1 then w - SOC (X) > 1.
3 If a Banach space (X, 11·11) has the nonstrict Opial property, then w - SOC (X) = w-AN(X).
4 The space £1'([0,1]) 1 < p < 00, p =I- 2, has w - usa property but it does not satisfy the nonstrict Opial condition.
1 5 w - SOC(ip) = 2;;.
234
6
w - SOC(X~) = max [1, min (2~' ~) ] . 7 If we take the space Z := X~ x f) endowed with the f)-norm then this space is
nonreflexive. Thus, AN(Z) = 1. But w - SOC(Z) = v'2.
Theorem 7.10 Let (X, 11·11) and (Y, 1·1) be isomorphic Banach spaces. Then we have
w - SOC(X) :S d(X, Y).w - SOC(Y).
Corollary 7.11 Let (X, II . II) and (Y, I .1) be isomorphic Banach spaces. If d(X, Y) < w - SOC(X) then w - SOC(Y) > 1 and hence (Y, I . I) has the uniform semi-OpiaZ property.
7.4. Stability of Orthogonal Convexity, [J 88]
If x, Y E X and), is positive put
M),(x,y) = {z EX: max{llz - xii, liz - yll} :S ~(1 + ),)llx - YII}.
If A is a bounded subset of X, IAI := sup{llxll : x E A}.
If (xn) is a bounded sequence in X recall that
D[(xn)] := limsup(limsup Ilxn - xmll). n m
Moreover we will use the notation
A),[(xn )]:= limsup(limsupIM),(xn,xm)l). n m
Definition 7.12 We say that a Banach space X is orthogonally convex (OC) if for every weakly null sequence (xn) with D[(xn)] > 0 there exists), = ),((xn)) > 0 such that A),[(xn)] < D[(xn)].
Notice that 1M), (xn, xm)1 is some measure of the size of the set of almost metric midpoints of the above segments [xm, xn], and we could say that this size is "small" when the unit ball B of X is rotund.
It is well known that in the unit ball of a uniformly convex space, if two points are far apart, then their midpoint must be within it. Similarly, property OC ensures that the "radial distances" IM),(xn,xm)1 are asymptotically strictly smaller than the length of the some linear segments [xm, xn]. This property was introduced in [J 88J, as a sufficient condition for the WFPP. Every uniformly convex Banach space is OC. Other examples of OC spaces given in [J 88] are Banach spaces with the Schur property (hence f1), co, c and James's space J. Moreover, Prus, Kutzarova and Sims have shown [Pr-K-S 93] that OC Banach spaces have the weak Banach-Saks property.
It turns that orthogonal convexity is reasonably well behaved under isomorphisms.
Definition 7.13 If (X, II . II) is a Banach space, and Q = (Qn), R = (Rn) are two bounded families of continuous linear operators from X to X, we shall refer to (Q, R) as a pair for DC if the following conditions hold
Stability of the FPP for nonexpansive maps 235
(a) For every weakly null sequence (xn) in X,
lim inf 11R;(xn)II = 0 n--+oo 't
and lim IIQi(Xn ) II = 0 for all i E Z+.
n .... oo
(b) The coefficient
P(Q,R)(X) := sup{lIxll : IIQi(X) II ::; 1, 11R;(x)ll::; 1 for some i E Z+}
is finite.
For simplicity we also write p(X) instead of p(Q,'R) (X).
In particular if (X, II . II) has a Schauder basis (en), a standard pair for OC is the sequence of natural projections
Theorem 7.14 [J-L 93] If X admits a pair for DC and
d(X, Y) < 2(pB)-1
then Y has DC and hence the WFPP.
Here B:= sup [{IIQill: i = 1,2, ... } u {IIR;II: i = 1,2, ... }].
Corollary 7.15 [J 88]
a) If X is a Banach space such that
d(X,co)) < 2
then X has DC and hence the WFPP. Moreover the bound 2 is sharp for DC.
b) If X is a Banach space such that
d(X,H) < v'2 where H is a Hilbert space, then X has DC and hence the WFPP. Moreover the bound V2 is sharp for DC.
Corollary 7.16 [J-L 93]
a) If X is a Banach space such that
d(X, fp)) < 21-~
for 1 < p < 00, then X has DC and hence the WFPP. This bound of stability for DC is sharp.
b) If X is a Banach space such that
d(X,fp,oo)) < 21-~
236
for 1 < p < 00, then X has OC and hence the WFPP.
c) Let V be the space £2 endowed with the norm (defined by D. van Dulst)
Ixl := max {~llxIl2' sup IXI + Xn + xn+1l}
where IIxI12 stands for the Euclidean norm of x E £2. If X is a Banach space such that d(X, V)) < v'2 then X has OC and hence the WFPP.
8. Summary
For X = .. inherits .. whenever d(X, Y) and the bound
References is sharp
H, a Hilbert space NS <V2 Yes [By 80][M 84)
H, a Hilbert space GGLD <V2 Yes [M 84]
LP, 1 < p < DO NS < min{2p, 2p} No [By 80]
i p .• , 1 < p < DO, 1 :'S q < DO GGLD < min{2",2 q }. Yes [By 80], [D-L-X 96]
Ef3, 1 < j3 < V2 GGLD <1} Yes [By 80), [A-X 93)
J GGLD < /f12 ? [J 92]
Ji (i= 1,2) GGLD <V2 ? [D-L-X 95]
Co WFPP <2 ? [B-S 84]
c WFPP <2 ? [B-S 84)
i p, 1 < p < 00 OC < 2' -" Yes [J 88)
i p.=, 1 < p < 00 OC < 21 p Yes [J 88]
V OC <V2 Yes [J 88]
Co OC <2 Yes [J 88]
H, a Hilbert space OC <V2 Yes [J 88)
Having property M WFPP <'¥ No [G-S 97]
i p,1 < p < 00 WUANS < 2" ? [B-K-R 98)
i p, 1 < p < 00 WSOC < 2" ? [B-K-R 98)
H a Hilbert space FPP < J5+f3 ? [L 99]
ip, 1 < p < 00 FPP --'-- =.c
< (1 + 2,,-1) " No [D 96) [J-L ~O] [L 99]
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Chapter 8
METRIC FIXED POINT RESULTS CONCERNING MEASURES OF NONCOMPACTNESS
T. Dominguez
Departamento de Analisis Matematicas
Universidad de Sevilla
Apdo 1160 Sevilla 41080, Spain
tomasdCDcica.es
M. A. Japon
Departamento de Analisis Matematicas
Universidad de Sevilla
Apdo 1160 Sevilla 41080, Spain
G. Lopez
Departamento de Analisis Matematicas
Universidad de Sevilla
Apdo 1160 Sevilla 41080, Spain
1. Preface
Fixed Point Theory has two main branches: on the one hand we can consider the results that are deduced from topological properties and on the other hand those which can be obtained using metric properties.
Among the former the most important result is Brouwer's theorem, which assures that the euclidean unit ball Bn(O, l) in ]Rn has the fixed point property for continuous functions, that is, if f : Bn(O, 1) -> Bn(O, 1) is a continuous mapping, there exists a point Xo E Bn(O, 1) such that f(xo) = Xo.
Schauder [57J generalized the Brouwer fixed point theorem to infinite dimensional Banach spaces for the case of compact operators. Compactness plays an essential role in the proof of the Schauder fixed point theorem. However, there are some important problems where the operators are not compact.
239
W.A. Kirk and B. Sims (etis.), Handbook of Metric Fixed Point Theory, 239-268. © 2001 Kluwer Academic Publishers.
240
The first step to extend the Schauder theorem to the setting of noncompact operators was given by G. Darbo [16]. The main idea is to define a new class of operators which map any bounded set in a "more compact" set. In order to state the property: A set is mapped into a "more compact" set, we need to define some "measure of noncompactness". The first such measure was defined by Kuratowski [50] in connection with certain problems of General Topology.
Probably the best known and important metric fixed point theorem is the Banach contraction principle, which assures that every contraction from a complete metric space into itself has a unique fixed point. A mapping T from a metric space (X, d) into itself is said to be a contraction if there exists k E [0, 1) such that d(Tx, Ty) :s: kd( x, y) for every x,y E X. Simple examples show that the Banach theorem does not hold letting k to be equal to 1. Such mappings are called nonexpansive. However in the sixties Browder [11], G6hde [32] and Kirk [44] proved that under suitable geometric conditions nonexpansive mappings have the fixed point property.
Although historically the two branches of the fixed point theory have had a separated development, in recent years measures of noncompactness have also been used to define new geometrical properties of Banach spaces which are interesting for fixed point theory for nonexpansive mappings (see for instance [4], [5], [48], [56], [59] and references therein).
Our goal in this chapter is to study the use of measures of noncompactness in metric fixed point theory.
2. Kuratowski and Hausdorff measures of noncompactness
In this section we define the Kuratowski and Hausdorff measures of noncompactness (MNCs) and study their basic properties. As we know, if B is a bounded non precompact set, there is a number e > 0 such that B cannot be covered by finitely many sets with diameter :s: e. Hence, we can give the following definition:
Definition 2.1 Let (X, d) be a complete metric space and B the family of bounded subsets of X. For every B E B we define the Kuratowski measure of noncompactness a(B) of the set B as the infimum of the numbers d such that B admits a finite covering by sets of diameter smaller than d.
Remarks:
1) As usual, the diameter of a set B is the number sup{d(x,y): x E B,y E B} denoted by diam(B), with diam(0) = O. It is clear that
o :s: a(B) :s: diam(B) < +00
for each nonempty bounded subset B of X and that diam(B) = 0 if and only if B is an empty set or consists of exactly one point.
2) The following properties are satisfied in any complete metric space and are a direct consequence of the definition:
(a) a(B) = 0 if and only if B is precompact.
(b) a(B) = a(B) for all BE B.
(c) a(BI U B2) = maxi a(BI), a(B2)}, VBI E B, VB2 E B.
Measures of noncompactness 241
(d) B1 C B2 implies a(B1) :S a(B2), VB1 E E, VB2 E E.
(e) a(B1 nB2):S min{a(B1),a(B2)}, VB1 E E, VB2 E E. (f) If B is a finite set, then a(B) = o.
The a-measure of noncompactness was introduced by Kuratowski in order to generalize the Cantor intersection theorem
Theorem 2.2 Let (X, d) be a complete metric space and {Bn} be a decreasing sequence of nonempty, closed and bounded subsets of X and limn-->oo a(Bn) = O. Then the intersection Boo of all Bn is nonempty and compact.
Proof. Let {xn} be a sequence such that Xn E Bn for all n E N and consider the decreasing sequence of sets {Cn} given by Cn = {Xi: i 2': n}. Obviously Cn C Bn and a(C1) = a(Cn) :S a(Bn) for every n E N.
Since limn-->oo a(Bn) = 0, it follows that a( C1 ) = 0 and so {xn} is a precompact set. Let x be the limit of a subsequence of {xn }. Obviously x E Bn for all n E N and hence Boo 0/0. Moreover, as a(Boo) :S a(Bn) for every n E Nand limn-->oo a(Bn) = 0, we obtain a(Boo) = 0 and so Boo is compact since it is a closed set. •
Horvath [35] has proved the following generalization of the Kuratowski theorem.
Theorem 2.3 Let (X, d) be a complete metric space and {B;}iEl be a family of nonempty closed and bounded subsets of X having the finite intersection property. If infiEl a(Bi) = 0 then the intersection Boo of all Bi is nonempty and compact.
Proof. Given n > 1 there exists i(n) E I such that a(BiCn)) < ~. Let Ak = n~=lBiCn)' then {Ak} is a decreasing sequence of nonempty, closed and bounded subsets of X and limn-->oo a(Ak) = O. It follows by the Kuratowski theorem that K = nk;::lAk is a nonempty compact subset of X.
Set Bi = Bi n K for each i E I. It will be enough to prove that the family {Bi};El has the finite intersection property. In this situation, by compactness, niElBi is nonempty, and since niElBi C niE1Bi the proof would be finished.
Let J be any non-empty finite subset of I and define
iEJ iEJ
By the Kuratowski theorem nk;::lBJ,k is nonempty and compact. But
n BJ,k = n Bi n Ak = n Bi n K. k;::l iEJ k;::l iEJ
So the family {BihEI has the finite intersection property and the proof is concluded .
• Corollary 2.4 Let (X, d) be a complete metric space and f : X -4 ~ a lower semicontinuous function such that infxEx a{y EX: f(y) :S f(x)} = O. Then f is bounded from below and there exists Xo E X such that f(xo) = infxEx f(x).
242
Proof. Let Bx = {y EX: f(y) ::; f(x)}, the family {BxhEX is in the hypothesis of the previous theorem. If Xo E nxExBx, then f(xo) ::; f(x) for all x EX. •
Corollary 2.5 Let (X, d) be a complete metric space and 9 : X -> X a function such that:
1) x -t d(x,g(x)) is lower semicontinuous.
2) infxEXd(x,g(x)) = O.
3) infxEx a{y EX: d(y,g(y)) ::; d(x,g(x))} = O.
Then there exists Xo E X such that g(xo) = Xo.
In the framework of linear spaces additional properties can be deduced for the Kuratowski measure of noncompactness. We omit their well known proofs (see for instance [5]).
Proposition 2.6 Let X be a Banach space, then we have:
(a) a(tB) = Itla(B) for any real number t and BEE.
(b) a(Bl + B2) ::; a(BIl + a(B2), VBl E E, VB2 E E.
(c) a(xo + B) = a(B) for any Xo E X and BEE.
(d) a(B) = a(co(B)).
Associated to the notion of measure of noncompactness we can consider the class of condensing operators which generalize the notion of compact operator.
Definition 2.7 Let D be a nonempty subset of a metric space X. A mapping
T:DcX->X
is an a-condensing mapping if T is continuous and satisfies that a(T(A)) < a(A) for every bounded subset A of D with a(A) > O.
For the class of compact operators we have the following generalization of Schauder theorem, usually called the Darbo and SadovskizTheorem [16J and [58J.
Theorem 2.8 Suppose M is a nonempty bounded closed and convex subset of a Banach space and suppose T : M -> M is condensing. Then T has a fixed point.
Proof. Fix m E M and denote by ~ the family of all closed and convex subsets K of M such that m E K and T(K) C K. Now set
B= n K KE'E
and let c = co(T(B) U {m}).
Obviously ~ oF 0, mE Band T : B -> B.
Mea.sure" of noncompactness 243
Moreover B = C. Indeed, since m E Band T(B) C B, it follows that C C B. This implies T(C) C T(B) C C and so C E~, and hence B C C. Using now the properties of a we obtain a(B) = a(T(B)). Since T is a-condensing, it follows that B is compact. Therefore T is a continuous mapping of the compact convex set B into itself. By the Schauder theorem T must have a fixed point. •
In the definition of the Kuratowski measure we can consider balls instead of arbitrary sets. In this way we have the definition of the Hausdorff measure.
Definition 2.9 The Hausdorff measure of noncompactness X(B) of the set B is the infimum of the numbers r such that B admits a finite covering by balls of radius smaller than r.
Remarks:
3) Recall that in a Banach space X a set SeX is called an r-net of B if
Be S+rB(O, I) = {s+rb: s E S,b E B(O,I)}.
So, the definition of the x-measure in Banach spaces is equivalent to the following:
X(B) = inf{r > 0: B has a finite r - net}.
4) The Hausdorff MNC enjoys the same properties as the Kuratowski MNC, appearing in Remark 2), Proposition 2.6 and Theorems 2.2, 2.3 and 2.8. However there is not a direct relationship between the Kuratowski and the Hausdorff MNCs as we will see in the following results.
Theorem 2.10 Let B(O, 1) be the unit ball in a Banach space X. Then
a(B(O, 1)) = X(B(O, 1)) = ° if X is finite dimensional, and a(B(O, 1) = 2, X(B(O, 1») = 1 otherwise.
Proof. If X is a finite dimensional Banach space, the result follows from the compactness of the unit ball.
It remains to consider the infinite dimensional case. We first prove the result for X. Obviously X(B(O, 1» ::::; 1. Suppose X(B(O, 1» = r < 1. Let us choose e > ° such that r + e < 1. Then there exist Xl, X2,"" Xm in X such that
m m
B(O,I) cUB (Xk, (r + e)) = U (Xk + (r + e)B(O, 1)). k=l k=l
From the properties of the MNC X, it follows that
r = X(B(O, 1» ::::; (r + e)X(B(O, 1» = r(r + e)
and this implies r = 0. So X(B(O, 1)) = ° and hence B(O,I) is precompact. This contradicts the infinite dimensionality of the space X. Therefore, X(B(O, 1» = 1.
244
To prove the result for a we make use of the Borsuk-Lyusternik-Shnirel'man theorem on antipodes (see [49]):
"If Sn(O, 1) is the unit sphere in an n-dimensional normed space and Ak (k = 1, ... , n) is a cover of Sn(O, 1) by closed subsets of that space, then at least one of the sets Ak contains a pair of diametrically opposite points, that is, diam(Ak) ~ diam(Sn(O, 1))".
Since diam(B(0,1)) = 2, it is obvious that a(B(0,1)) ::,; 2. Suppose a(B(0,1)) < 2. Then we can find a finite number of closed subsets {Bl,B2, ... ,Bn } of X with diam(Bk) < 2 for all k = 1, ... , n such that B(O, 1) c U~=l Bk. Now, taking the section of B(O, 1) with an arbitrary n-dimensional subspace Xn and setting Ak = Bk n X n, we obtain a contradiction with the theorem on antipodes. •
Since a and X are invariant under passage to the convex hull, we obtain the following corollary:
Corollary 2.11 Let S(O, 1) be the unit sphere in a Banach space X. Then
a(S(O, 1)) = X(S(O, 1)) = ° if X is finite dimensional, and a(S(O, 1)) = 2, X(S(O, 1)) = 1 otherwise.
Theorem 2.12 The Kuratowski and Hausdorff MNCs are related by the inequalities
X(B) :S a(B) ::,; 2X(B).
In the class of all infinite dimensional Banach spaces these inequalities are the best possible.
Proof. The inequalities are obvious from the definitions of a and X. The sharpness of the second inequality follows from Theorem 2.10. The following example shows that the first inequality is also sharp. Let B = {ek : k ~ 1} the set of standard basis vectors in co. Since for all i # j, lie; - ejll = 1, we have a(B) = L On the other hand, X(B) = 1 because the distance from any infinite subset of B to any element of CO cannot be smaller than L •
Remark 5) Even though in general a and X are different MNCs, in some Banach spaces we can find a close relation between them. This is, for instance, the case of hyperconvex spaces (see chapter by Khamsi-Espinola).
3. cf>-minimal sets and the separation measure of noncompactness
The notion of a ¢-minimal set, where ¢ is the a or X MNCs, was introduced in [22] in order to study the relationships between condensing mappings for Kuratowski and Hausdorff's measures of noncompactness. The notion has become very useful to study some properties of the separation measure of noncompactness, to obtain simpler forms for some moduli of noncom pact convexity and to simplify several proofs.
Definition 3.1 Let X be a metric space and B the family of all bounded subsets of X. An infinite subset A E B is said to be minimal for the measure ¢ (or, in short, ¢-minimaQ if ¢(A) = ¢(B) for every infinite subset B of A.
Measures of noncompactness
Examples:
(1) Every infinite precompact set is obviously ¢-minimal.
(2) In particular, every Cauchy sequence with infinite range is ¢-minimal.
(3) Every infinite subset of a ¢-minimal set is a ¢-minimal set.
( 4) The standard bases in £P or Co are minimal sets.
(5) A nonprecompact convex subset of a Banach space fails to be ¢-minimal.
We are now going to prove the existence of ¢-minimal sets in bounded sets.
Theorem 3.2 Let X be a bounded metric space. Then:
(a) There is a subset A of X such that A is ¢-minimal.
(b) If X is not a precompact set, A can be chosen such that ¢(A) > O.
Proof. Let Ao = X; recursively let
¢n+l = inf{¢(A) : A c An, A infinite}
and let An+l be chosen to be an infinite subset of An with
1 ¢(An+l) < ¢n+l + --1' n+
245
Since An is an infinite set, for every n we can choose an E An such that an # ak for k = 1, 2, ... , n - 1. Let A be the infinite set {an: n E N}. Then A \An is finite for each n E N. Let us see that A is ¢-minimal. Indeed, let A' be an infinite subset of A. Since A'\An- I is a finite set for each n > 1, we have
111 ¢(A) ::; ¢(An) < ¢n + - ::; ¢(A' nAn-I) + - = ¢(A') + -.
n n n
Hence ¢(A') = ¢(A) and A is ¢-minimal. This argument concludes part (a).
We now assume that ¢(A) = 0 for every ¢-minimal subset A of X. Let {xn} be a sequence in X and we assume that the set {Xn : n E N} is infinite. Then there exists a ¢-minimal subsequence {Yn} of {xn}. Since ¢({Yn : n EN}) = 0, {Yn : n E N} is precompact and so {Yn} has a Cauchy subsequence. Thus every sequence in X has a Cauchy subsequence. Hence X is a precompact set and the proof is complete. •
Theorem 3.3 Let X be a separable metric space and B a bounded subset of X. Then there exists a x-minimal s1Lbset A of B such that X(A) = X(B).
Proof. Without loss of generality, assume that X(B) > O. Let {Pk} be a dense sequence in X and let {Bm} be the sequence formed by all balls centered at each Pk with radius a rational positive number less than X(B). Then for every positive integer n there are infinitely many points which are not in Mn = U;':'=lBm .
Choose Xn E B \ Mn and denote by A the set {xn : n EN}. For every n there exists k(n) > n such that Xj E Mk(n) for j = 1, ... , n. Hence Xk(n) # Xj, j = 1, .... , n. Therefore A is an infinite set.
246
Let C be an infinite subset of A. For each n E N, C is not contained in Mn because nMn is a finite set. Now it is easy to prove that X(C) 2: X(B) 2: X(A). Thus A is x-minimal and X(B) = X(A) •
Remark 6) This result holds for a wide class of spaces. We recall that a Banach space X is said to be weakly compactly generated if it contains a weakly compact set which is linearly dense in X. Obviously, reflexive and separable Banach spaces satisfy this condition. S. Prus ( see [5]) has proved that the previous theorem is also true for weakly compactly generated Banach spaces. However, this result does not hold for foo (se [22]).
The situation for the Kuratowski MNC is different as we are going to show. We need some technical results
Lemma 3.4 Let A be an a-minimal subset of a metric space (X,d). Then, for every positive number e, there exists an infinite subset B of A such that
a(A) - e < d(x,y) < a(A) + e
for every x E B, Y E B, x =f. y.
Proof. Without loss of generality we can assume that card(A)= XO and
d(x,y) < a(A) + e
for every X,y E A. By Ramsey's theorem (see, for example, [6], pag. 392) there exists an infinite subset B of A such that d(x, y) :s; a(A) - e for every x, y E B or d(x, y) 2: a(A) - e for every x, y E B. Since the first possibility contradicts the aminimality of A and the proof is concluded. •
Remark 7) For a proof of the above lemma without using Ramsey's theorem see ( [5], Lemma 3.1.3).
The following theorem which can be deduced from Lemma 3.4 is known ( see [25J, pag. 235).
Theorem 3.5 Let {xn } be a bounded sequence in a metric space (X,d). Then there is a subsequence {Yn} of {xn } such that limn,m;n#md(Yn,Ym) exists.
Let {xn} be a bounded sequence in a Hilbert space H. Since H is reflexive, there is a subsequence {Yn} of {xn} and a vector v E H such that {Yn} is weakly convergent to v. Taking a subsequence if necessary, we can also assume that limn-->oo llYn - vii exists.
Consider the mapping <I> : H --> lR : <I>(z) = limsuPn-->oo llYn - zll. This map takes its unique absolute minimum in v. This result is a consequence of the following lemma due to Opial [54J:
Lemma 3.6 If in a Hilbert space H the sequence {xn} is weakly convergent to xo, then for any x =f. Xo we have
lim sup IIxn - xII > lim sup IIxn - xoll· (1) n-->oo n-->oo
Measures of noncom-pactness 247
Proof. Since every weakly convergent sequence is necessarily bounded, both limits in (1) are finite. Thus, to prove this inequality, it suffices to observe that in the equality
the last term tends to zero as n tends to infinity. • Now it is easy to deduce that X({Yn: n EN}) = iP(v). Indeed, since
iP(v) = lim llYn - vII n--->oo
it follows that for every c: > 0 there exists no EN such that Yn E B(v, iP(v) + c:) for all n;::: no, and hence X({Yn: n EN}) ::; iP(v).
Conversely, let us suppose that {Yn : n E N} can be covered by finitely many balls with radius r < iP(v). Then there is a ball B(u,r) containing infinitely many elements of this sequence. We still denote the subsequence contained in this ball by {Yn : n EN}. This sequence is weakly convergent to v, iP(v) is still given by limn--->oo llYn - vII and the function iP' : H --4 lR : iP'(z) = lim sUPn---> 00 llYn - zil takes its unique absolute minimum in v. Therefore we obtain llYn - ull ::; r < iP(v) for all n E N and thus iP'(u) = limsuPn--->oo llYn - ull ::; r < iP(v) = iP'(v) contradicting the fact that iP' has an absolute minimum at v.
Lemma 3.7 Let {Xn} be an a-minimal sequence in a Hilbert space H. Assume that Xn # Xm if n # m. If {Yn}, v and iP are constructed as above, then
a({xn: n EN}) = a({Yn: n EN}) = iP(v)J2 = X({Yn: n E N})J2.
Proof. We only have to prove the equality 0'( {Yn : n E N}) = iP(v)y'2.
Let c: be an arbitrary positive number and define 71 = 0'( {Yn : n E N}). By Lemma 3.4 there is a subsequence {Zn} of {Yn} such that
(2)
Since limn--->oo llYn - vii = iP(v), we can choose a positive integer k such that
IlIzn - vii - iP(v)1 < c:, n;::: k. (3)
Fixing kEN, let n > k be large enough so that I(zn - v) . (Zk - v)1 < c:. From the identity
Ilzk - znl1 2 = IIzk - vl1 2 + Ilzn - vll2 - (Zk - v) . (zn - v) - (zn - v) . (Zk - v)
we obtain by (2) and (3)
2(iP(v) + c:)2 + 2c:;::: (1) - c:)2 and (1) + c:)2 ;::: 2(iP(v) - c:)2 - 2c:.
Hence, letting c: --4 0 we obtain 1) = y'2iP(v). • Theorem 3.8 Let H be an infinite dimensional Hilbert space and A an a-minimal subset of the unit sphere 5(0,1). Then
a(A) ::; J2 < 2 = 0'(5(0,1)).
248
Proof. Let {xn} be a sequence in A with Xn =f. Xm for all n =f. m. By Lemma 3.7 we have o(A)jJ2 = inf{<I>(z) : z E H}, where <I>(z) = limsuPn->oo llYn - zll for a subsequence {Yn} of {xn}. Then o(A)jJ2 ::; <I>(O) = limsuPn->oo IIYnl1 = 1. On the other hand we know from Corollary 2.11 that 0(8(0,1)) = 2. •
The notion of minimal set is strongly related to the separation measure of noncom pactness which was first introduced by Istra~escu [37]. The set B is said to be r-separated if d(x,y) ~ r for all X,Y E B, x =f. y. The set B will be call an r-separation.
Definition 3.9 The measure of noncompactness f3(B) of the subset B of the metric space X is the infimum of the numbers r > 0 for which B does not have an infinite r-separation or, equivalently, the supremum of those r > 0 for which B has an infinite r-separation.
Remarks:
8) It is easy to prove that
f3(B) = sup{a(A) : A c B and A is a-minimal }.
9) The properties of the separation measure of noncompactness and related fixed point theorems has been studied by many authors (see [5]). The permanence under passage to the convex hull was proved independently by J. Arias [3] and Erzakova [26].
4. Moduli of noncompact convexity
A generalization of the moduli of k-uniform convexity, (see chapter by S. Prus), considering infinite dimensional subspaces, was given by Goebel and S~kowski in 1984 [30]. The corresponding concept to k-uniformly convex space is, in this case, that of nearly uniformly convex space. Goebel and S~kowski use the Kuratowski MNC as a measure of the volume of the sets. The use of the measure of noncompactness is relevant since it implies a strong connection between topological methods and metric methods.
We are going to start this section with the definition of the moduli of noncompact convexity. Throughout this section ¢ will denote the a, X Dr f3 measure of noncompactness.
Definition 4.1 Let X be a Banach space. We define the modulus of noncompact convexity associated to ¢ in the following way
~x,,,,(e) = inf{1 - d(O, A) : A C Bn(O, 1) is convex, ¢(A) > e}
= inf{l- d(O,co(A)) : A C Bn(O, 1),¢(A) > e}.
We define the NUC-chamcteristic of X associated to the measure of noncompactness ¢ to be
e",(X) = SUp{e 2 0 : ~x,,,,(e) = OJ.
The function ~X,a(e) has been considered by Goebel and S~kowski [30], ~X,x(e) by Banas [7] and ~X,.B(e) by Dominguez and Lopez [20].
Remarks:
Measures of noncompactness 249
10) A Banach space is NUC if and only if c:¢>(X) = 0.
11) The following relationships among the different moduli are easy to obtain
and consequently
We are going to introduce some equivalent definitions for the moduli Llx,x(C:) and Llx,/3(C:) in the case of reflexive spaces. The following technical lemma can be easily proved.
Lemma 4.2 Let X be a Banach space and {xn} a sequence in X weakly convergent to w. Let An = co({xkh<,:n)' Then
Theorem 4.3 Let X be a reflexive Banach space. Then
Llx,/3(C:) = inf{1 -lIwll : {xn} C Bn(O, 1),xn ~ wand Sep( {xn}) > c:}
where Sep({xn}) = inf{c:: IIxn -xmll2 c:: n of. m}
and
Proof. Both equalities have a similar proof. So we are only going to prove the first one.
First we note that if A C Bn(O, 1) with ,B{A) > c:, there exists a sequence {Xn} such that Sep{ {xn}) > c:, {xn} C A and {Xn} ~ w. Thus
Llx ,/3(C:) = inf {1- d (CO({Xn}), 0) : Xn ~ w, {xn} C Bn(O, 1) and Sep({xn}) > c:}
Now using the previous lemma it is easy to deduce the first inequality:
Llx,/3{C:) 2 inf {I - IIwll : Xn ~ w, {xn} C Bn(O, 1) and Sept {xn}) > c:J .
Conversely, let Xn ~ W with {xn} C Bn(O, 1) and Sep({xn}) c> c:. As
1 - Llx,/3(c) = sup {d(O, cot {Yn}) : {Yn} C Bn(O, 1) Sep( {Yn}) > c:} ,
Let TJ > 0. For all k there exists
such that
250
Let Znk be a weakly convergent subsequence of {zn}, and En = co{xnkh>n' Then En c An. So if Znk ~ Z we obtain -
00 00 ZEn En C n An = {w}. n=l n=l
Thus Zn ~ wand so
Since the last inequality is true for every Tj, we obtain
and again this inequality is true for every Xn ~ w satisfying {xn} C En(O,l) and Sep({xn}) > e. So
L1x,,B(e) :.S inf{l - Ilwll : Xn ~ w, (xn) C En(O, 1) Sept {Xn}) > e}
and the proof is now complete. • Remarks:
12) The function
PX(e) = inf {l- Ilwll : Xn ~ w, {Xn} C En(O, 1) and Sept {xn}) > e}
has been considered by Partington [55] as a modulus for the UKK property. Theorem 4.3 shows that for reflexive spaces Partington's modulus is identical to the modulus of noncom pact convexity associated to {3.
13) Let X be a Banach space and T a topology on X. Bearing in mind Theorem 4.3 it is natural to consider the following moduli (for more details see [38])
L1X,,B,T(e) = in£{ 1 - Ilwll : T -lim{xn} = w, {xn} C En(O, 1), and SePt {xn}) > e}
and
L1X,X,T(e) = inf {l- Ilwll : T -lim{xn} = w, {Xn} C En(O, 1), and X( {xn}) > e} ,
where T - lim{ xn} denotes the limit of a sequence {xn} with respect to the topology T
Notice that a Banach space X has the UKK(T) property (see definition 8 in Chapter Prus) if and only if L1X,,B,T(e) > 0 for all e > O.
Example 6. Let (n, ~,fl) be a positive a-finite measure space, X = (Lp(fl) , 1I.llp) and T the convergence locally in measure topology (elm) (see Chapter by S. Prus for more details). We are going to prove that:
1
L1X,,B,clm (e) = 1 - (1 - (e)P)p .
for 1 :.S p < +00.
We use the following result which can be derived from [9]:
Measures of noncompactness 251
If {In} is a elm-null sequence in Lp(/-L), 1 ~ p < +00, and f is a function in Lp(/-L), then
lim sup IIfn - fll~ = Ilfll~ + lim sup IIfnll~· n n
(4)
Let {In} be a sequence which is elm-convergent to f E Lp(/-L). Suppose that IIfnllp ~ 1 and Sep( Un}) > 6. Take a subsequence {gn} C Un} such that limn IIgn - flip and limn IIgnlip exist. If we denote hn = gn - f and apply (4) to the sequence {hn} and - f, we obtain:
IIfll~ = lim Ilhn + fll~ -lim IIhnll~ = lim IIgnll~ -lim IIgn - fll~ n n n n
~ 1 -lim IIgn - fll~· n
Denote I = limn IIgn - flip and let mEN. We have
6P ~ lim sup IIgn - gmll~ = lim sup IIgn - f - (gm - J)II~· n n
Applying again (4) we obtain
6P ~ Ilgm - fll~ + lim IIgn - fll~· n
1
Taking the limit when m tends to infinity, we deduce I 2: 6/2v. Hence
1
IIfll~ ~ (1- 6P /2)v
and 1
t.X,,B,clm (6) 2: 1 - (1- (6)P)V .
Let us now study the converse inequality.
Suppose n = U~l nn with /-L(nn) < 00. Given TJ > 6 consider the function f = aX{1, where
aP = (1- TJP/2) /-L(n1)
and the sequence fn = f + bnX{1n where lfn = if /(2/-L(nn)) for n 2: 2. It is elear that
Since TJ is any real number greater that 6, we finally obtain the desired equality.
Using similar arguments ( see [38]) we can also compute the modulus for the case of the measure X
_ pl t.X,x,clm (6) - 1 - (1 - (6) )p .
Remarks:
14) Suppose that n = Nand /-L is the counting measure. Then we obtain the space £P and elm-convergence of a sequence is equivalent to weak convergence for bounded sequences if p > 1. So
1
t.fY,,B (6) = 1 - (1 - (6)P)V .
and 1
t./.P,x (6) = 1 - (1 - (6)P)V ,
for 1 < p < +00.
252
If p = 1 elm-convergence is equivalent to weak* convergence for bounded sequences if £1 is seen as the dual space of co.
15) Goebel and S~kowski [30] computed the value of i1ep ,a obtaining
which, for p 2: 2, coincides with the value of the Clarkson modulus of convexity (see Chapter by S. Prus ).
5. Fixed point theorems derived from normal structure
In this section we are going to study fixed point results for nonexpansive mappings. Recall that if (C, d) is a metric space, a mapping T : C -> C is said to be nonexpansive if d(Tx,Ty) ::; d(x,y) for every x,y E C. The framework where we are going to work is the following: Let X be a Banach space and T an arbitrary topology considered on X. We shall say that X has the fixed point property with respect to T (T-FPP), if every nonexpansive mapping defined from a T-sequentially compact bounded convex subset of X into itself has a fixed point. Notice that according to the Eberlein-Smulian Theorem, which proves the equivalence between weak compactness and weak sequential compactness, the previous definition is equivalent to the usual definition of the weak fixed point property whenever T is the weak topology. Our main goal in this section is to obtain fixed point results derived from normal structure. We will finish using the moduli of noncompact convexity to deduce fixed point results for nonexpansive mappings.
We shall say that a Banach space X has normal structure with respect to a topology T
(T-NS), if every T-sequentially compact bounded convex set C C X with diam(C) > 0 contains a nondiametral point, i.e., there is Xo E C such that
sup{llx - xoll : x E C} < diam(C).
This notion has been defined previously in Chapter by S. Prus.
Before starting the main theorem of this section we need to prove the following Lemma which can be found essentially in [28]' although the proof that we give is from [45]. Recall that the Chebyschev radius of a set C is defined as
r(C) = inf {sup Ilx - yll : y E C} xEC
Lemma 5.1 Let X be a Banach space and C a bounded subset. Let S be a class of subsets of C which is stable under arbitrary intersections and it contains B n C for every closed ball B of X. If T : C -> C is a non expansive mapping, for every E > 0 there exists C(E) E S such that T : C(E) -> C(E) and diam(C(E)) ::; r(C) + E.
Proof. If diam(C) = 0, take C(E) = C. Otherwise, let p = r(C) + E. We are going to construct C(E).
Firstly, by definition of p, the set M = {z E C: C C B(z,p)} is a nonempty set. Let
:F = {D E S: M C D,T: D -> D}.
Measures of noncompactness 253
Notice that F =I 0 since C E F. Define
L= nD DEF
Then L E S, MeL and T : L ---+ L. Let A = M U T(L) . Let us prove that
L=n{DES:ACD}.
Since A c L it is clear that n{ DES: A C D} c L. This implies that
T(n{D E S: A cD}) c T(L) cAe n{D E S: A cD}.
Also MeA c n{ DES : A cD}. Thus the last set belongs to F and contains the set L.
Finally we define
C(e) = {x E L: L C B(x,p)} = {n B(U,P)} nL uEL
Notice that C(e) =I 0 since M C C(e). On the other hand, C(e) E S because it is an arbitrary intersection of sets in S. It is also clear that diam(C(e)) ::; p. We need to prove that C(f) is T-invariant.
Let x E C(e). Obviously T(x) E L so we have to prove that L C B(Tx,p). Firstly, we shall see that A C B(Tx, p). If u E M we have u E B(Tx, p) because C C B(u, p). Otherwise, if u E T(L) there exists y E L such that u = T(y). Hence
d(u, Tx) ::; d(y, x) ::; p
because x E C(e). Consequently, A C B(Tx,p) n L and by the equivalent definition of L we finally deduce that L C B(Tx,L) and C(e) is T-invariant. •
Recall that if X is a Banach space and T is a topology on X, it is said that the norm is if IIxll ::; lim infn IIxnll whenever T -limn Xn = X.
Theorem 5.2 Let X be a Banach space and T a topology on X such that the norm II ·11 is a T-slsc function. If X has T-NS, then X satisfies the T-FPP.
Proof. Let C be a nonempty T-sequentially compact bounded convex subset of X and let T : C ---+ C be a nonexpansive mapping. Denote by A the following family of subsets of C:
A = {K c C: K =I 0 T-sequentially closed, convex and T : K -> K} .
Notice that A =I 0 since C E A. For every K E A we define
8(K) = inf{ diam(F) : F E A, F c K}
Set Kl = C, and with Kl, ... , Kn given, select Kn+l E A such that Kn+l C Kn and
diam(Kn+l) ::; 8(Kn) + ~ n
for every n E N.
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Let Ko = n~=lKn. Firstly, we are going to check that Ko =f. 0: Take Xn E Kn for every n E N and consider the sequence {xn }. Since C is a T-sequentially compact set, there exists a subsequence {xn.} C {xn} which is T-convergent to a point x. Since the sets Kn are T-sequentially closed and Kn+1 C Kn, we know that x E Kn for every n E N and so, Ko =f. 0. On the other hand, Ko is convex, T-sequentially compact and T-invariant.
Define
S = {D C Ko: D is convex and T-sequentially closed}
Clearly, S =f. 0 since Ko E S, S is stable under arbitrary intersections and B neE S for every closed ball in X due to the T-sequentiallower semi-continuity of the norm.
The above lemma implies that for each f > 0 there exists KO(f) EST-invariant and such that
diam(Ko) - .!. ::; diam(Kn+d - .!. ::; 8(Kn) ::; diam(Ko(f)) ::; r(Ko) + f n n
Letting n --> 00
diam(Ko) ::; r(Ko) + f Since this is true for each f > 0 we deduce that diam(Ko) = r(Ko) which implies that Ko = {x} since X has T-NS. Therefore T(x) = x. •
Remark 16) Notice that Theorem 5.2 generalizes Kirk's Theorem [44] for general topologies because the norm is always a w-slsc function.
On the other hand, the existence of fixed points for nonexpansive mappings defined on domains which are compact with respect to certain topology, has been studied in several papers [8], [53], [51], [42], [43]. All of them require, as additional assumptions, the T-sequential compactness of the domain. From Theorem 5.2 we check that the T-compactness is not necessary to assure fixed point result in the above hypothesis
Next we are going to define a geometric property in a Banach space which implies normal structure with respect to a topology T in very general conditions.
Definition 5.3 Let X be a Banach space and T a topology on X. We shall say that X has the T-GGLD property if:
lim IIxnll < lim Ilxn - xmll n n.mjn#:m
for every bounded T-null sequence {xn} such that both limits exist and limn IIxnll =f. O.
Theorem 5.4 Let X be a separable Banach space. If T is a linear topology and X has the T-GGLD property then X has T-NS. In case T == w the separability assumption is not necessary.
Proof. If X has not T-NS there is a bounded convex T-sequentially compact subset B which satisfies r(B) < diam(B) where r(B) is the Chebyshev radius of B. As in Corollary 52 in Chapter (Prus) we can obtain a sequence {xn} C B which converges to x with respect to T and
Measures of noncompactness 255
for every mEN. Applying Theorem 3.5 there exists a subsequence {Yn} C {xn} such that limn,m;n;lm llYn - Ymll exists. Finally, if we set z", = Yn - X it is clear that the sequence {z",} is bounded, T-null convergent since T is a linear topology and also
lim IIznll = lim liz", - zmll n n,m;n#m
which contradicts the T-GGLD property.
Notice that if T is the weak topology the separability is not necessary in the same way as in Corollary 52 in Chapter (Prus). •
If X is a Banach space and T a topology on X the following coefficient was defined in [18]
TGS(X) = inf { limn,m;n;lm IIxn - xmll } limn Ilxnll
where the infimum is taken over all norm bounded sequences which converge to 0 in T, both limits exist and lim IIxnll =I- O. Property T-GGLD is related to the coefficient TGS(X) in the following way: X has the T-GGLD property if TGS(X) > 1. Consequently, we can state the following result which relates the T-FPP and the coefficient TGS(X).
Corollary 5.5 Let X be a separable Banach space and T a linear topology on X such that the norm is a T-slsc function. If TGS(X) > 1 then X has the T-FPP. If T == w the separability assumption is not necessary.
Remark 17) If X = (Lp(fL) , 1I.llp ) and T is the clm-topology then TGS(X) = 21/ p for 1 ::; p < 00 ( see chapter by S. Prus). So as a consequence of Corollary 5.5 we can deduce that L 1(fL) has the clm-FPP although fails the weak-FPP (see [1]). This-result had been proved previously by Lennard [51].
To finish this section, we shall see how the coefficient TGS(X) is related to the moduli of noncompact convexity defined in Section 4, Remark 13. As a consequence the moduli of noncompact convexity are also useful to deduce fixed point results for nonexpansive mappings.
Lemma 5.6 Let X be a Banach space and T a linear topology on X. Then:
TGS(X) 2: lim 1 ~ 1 () g--+l- - X,{3,T e
1
Proof. Let e < 1 be an arbitrary positive number and {xn} a sequence T-convergent to zero, such that limn IIxnll exists and limn,m;n;lm IIxn - xmll = (t + 1)/2. If we take t < '1/ < (e + 1)/2, we can find ko E N such that '1/::; IIxn - xmll ::; 1 if m,n 2: ko.
Consider m 2: ko and the sequence {Yn} = {xn - Xm}n~ko which is T-convergent to the point -Xm. In addition, the sequence {Yn} is contained in the closed unit ball of X and Sep( {Yn}) 2: '1/ > e. From the definition of the modulus ~X,{3,T('), we deduce that Ilxmll ::; 1 - ~X,{3,T(t). Letting m -> +00 we obtain:
limn,m;n;lm Ilxn - xmll > ~ limn IIxnll - 1 - ~X,{3,T(e)
Finally, taking the limit as e -> 1- we deduce the desired inequality. •
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Remark 18) If X is UKK(T), from Lemma 5.6, TGS(X) > 1. Therefore, if X is a separable Banach space, T a linear topology such that the norm is a T-slcs function and X is UKK(T), then has the T-FPP [51]. Notice that from Lemma 5.6 we may deduce the same conclusion considering Llx ,j3,T(I-) > O.
Recall that a Banach space X is said to have the nonstrict Opial property with respect to a topology T is
lim inf IIxn II ::; lim inf IIx + Xn II n n
for every T-null bounded sequence {xn} in X and x E X. Before Lemma 3.7 it was proved for Hilbert spaces that X( {Yn}) = lim llYn - vII if {Yn} is w-convergent to V and lim llYn - vII exists. Following the same argument it is easy to deduce the following fact: If T is a linear topology on X and X satisfies the nonstrict Opial property with respect to T then X( {Yn}) = lim llYn - vII if {xn} is a sequence in X r-convergent to v and lim llYn - vII exists. This property will be used in the proof of the next result:
Lemma 5.7 Suppose that X verifies the nonstrict Opial property with respect to a linear topology T defined on X. Then rGS(X) 2: 110 where
ho = sup { t 2: 1 : ~ + Llx,x,r C:) 2: 1 } .
Proof. We define the function
1 (1-) h(t) = t + LlX,)(,T T .
Since LlX,X,T(·) is a nondecreasing function, h is increasing.
To obtain a contradiction, suppose that rGS(X) < ho. Take I E (TGS(X),ho) and a normalized sequence which converges to zero for the topology r and such that limn,m;ni"'m IIxn - xmll ::; I.
Let 10 be an arbitrary positive number with I + 210 < 110. There exists no E N such that Ilxn - xmll ::; I + 10 if n, m 2: no. Fix k 2: no and consider the sequence Yn = (Xl.; - xn)/(I + e) with n 2: no. Then, {Yn} is contained in the closed unit ball of X, T-converges to the point xk/(I + e) and
for every n E N.
From the non-strict Opial condition with respect to r we deduce
1 1 X( {Yn}) = 1+10 > 1+ 2e"
Therefore, if we put I' = 1+ 210, we obtain:
~ < I ~ 10 = III ~ ell::; 1 - LlX,)(,T C : 210) = 1 - LlX,)(,T G) . Hence h(/') < 1, which contradicts the definition of 110 since I' < 110. • Remark 19) The lower bounds for TGS(X) given in the above lemmas are the best possible in general. Notice that for the space X = (Lp(/-L), 1I·lIp) endowed with the clm topology we have:
Measures of noncompactness 257
This corresponds to the actual value of (clm)CS(Lp(J.L)). However, the inequalities can be strict in some Banach spaces (see Chapter VI in [5]).
6. Fixed points in NUS spaces
In Section 5 we studied the w-FPP as a property which is implied by either the GGLD property or normal structure. However, there are Banach spaces without these properties which have the FPP. For instance, Cp,co does not have normal structure (see [5], Example VI.2), but this space has the FPP. This fact can be proved, for instance, checking that theBanach-Mazur distance between £P,co and £P is 21/ p and applying a stability result in [14].
On the other hand, NUS spaces (see definition in Prus chapter) can also fail to have normal structure. The space Cp,rX) is again an example of this assertion. However, we will prove that NUS spaces have the FPP.
We are going to prove a basic result in metric fixed point theory: Goebel-Karlovitz's lemma. We need some previous definitions.
Definition 6.1 Let (X,d) be a metric space and T a mapping from X into X. A sequence {xn} is called an approximated fixed point sequence for T if d(xn , TXn) --> 0 as n --> 00.
If we assume that C is a convex bounded closed subset of a Banach space X and T : C --> C a nonexpansive mapping, it is easy to prove that an approximated fixed point sequence exists in C.
Definition 6.2 The asymptotic radius and center of a sequence {xn} in a Banach space X will be defined by:
ra({xn},B) = inf {lim sup Ilxn - yll : y E B}, n~co
for a subset B of X. When B = co({Xn}) we will denote ra({xn},co({xn})) and Za({Xn},co({Xn})) respectively by ra({xn}) and Za({xn}).
Proposition 6.3 Let K be a weakly compact convex subset of a Banach space X, and T : K --> K be a nonexpansive mapping. Assume that K is minimal for T, that is, no closed convex bounded proper subset of K is invariant for T. If {xn} is an approximated fixed point sequence in K, then
Proof. Let
Za,E( {Xn}, K) = {y E K : lim sup Ilxn - yll ::; ra( {xn}, K) + c} . n~oo
It is easy to check that Za,E( {xn}, K) is noncmpty, closed, convex and invariant for T. Thus Za,£({xn},K) = K and Za({xn},K) = n£>oZa,£({xn},K) = K. •
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Lemma 6.4 (Goebel-Karlovitz) Let K be a weakly compact convex subset of a Banach space X, and T : K ~ K a nonexpansive mapping. Assume that K is minimal for T and {xn} is an approximated fixed point sequence for T. Then
lim IIY - xnll = diam(K) n~oo
for every y E K.
Proof. Since T(K) C K we have co(T(K)) C K. Thus co(T(K)) is a weakly compact convex subset of K which is also invariant under T. The minimality of K implies K = co(T(K)). Since K is a weakly compact and convex set we know that Z(K) is a nonempty set. Let x E Z(K). For every y E K we have IITy - Txll ::; Ily - xII ::; r(K). Thus T(K) is contained in the closed ball B(Tx,r(K)) which implies that
co(T(K)) = K c B(Tx,r(K)).
Hence sup{lIy - Txll : y E K} ::; 1'(K) which means Tx E Z(K). Thus Z(K) is a convex weakly compact subset of K and is invariant under T. Again the minimality of K implies Z(K) = K and so K is a diametral set.
We claim now that limsuPn~oo lIy - xnll = diam(K) for every y E K. Indeed, we assume that a vector y E K exists such that limsuPn~oo Ily - Xnll < diam(K). We let l' = limsuPn~oo lIy - xnll and d = diam(K) and consider the family
{B(z, (1' + d)/2) n K : z E K}.
Choose any positive c: < (d - 1')/2. From Proposition 6.3 we know that
lim sup IIxn - zll = l' n~oo
for any z E K. Thus, for any finite subset {Zj, ... , zd of K an integer N exists such that IlxN - zill ::; r+c: = (1'+d)/2 for i = 1, ... , k. Hence XN belongs to nf=jB(Zi' (r+d)/2). The weak compactness of K implies the existence of
Xo E n B(z, (1' + d)/2) n K zEK
and this point is not diametral because
sup liz - xoll < l' + d < d = diam(K). zEK 2
This contradiction proves the claim. If liminfn~oo Ily-xnll < diam(K) for some y E K there is a subsequence {Yn} of {xn} such that
lim sup llYn - yll = liminf Ilxn - yll < diam(K), n~oo n~oo
contradicting the claim, because every subsequence of an approximated fixed point sequence is also an approximated fixed point sequence. •
Lemma 6.5 Let K be a weakly compact convex subset of a Banach space X, and T: K ~ K be a non expansive mapping. Assume that K is minimal for T, diam(K) = 1 and {Xn} is an approximated fixed point sequence for T which is weakly null. Then, for every c: > 0, there exists a sequence {zn} in K such that:
Measures of noncompactness
(i) {z,.} is weakly convergent to a point Z E K.
(ii) IIznll > 1- e for every n E N.
(iii) IIzn - zmll ::; 1/2 for every n, mEN.
(iv) limsuPn Ilzn - xnll ::; 1/2.
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Proof. Let 0 E K. If {wn } is an approximated fixed point sequence in K, we know from Lemma 6.4 that limn IIwnll = 1. Therefore, for every e > 0 there exists See) > 0 such that IIxll > 1 - e if x E K and IITx - xII < See). Indeed, otherwise, there exists eo > 0 such that we can find Wn E K with IITwn - wnll < l/n and IIwnll ::; 1 - eo for every n E N. Thus, the sequence {wn } is approximated fixed point sequence in K which satisfies limsuPn IIwnll ::; 1 - eo.
Let e > O. We choose 'Y < min{l,S(e)} and for every n E N we define the contractive mapping Sn : K -> K by
Sn(X) = (1 - 'Y)T(x) + 'Y/2xn.
Banach's Contraction Principle assures the existence of a (unique) fixed point Zn for Sn. Since K is weakly compact, we can assume, taking a subsequence if necessary that {zn} satisfies (i). Since
liz,. - TZnl1 ::; 'Y IITzn - ~xnll ::; 'YS(e)
we know that {zn} satisfies (ii).
Condition (iii) easily follows and (iv) is a consequence of the inequalities
Thus,
IIzn - xnll ::; 11(1 - 'Y)Tzn + 'Y;n - Xnll
'Yllxnll ::; (1- 'Y)IITzn - TXnl1 + (1- 'Y)IITxn - xnll + -2-'
1 1-'Y liz,. - xnll ::; 2" + -'Y-IITXn - xnll·
Taking limit sup as n tends to infinity we obtain (iv).
The following definition was given in [27):
Definition 6.6 Let X be a Banach space. We define the coefficient
R(X) = sup {lim inf IIxn + xII} n--+oo
•
where the supremum is taken over all x E X with IIxll ::; 1 and over all weakly null sequences in B(O, 1).
Theorem 6.7 Let X be a Banach space and assume that R(X) < 2. Then X has the weak fixed point property.
Proof. If we assume that X fails to have the w-FPP, we can find a weakly compact and convex subset K of X such that diam(K) = 1 and K is minimal invariant for a
260
nonexpansive mapping T and an approximated fixed point sequence {xn} in K. We can assume that {xn} is weakly convergent, and by translation that {xn} is weakly null. Consider a sequence {Zn} satisfying (i)-(iv) in Lemma 6.5. Taking again a subsequence, if necessary, we can assume that limn Ilzn - zil exists. Furthermore,
lim IIzn - zil ::; lim sup lim sup IIzn - zmll ::; -21 • n n m
A positive 1) can be chosen such that 1)R(X) < 1 - R(X)/2. For a large enough n, we have IIzn - zil ::; 1/2 + 1). Furthermore IIzil ::; lim infn-<oo IIzn - xnll ::; 1/2. Hence
111/;: 1)11 = 11172 ~z1) + 1/2z+ 1)11::; R(X).
Thus limsuPn-<oo IIznll ::; R(X)(1/2 + 1) < 1 which is a contradiction because 0 E K .
• Remark 20) Assume X = co. It is easy to check that R(co) = 2 and so CO has the w-FPP although CO does not satisfy the GGLD property.
Theorem 6.8 Let X be an NUS Banach space. Then R(X) < 2.
Proof. Since X is NUS we know that for any e > 0 there exists 8 > 0 such that for any weakly null sequence in Bx and any x E Bx a positive integer k exists such that IIx + tXkll ::; 1 + et if 0 < t < 8. Hence,
IIx + xkll ::; IIx + tXkll + (1 - t)IIXkll ::; 2 + t(1 - e). Furthermore k can be chosen arbitrarily large because the same argument can be applied to the sequence {Yn} where Yl = x and Yn = Xk+n for n > 1. Hence R(X) < 2. •
7. Asymptotically regular mappings
We shall study in this section the existence of fixed points for a different class of mappings, called asymptotically regular mappings. The concept of asymptotically regular mappings is due to Browder and Petryshyn [10]. The fixed point theorems which we shall study are based upon coefficients concerning the modulus of NUC or Opial modulus.
Definition 7.1 Let (X,d) be a metric space. A mapping T M ..... M is called asymptotically regular if
lim d(Tnx,rn+1x) = 0 n-<oo
for all x E M.
Example 7: Let T : [0,1] ..... [0,1] be an arbitrary nonexpansive mapping. It is easy to check that S = (I + T)/2 is also nonexpansive. Thus
ISn+1x - snxi ::; ... ::; IS2x - Sxl ::; ISx - xl.
Furthermore S is a nondecreasing function. Indeed, if x ::; Y and Sx > Sy we have (x + Tx)/2 > (y + Ty)/2 which implies ITx - TYI ~ Tx - Ty > y - x = Ix - yl. Thus
n
1 ~ ISn+1x - xl = L ISk+1x - Skxl ~ nlsn+1x - Snxi k=l
Measures of noncompactness 261
which implies Isn+1 x - snxi :::; lin. Then S is asymptotically regular. In fact, it can be proved (see [29], Theorem 9.4, page 98)·that if T is a nonexpansive mapping from a bounded convex subset C of a Banach space into C, then T>. = >"1 + (1 - >..)T is asymptotically regular for all 0 < >.. < 1.
We will show the possibility of using a modulus of noncompact convexity and Opial modulus to obtain fixed point theorems for asymptotically regular mappings. We need some notations: If S is a mapping from a set C into itself, then we use the symbol lSI to denote the Lipschitz constant of S, that is,
{ IISx - Syll } lSI = sup IIx-yll :x,yEC,x=j.y.
Now if T is a mapping on C, we set
seT) = liminf ITnl. n .... oo
Theorem 7.2 Let X be a Banach space, C a nonempty closed r-sequentially compact subset of X, and T : C -t C an asymptotically regular mapping. Assume that one of the following conditions is satisfied:
(a) seT) < 1 + rX,r(l)
(b) seT) < vrCS(X)
(c) seT) < 1/(1- ~x,,8,r(l-))
Then T has a fixed point.
Proof. Choose a subsequence {nk} of positive integers such that seT) = limk Imkl· For any x E C we can construct a subsequence {nk(x)} C {nd such that {Tnk(x)xh is r-convergent, say to lex), and liml,k;l# IITnk(X)X - Tn,ex)xll and limk Ilmk(x)x -1(x)1I exist. We construct a sequence {xm} C C and sequences of positive integers {nk(xm )}
in the following way: Xo E C arbitrary,
Xm = I(Xm-l) = r -limTnkex=-l)Xm_l, k
{nk(xm)h C {nk(xm-l)h for all k ~ 1.
By using a diagonal argument we consider the sequence of positive integers 1li = ni(xi) for all i E N. In this way, we have that the sequence {mix;"'} is weakly convergent to xm+l, limi,j;#j IITn'xm - mixmll and limi IIm'xm - xm+lll exist for all mEN and seT) = limi ITn , I.
We are going to prove that {xm } is norm convergent to a vector in C. Denote Bm = lim; IIm'xm - xm+1l1. Our proof is based upon the following claim:
Claim. Assume that seT) satisfies either (a) or (b) or (c). Then, there exists a < 1 such that Bm < aBm-l for any mEN.
Using this claim we can go on the proof. Firstly, we know that limmBm = O. By using that the norm on X is r-sequentially lower semicontinuous we deduce
Ilxm - Xm +111 :::; lim sup IIxm - rn'xmll + lim sup IITn'xm - xm +111 i i
262
::; limisup (lim/nf lIT"ixm-l - T"ixmll) + Bm
::; s(T)Bm_l + Bm.
which implies that {xm} is a Cauchy sequence. Let z = limm X m. Since G is closed, z E G. Furthermore
/lz - T"i zll ::; liz - Xm+lll + IIxm+l - Tn'xmll + 11T"'xm - Tn, zll
::; liz - Xm+lll + IIxm+l - Tnixm/l + Lillxm - z/l.
Taking the limit as i -4 00 yields
lim sup liz - T"'zll ::; liz - xm+lll + Bm + s(T)/lxm - z/l i
...... 0 as m ...... 00.
Hence Tn, z -4 z. From the assumption on s(T) there exits a positive integer k such that ITkl < +00. If we choose a positive number E, the asymptotic regularity of T implies that there exists io E N such that I/Tn,+k z - T"izl/ < € if i ~ io. We also assume /IT'" z - zll ::; €.
Hence, for i ~ io we obtain
IITkz - zll ::; IITkz - T"'+kzl/ + /IT'''+kz - T"'zll + IIT"'z - zll
::; (ITkl + 1 + I)E
-40 as E ...... O.
yielding Tkz = z. It is easily verified by induction that Tksz = z for s = 1,2,···. Therefore
IITz - zll = IITks+l - Tkszll ...... 0 as s -4 00.
So z is a fixed point for T in G.
It remains to prove the claim. From the definition of rGS(X) we obtain
lim· ... oJ. . liT'" X - mix II B < '.J.'7'"J m m m - rGS(X)
The asymptotic regularity of T on G ensures that
lim lIT"i-lxm - xmll = lim I/Tn;xm - xmll J J
for any lEN. Thus
Therefore
lim ./lTn'xm - Tn;xm/l = lim sup (lim sup /lTn'xm - Tn;xmll) 'L,1j14=J i j
::; lim sup IT'" I lim sup I/xm - T";-n'xmll i j
= s(T) lim sup IIxm - T";xmll· j
Measures of noncompactness 263
Now, we split the proof in two parts. For (a) denote
a = sup{c > 0: rx,r(c) ::; seT) - I}.
We claim that a < 1. Indeed, otherwise sup{c > 0 : rx,r(c) 5 seT) - I} 2': 1 which implies rx,r(c) 5 seT) - 1 for every c < 1. The continuity of rx,rO implies rx,r(l) 5 seT) - 1 < rx,r(l), a contradiction. Consider the sequence
which is null convergent with respect to the topology rand limj IIZjll = 1. We can apply the definition of the uniform Opial modulus rx,r(-) to the sequence {Zj} and the vector (xm - Tn'xm)/Bm_l . In this way we obtain
Thus
( II Tn'xm - xmll) 1 .. II""" II L l+rx,r B 5 -B L;hmlnf -' 'Xm_I-Xm = ; m-I m-I J
which implies, using the continuity of rX,r(')' that
1 ( limsup;IITn'xm-xmll) < (T) +rx,r B _ S m-I
and limsuPi IIT"'xm - xmll 5 aBm-l. Since seT) < 1 + rx,r(l) 5 rCS(X) we obtain Bm 5 aBm-l.
For (b) and (c) we consider the sequence
{Zj} = {rn'xm ~ TnjXm_l} ",L,Bm-1 j
where L; = ITn'l, i EN is fixed and", > 0 is arbitrary. Since
lim sup IITn'xm - rnjxm-Ill 5 ITn, I lim sup Ilxm - Tnjxm_11l = L;Bm-I. j j
there exists jo EN such that {Zj}j::::jo lies in the unit ball of X. On the other hand, this sequence is r-convergent to the vector
Tn'xm-xm L;Bm_l(l + ",)'
Furthermore
264
Since
1 s(T) < ( ) S TCS(X),
1 + fj.X,(3,T 1
choose i ENlarge enough such that (1 + 2",)L; < (1 + 3"')TCS(X). Applying the definition of the modulus fj.X,(3,T(-) we obtain
IITnixm - xmll < 1 fj. ( TCS(X) ) < 1 fj. (1) (1 + TJ)Li Bm - 1 - - X,(3,T (1 + 2TJ)L; - - X,(3,T 1 + 3TJ
Letting i --> 00 and TJ --> 0 we get Bm S aBm- 1 with
S(T)2 _ 0= TCS(X) (1 - fj.x,(3T(1 »).
Now notice that a < 1 in case (a) because from the T-sequentiallower semi-continuityof the norm we derive that the function fj.X,(3,TO is nonnegative and in case (b) we can use (*). •
8. Comments and further results in this chapter
(1) Lemma 6.4 was independently proved by Goebel [31] and Karlovitz[41]. Theorems 6.7 and 6.8 were first proved in [271. We follow a more direct proof using Lemma 6.5, which is taken from [39]. Theorem 7.2 is basically proved in [191. Previous results in this direction can be found in [15], [33], [34], [46] and [24].
(2) The converse of Theorem 5.4 is not true. Indeed, on the space Ll([O,I]) we introduce the equivalent norm
IIlxlW = IIxlli + (~IX~:)I) 2
where x = ~~1 x(k)ek, {ed is a Schauder basis in Ll([O, 1]) (for instance, the Haar system) and 1I·liI is the usual norm on Ll([O, 1]). Since (Ll([O, 1]), 1I·liI) fails to have the w-FPP [1], LIlO, 1] does not have the w-GGLD property. Then, there exists a weakly null sequence {xn} such that limn IIxnliI = limn,m;n¥m IIxn -Xmlll. It is easy to check that if {Xn} is a weakly null sequence then
Thus
1· ~ IXn(k)1 = 0 li?~ 2k .
k=1
limn,m;n¥m Illxn - xmll12 _ limn,m;n¥m IIxn - xmlli _ 1 limn IIIxnl1l2 - limn IIxnlli -.
But (Ll([O, 1]), III·IID is U.C.E.D. ([17], Corollary 6.9, page 66) and consequently this space has the w-FPP.
(3) The coefficient WCS(X) is introduced in [14] using a different definition:
WCS(X) = inf {di;~{~:;)}) : {Xn} is a weakly convergent sequence
which is not norm convergent}
Measures of noncompactness 265
where diama ( {xn}) = limk sup{lIxn -xmll : n, m 2 k} is the asymptotic diameter of {xn }. However, both definitions are equivalent as proved in [5].
(4) In some spaces, WCS(X) can be strictly greater than the lower bound obtained in Lemma 5.6, as the following example shows. Let X be the space £2 renormed by
Il(x') II ~ m~ { lx' I. (~(X')' ) I} . If {xn} = {(x~)} is a weakly convergent sequence, taking a subsequence and by translation we can assume that {x~} --> O. So WCS(X) = WCS(£2) = .../2. However, considering the sequence {xn} = {(x~)} where x~ = 1 for every n EN, and x~ = 8kn for k 2 2, it is clear that AX,p(l) = O.
(5) It is known that a Banach space X is reflexive if N(X) > 1 (N(X) is the normal structure coefficient defined in [14]). However we can find nonreflexive Banach spaces, without the Schur property, such that WCS(X) > 1. Indeed, let X be the space £1 x £2 with the norm lI(x,y)11 = vllxlli + IIYII~ and let {(Xn,Yn)} be a normalized weakly null sequence such that limn,m;n#m II (Xn,Yn) - (xm,Ym)11 exists. Since {xn} is weakly null and £1 has the Schur property we know {xn} converges to zero. Therefore
lim II(xn,Yn) - (xm,Ym)1I = lim llYn - Ymll n,mjn=fm n,mjn=fm
and limn llYn II = 1. Since WCS(£2) = .../2 it is easy to check that WCS(X) = .../2 although X is not reflexive.
(6) The following example shows that WCS(X) (and so N(X)) can be 1 in a Banach space with normal structure: Let X be the £2-direct sum of the sequence spaces {.en }n2:2. This space is reflexive and it has normal structure because it is a UCED space. However, WCS(X) = inf{WCS(.en) : n 2 2} = inf{21/n : n 2 2} =L
(7) Instead of the coefficient R(X) a uniparameter family of coefficients for any nonnegative number a are defined in [23] by
R(a,X) = sup{liminf IIxn + xII} n ..... oo
where the supremum is taken over all x E X with IIxll :S a and over all weakly null sequences in 13(0,1) such that limn,m;n,<m IIxn - xmll :S 1 and it is proved that X has the weak fixed point property if R( a, X) < 1 + a for some a. The coefficient M(X) is defined now as
SUP{R~:;) :a2o}. This coefficient is specially useful to obtain stability 'results of the w-FPP. In particular, M(X) can be bounded from below using the modulus of k-uniform smoothness and the modulus of nearly uniform smoothness ( [5], Chapter VII). Using the coefficient M(X) it is proved in [23] that X has the FPP if X is isomorphic to £2 and the Banach-Mazur distance between these spaces is less than v'3. A better stability bound for £2 has been obtained in [52]
(8) An interesting open question is to know for which topology T, Theorem 6.8 holds when the space X is NUS(T). The proof of Lemma 6.4 is heavily based upon the weak lower sequential semi-continuity of the type null functions, i.e. assume that {xn} is a weakly null sequence and consider the function </J(x) = lim sup IIxn - xII·
266
Then ¢(.) is weakly lower sequentially semi-continuous. It is unknown if this property is preserved for some other topologies, in particular for the weak star topology. In [38] it is proved that </J(.) is elm-lower semi-continuous in the spaces Lp(D) for 1 < p < +00.
(9) The existence of fixed points for asymptotically regular mappings can be also studied using the coefficient KT(X) defined in the following way:
(i) A number b ~ 0 is said to have the property (PT ) with respect to M if there exists a > 1 such that for all x, y E M and r > 0 with IIx - yll ~ r and each r-convergent sequence {~n} eM for which
lim sup II~n - xii::; ar n~oo
and limsuPn~oo II~n - yll ::; br, there exists some z E M such that
lim inf II~n - zll ::; r. n~oo
(ii) KT(M) = sup{b > 0: b has property (PT) with respect to M}.
(iii) KT(X) = inf{KT(M) : M is as above }.
The following result is proved in [24]:
Theorem 8.1 Suppose X is a Banach space and C is a r-sequentially compact subset of X. If T : C -t C is an asymptotically regular mapping and s(T) < Kw ( C), then T has a fixed point.
Connections between KT(X) and either moduli of noncompact convexity or the coefficient rCS(X) can found in [38]
(10) The modulus of noncompact convexity corresponding to the Hausdorff measure of noncompactness can be also used to to obtain fixed point for asymptotically regular mappings. Indeed, in [19] the following result is proved: Let X be a Banach space, C a nonempty weakly compact subset of X, and T : C -; C an asymptotically regular mapping and h = sup{ t ~ 1 : t + Llx,x (t) ~ I}. If X satisfies the non-strict Opia! condition and s(T) < h, then T has a fixed point. On the other hand, Opia! modulus also provides a lower estimate for the coefficient Kw(X) above defined (see [5], Chapter IX).
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Chapter 9
RENORMINGS OF £1 AND Co AND FIXED POINT PROPERTIES
P. N. Dowling
Department of Mathematics and Statistics
Miami University
Oxford OH 45056
C. J. Lennard
Department of Mathematics and Statistics
University of Pittsburgh
Pittsburgh, PA 15260
B. Thrett
Department of Mathematics and Statistics
Oakland University
Rochester, MI48909
1. Preliminaries
As has been noted in previous chapters, there are many geometric conditions on a Banach space strong enough to imply that the Banach space has the fixed point property. Geometric conditions such as uniform rotundity, uniform smoothness, or normal structure together with reflexivity are sufficient to imply the fixed point property. Each of these conditions also implies (or assumes in the last case) that the Banach space is reflexive.
When one considers the classical Banach spaces, the reflexive ones (LF or fP where 1 < P < CXJ) have the fixed point property since each is uniformly rotund. The classical nonreflexive Banach spaces (CO, gl, L1, Loo , and e[0,1]) fail to have the fixed point property since each of these spaces contains an isometric copy of gl or Co and we have already seen easy examples of isometries on nonempty closed bounded convex subsets of gl or CO without fixed points. Thus it is natural to ask the following questions:
269 W.A. Kirk and B. Sims (eds.), Handbook o/Metric Fixed Point Theory, 269-291. © 2001 Kluwer Academic Publishers.
270
Question 1.1 If a Banach space X has the fixed point property, is X reflexive?
Question 1.2 If a Banach space X is reflexive, does X have the fixed point property?
In considering the first question, whether the fixed point property implies reflexivity, it is tempting to try to construct nonreflexive spaces with the fixed point property. Where however should one look for such spaces? Perhaps the most natural place to look is among the renormings of CO or e1 since these are, in a certain sense, the simplest of the nonreflexive Banach spaces. There is another advantage in considering these two classical spaces. It is well-known (Theorems l.c.12 in [33] and l.c.5 in [34]) that a Banach lattice or a Banach space with an unconditional basis is reflexive if and only if it contains no isomorphic copies of Co or e1. Thus, if it can be shown that neither Co
nor e1 can be renormed to have the fixed point property, it would follow that the fixed point property in either a Banach lattice or in a Banach space with an unconditional basis would imply reflexivity. Thus, success in considering the isomorphism classes of these two spaces would have extremely pleasant consequences for several large classes of Banach spaces.
If one attempts to renorm e1 and CO to have the fixed point property or if one attempts to show that e1 and CO can not be renormed to have the fixed point property, one is interested in knowing how distorted renormings of e1 and Co can be? Intuitively, if renormings of e1 and Co are not much different from e1 and Co with their usual norms, one may have the feeling that perturbations of the usual examples showing that e1 and CO fail the fixed point property may show all renormings of e1 and CO fail the fixed point property. If renormings of e1 and CO can look significantly different than e1 and Co with their usual norms, then one may have the feeling that one of these renormings may have the fixed point property. In fact, R. C. James has shown that neither e1 nor Co can be distorted (in the sense that any renormings of e1 and CO contain almost isometric copies of e1 and CO respectively). To be specific, James [29] proved the following theorems.
James's Distortion Theorems. A Banach space X contains an isomorphic copy of e1 if and only if, for every 0 < e < 1, there exists a sequence (xn) in X such that
for all (tn ) E £1.
A Banach space X contains an isomorphic copy of CO if and only if, for every 0 < e < 1, there exists a sequence (xn) in X such that
(1 - e) s~p Itnl :S II~ tnxnll :S (1 + e) s~p Itnl
holds for all (tn) E co. In fact, the proofs that James provides shows a bit more.
James's Distortion Theorems - Stronger versions. A Banach space X contains an isomorphic copy of e1 if and only if, for every null sequence (en) in (0,1), there exists a sequence (xn) in X such that
Renormings of £1 and Co and fixed point properties 271
for all (tn) E £1 and for all kEN.
A Banach space X contains an isomorphic copy of CO if and only if, for every null sequence (t:n) in (0,1), there exists a sequence (xn) in X such that
(1 - t:~,) ~~1Itnl ::; II~ tnxnll ::; (1 + t:k) ~~1Itnl holds for all (tn) E Co and for all kEN.
2. Renormings of f.1 and Co and fixed point properties
To see how James's theorems can be used to get results concerning fixed points, consider the class of uniformly Ii pschitzian maps on closed, bounded, convex subsets of X. Fixed point theorems for this class of mappings were considered in [24, 25].
Let X be a Banach space containing an isomorphic copy of £1 and consider a null sequence (t:n ) in (0,1) and a sequence (xn) guaranteed by the stronger version of James's theorem for £1. Define C = {L::::"=1 tnxn : 0 ::; tn and L::::"=1 tn = I} and T(L::::"=l tnxn) = L::::"=1 tnXn+1· Let z = L:~1 tnxn and w = L::::"=18nXn be elements of C. Then C is a closed, bounded, convex subset of X and, for any kEN,
00
n=l
::; 1 ~ t:1 11~(tn -8n ) xn ll
1 =--lIz-wll·
1- t:1
Thus T is uniformly lipschitzian and it is easily checked that T has no fixed point. This proves the following:
Proposition 2.1 If X is a Banach space containing an isomorphic copy of £1 and t: > 0, then there exists a closed, bounded, convex subset C of X and an affine, uniformly lipschitzian map T : C -> C such that the uniform Lipschitz constant of T is less than 1 + t: and T has no fixed point. Consequently, £1 cannot be renormed to have the fixed point property for uniformly lipschitzian mappings.
A similar computation using the map defined in the proof of Theorem 2.27 below gives the analogous result for co: in particular, CO cannot be renormed to have the fixed point property for uniformly lipschitzian mappings. This combines with Proposition 2.1 and the results mentioned in the introductory paragraphs to show:
Proposition 2.2 Let X be a Banach lattice or a Banach space with an unconditional basis. If, for some k > 1, X has the fixed point property for k-uniformly lipschitzian mappings, then X is reflexive.
In studying the fixed point property for the class of nonexpansive self-maps of closed, bounded, convex subsets of a Banach space, it has been found to be useful if the
272
terms (1 ± ek) in the stronger version of James's theorems are "inside" the sum or the supremums.
Definitions. A Banach space X is said to contain an asymptotically isometric copy of £1 if there is a null sequence (en) in (0,1) and a sequence (xn) in X such that
for all (tn ) E £1.
A Banach space X is said to contain an asymptotically isometric copy of CO if there is a null sequence (en) in (0,1) and a sequence (xn) in X such that
for all (tn ) E CO.
It is easy to see that there is no difference in the above definitions if the phrase "there is a null sequence (en) in (0,1) and a sequence (xn) in X" is replaced by the phrase "for every null sequence (en) in (0,1), there exists a sequence (xn) in X."
These "better" copies of £1 or CO can be used to investigate the fixed point property for nonexpansive mappings.
Theorem 2.3 ([14)) If a Banach space X contains an asymptotically isometric copy of £1, then X fails the fixed point property for nonexpansive mappings on closed bounded convex subsets of X.
Proof. By assumption, there is a null sequence (en) in (0,1) and a sequence (xn) in X so that
for all (an) E £1. Let (An) be a strictly decreasing sequence in (1,00) with limn---+oo An = 1. By passing to subsequences if necessary, we can assume that
Define Yn = Anxn' for all n E N, and let C = cO{Yn : n EN}, the closed convex hull of the sequence (Yn). Clearly, C is a closed bounded convex subset of X whose elements are of the form z = I:~=1 tnYn, where tn ~ 0 for all n E N and I:~=1 tn = 1. Define a mapping T : C ---+ C by T(I:~=l tnYn) = I:~=1 tnYn+l> where tn ~ 0 for all n E Nand I:~=1 tn = 1. It is easily seen that T has no fixed points in C. We now show that T is nonexpansive. (In fact, we show that T is contractive).
Let z = I:~=1 tnYn and w = I:~=1 SnYn be elements of C with z i- w. Then
Renormings of £1 and Co and fixed point properties 273
00
:::; L Itn - Sn11lYn+11l n=l 00
n=l 00
< L Itn - snlAn(l - en) n=l
:::; 11~(tn - sn)Anxnll
= IIz-wll· This completes the proof of Theorem 2.3. • Similarly we obtain:
Theorem 2.4 ([15]) If a Banach space X contains an asymptotically isometric copy of co, then X fails the fixed point property for nonexpansive mappings on closed bounded convex subsets of X.
Proof. By assumption there is a null sequence (en) in (0,1) and a sequence (xn) in X so that
s~p(l - en)lltnl :::; II~ anxnl/ :::; s~p lanl,
for all (an) E co. Let (An) be a strictly decreasing sequence in (1,00) converging to l. By passing to subsequences if necessary, we can assume that An+l < (1 - en)An.
Define Yn = AnXn for all n E Nand
c = {~tnYn: (tn) E co, 0:::; tn :::; 1 for all n EN}.
C is clearly a closed bounded and convex subset of X. Define T : C --> C by
00
for L tnYn E C. n=l
It is easily seen that T has no fixed points in C. To see that T is nonexpansive (in fact, contractive), let z = E:::"=l tnYn and w = E:::"=l SnYn be elements of C with z 1= w.
Then
IITz - Twll = 11~(tn - sn)Yn+l11
= 1I~(tn - sn)An+lXn+11/
:::; sup Itn - snlAn+l n
< sup It~ - snlAn(l - en) n
274
~ 11~(tn - sn)Anxnll
= IIZ-Wll·
This completes the proof of Theorem 2.4. • Given the preceding theorems, it is of interest to determine several things. First, do all renormings of £1 or CO contain asymptotically isometric copies of £1 or Co respectively? If the answer were "yes", the preceding results would show that all renormings of £1 and CO fail the fixed point property. Second, if the answer to the first question is "no", we would like to determine which Banach spaces contain asymptotically isometric copies of £1 or Co since these Banach spaces would fail to have the fixed point property.
A natural place to expect to find asymptotically isometric copies of £1 or CO is among the infinite-dimensional subspaces of £1 and Co. There one isn't disappointed. Part (b) of the following theorem is from [16) .
Theorem 2.5 (a) IfY is a closed infinite-dimensional subspace of (£1, 11·111), then Y contains an asymptotically isometric copy of £1.
(b) If Y is a closed infinite-dimensional subspace of (co, II ·1100), then Y contains an asymptotically isometric copy of co.
Thus, a closed subspace of (£1, II ·lId or (CO, II· 1100) has the fixed point property if and only if it is finite-dimensional.
Note the last sentence in the statement of Theorem 2.5 could just as well read: a closed subspace of £1 or Co has the fixed point property if and only if it is reflexive. This phrasing will occur frequently in this chapter.
Proof of (b). Let (en) denote the standard unit vector basis in Co. Let Zn denote the closed linear span of (ej)j~n in (CO, 11·1100). Since Y is an infinite-dimensional subspace of CO, Y n Zn oF {OJ for all n E N. Thus we can choose Xl E Y n Zl with Ilxllioo = 1. We can write Xl = ~~1 a~en, where a~ E lR for all n E N. Let no = 1. Choose n1 > 1 so that sUPn~nl la~1 < 2-2. Since YnZn, oF {OJ, choose X2 E YnZn, with IIx21100 = 1. We can write X2 = ~::"=nl a;en , where a; E lR for all n E N. Choose n2 > n1 so that sUPn~n2Ia~1 < 2-1 .2-3 for i = 1,2. Since Y n Z~ oF {OJ, choose X3 E Y n Zn2 with IIx31100 = 1. We can write X3 = ~::"=n2 a~en, where a~ E lR for all n E N. Continuing in this manner, we obtain a strictly increasing sequence (nk) in N and a sequence (Xk) in Y with Xk = ~::"=nk-l a~en and sUPn~nk_l la~1 < (k _1)-1. 2-k- 1 for alII ~ i ~ k-I with k 2: 2. Fix (tk)k:,l E Co and consider the element z = ~~1 tkxk E Y. Then
Since z E CO, we can also write z = ~::"=1 Znen. Hence, for nk-1 ~ n ~ nk - 1, Zn = ~~=l tia~. Let bk = ~:~-;'~-l a~en and note that Ilbklloo = 1 for all kEN. Thus, for each kEN, we obtain
Renormings of £1 and Co and .fixed point properties
= nk_l~:~nk-ll~ tia~1 ::; (k - 1) Iltll oo Ck_lr;::;;nk-1 la~l)
1::;i::;k-1
::; (k - 1) Iltll oo Ck+1(~ _ 1J = 2-k - 1 IItll oo.
Also, for nk-l ::; n ::; nk - 1, we have:
Therefore,
and so
Thus
and therefore
IZnl = It tia~1 ~ 'tka~'-I~ tia~1 ~ Itklla~l- (k - 1) Iltll oo max la~1
1::;,::;k-1
~ Itklla~l- (k - 1) Iltll oo ((k _ ~)2k+1 ) = Itklla~l- r k-11Itll oo.
II nfl Znen - tkbkll ::; 2-k- 1 1Itlloo ::; 2-kllzlloo n=nk_! 00
In particular, for each kEN,
Choose ko E N so that nko~l
II L znenll = IIzlloo . n=nko-l 00
Then Ilzlloo - Itko I ::; 2-ko Ilzlloo . Consequently,
Ilzlloo::; 1- ~-ko Itkol ::; rr:r: (1_12_k ) Itkl·
275
276
We also have that, for all kEN,
Itkl -II nf1 znenll :::: 2-k IIzlloo. n=nk_l 00
Hence
Therefore
~~r:. (1 +12_k ) Itkl:::: IIzlloo.
Putting the above inequalities together we get that, for all (tk)~l E Co,
This means that (Xk) spans an asymptotically isometric copy of CO in Y.
Since the proof of (a) is similar, this completes the proof of Theorem 2.5. • For the next "example", consider a result of Kadec and Pelczynski [30]: all nonreflexive subspaces of L1 [0,1] contain isomorphic copies of £1. The form of the Kadec-Pelczynski result stated below can be proved by analyzing the proof of the original result (see [10]).
Theorem 2.6 ([14]) If X is a nonreflexive subspace of (L1[0, 1], II . lit), then X contains an asymptotically isometric copy of £1. In particular, every nonreflexive subspace of (L1[0, 1], II· lit) fails the fixed point property.
Since B. Maurey [35] has shown using ultrapower techniques that each reflexive subspace of L1[0, 1] has the fixed point property, Theorem 2.6 used in tandem with Maurey's result yields:
Corollary 2.7 Let X be a subspace of (L1[0, 1], II· lit). Then X has the fixed point property if and only if X is reflexive.
Before continuing with other examples of spaces that have or fail to have the fixed point property using these methods, let us note that, for better or worse, there do exist renormings of £1 which fail to contain asymptotically isometric copies of £1 and renormings of CO which fail to contain asymptotically isometric copies of co. Thus the hope that this method will lead to a proof that all renormings of £1 or Co fail to have the fixed point property is forlorn. In spite of this depressing sounding statement, we shall see that the use of asymptotically isometric copies of £1 and CO yields rich and interesting dividends.
Example 2.8 Let hn) be a sequence in (0,1) strictly increasing to 1. Define a norm on £1 by
00
IIIXIII = sup 'Yn L I~kl n k=n
Renormings of £1 and Co and fixed point properties 277
It is easy to check that III . III is equivalent to the usual £l-norm. We will show that (£\ III . III) does not contain an asymptotically isometric copy of £1.
Let us assume that (£1, III . III) contains an asymptotically isometric copy of £1. Then there is a null sequence (en) in (0,1) and a III· III-normalized sequence (xn) in £1 so that
I:(l-ej)ltjl ~ !!!I:tjXj!!! ~ t Itjl for all t= (tj) Eel. (*) J=l J=l J=l
Without loss of generality we can assume that the sequence (xn) is disjointly supported; i.e., that the support of Xm is disjoint from the support of Xn if m =I n. This is possible because, since the closed unit ball of £1 is weak-star sequentially compact with respect to the predual Co, by passing to a subsequence, we may suppose that (xn) converges weak-star (and hence pointwise with respect to the canonical basis (en)of £1) to some y E £1. By replacing (xn) by the III· III-normalization of the sequence «X2n -X2n-l)/2), we may suppose that y = O. By the proof ofthe Bessaga-Pelczynski Theorem [5, lOJ, we can pass to a subsequence of (xn) which is essentially disjointly supported. Truncating this subsequence appropriately, we obtain a disjointly supported sequence which, when normalized, satisfies (*). Consequently, we can and do assume that (xn) is disjointly supported.
By passing to subsequences if necessary, we can also assume that en < f.n for all n E N.
Let (m(k))~o be a strictly increasing sequence in N U {O} with m(O) = 0 and (~j).i=l a sequence of scalars such that, for each kEN,
For each N E N, we have
m(k)
Xk = L ~jej. j=m(k-l)+1
{ m(l) N
sup Ij L I~kl + Im(l) l:5j:5m(l) k-' Im(N-l)+1
m(N-l)+1:O;i:O;m(N) -J
~ max {1 + N'm(l) , N} Im(N-l)+1
278
Thus N + 1 - El - NEN ::; max {1 + N1'm(l} , N} for all N E N. Since El < -21 and 1'm(N-l}+1
NCN < !, we have N + 1- El - Ncn > N, and hence
Therefore
N"fm(l) N + 1 - El - NCN ::; 1 + ---'--'-
"fm(N-l)+! for all N E N.
1 + ~ _ 101 - ION < ~ + "fm(l) N N - N "fm(N-l)+!
Letting N --> 00 yields 1 ::; "fm(l), a contradiction since "fn < 1 for all n E N.
In order to see the precise difference between a Banach space containing isomorphic copies of fl and a Banach space containing asymptotically isometric copies of fl, consider the following two theorems. The first is a classic due to A. Pelczynski [36] in the separable case and J. Hagler [28] in the nonseparable case. The second result was recently proven by S. Dilworth, M. Girardi, and J. Hagler.
Theorem 2.9 Let X be a Banach space. The following conditions are equivalent:
(a) X contains an isomorphic copy of fl.
(b) X' contains an isomorphic copy of Ll[O, 1].
(c) X' contains an isomorphic copy of C[O, 1]*.
If, in addition, X is separable, (a), (b), and (c) are equivalent to:
(d) X' contains an isomorphic copy of fl(r) for some uncountable set r.
Condition (a) in Theorem 2.9 always implies condition (d), but the converse can sometimes fail. For example, if X = CO(r) for an uncountable set r, condition (d) clearly holds and yet X contains no isomorphic copy of fl. Indeed, if CO(r) contains a subspace Y isomorphic to fl, then there would exist a countable subset ro of r such that Y is a subspace of X = co(ro). Since CO(ro) is isometric to CO and each infinite-dimensional subspace of CO contains another subspace isomorphic to co, this would imply that fl contains a subspace isomorphic to co, an impossible situation for fl.
Banach spaces containing asymptotically isometric copies of fl are characterized in the next theorem. Other equivalent conditions can be found in [11].
Theorem 2.10 ([11)) Let X be a Banach space. The following conditions are equivalent:
(a) X contains an asymptotically isometric copy of fl.
(b) X· contains an isometric copy of Ll [0, 1].
(c) X· contains an isometric copy ofC[O,l]·.
It is possible to use Theorem 2.10 to construct renormings of fl other than the one in Example 10 which contain no asymptotically isometric copy of fl. For example, see
Renorrnings of £1 and Co and fixed point properties 279
Corollary 12 in [11] or take any renorming of £1 such that its dual space is a rotund renorming of £00 (as in Example 2 in [44]). In a similar vein, Dowling [22] has shown that a dual space X' contains an asymptotically isometric copy of Co if and only if X' contains an isometric copy of £00. Thus any rotund dual renorming of £00 fails to contain an asymptotically isometric copy of co. Specific renormings of £1 which contain no asymptotically isometric copy of £1 or specific renormings of CO which contain no asymptotically isometric copy of CO can be found in [13, 16].
Theorem 2.10 and Theorem 2.3 combine to show:
Corollary 2.11 If a Banach space X contains an asymptotically isometric copy of £1, then neither X nor X' has the fixed point property.
In fact, as we shall see in Corollary 2.15 below, a much stronger statement can be made.
Moreover note that Theorem 2.10 combines with Alspach's example [1] to show that if a Banach space X contains an asymptotically isometric copy of £1, then X* also fails to have the weak fixed point property.
In order to give a proof of Theorem 2.10, we need some equivalent reformulations of the definitions of a Banach space containing an asymptotically isometric copy of £1 or an asymptotically isometric copy of CO. These reformulations will give us alternative ways to recognize when a Banach space contains asymptotically isometric copies of £1 or Co will prove extremely useful in many of the results below.
Theorem 2.12 Let X be a Banach space. The following conditions are equivalent:
(a) X contains an asymptotically isometric copy of £1.
(b) There is a sequence (xn) in X and constants 0 < m ::; M < 00 such that, for all (tn) E £1 ,
and lim IIXnl1 = m.
n ..... oo
(c) There is a null sequence (en) in (0,1) and a sequence (xn) in X such that, for all (tn ) E £1 and for all kEN,
Proof. Suppose condition (a) holds. Then there is a null sequence (en) in (0,1) and a sequence (Yn) in X so that
280
Also, since 1 ::; IIXnll ::; (1 - en)-l, Iimn--->oo Ilxnll = 1. Thus conditions (b) holds.
Conversely, suppose that condition (b) holds. Fix a null sequence (en) in (0,1). By scaling if necessary, we can assume that m = 1. Since limn--->oo Ilxnll = m = 1, and Ilxnll 2: m = 1 for all n E N, by passing to subsequences, if necessary, we can assume that 1 ::; Ilxnll ::; 1 + en for all n E N. Define Yn = (1 + en)-lxn for all n E fiT. Then, since IIYnl1 ::; 1, we have
II~ tnynll ::; ~ Itnl for all (tn) E 1!1.
Also
00
n=l 00
2: L (1 - en)ltnl· n=l
Thus X contains an asymptotically isometric copy of £1 so that (b) implies (a).
Since the equivalence of (b) and (c) is immediate, this completes the proof of Theorem 2.12. •
Similarly:
Theorem 2.13 ([16]) Let X be a Banach space. The following conditions are equivalent:
(a) X contains an asymptotically isometric copy of co.
(b) There is a sequence (xn) in X and constants 0 < m ::; M < 00 such that, for all (tn) E Co ,
and
lim Ilxnll = M. n--->oo
(c) There is a null sequence (en) in (0,1) and a sequence (xn) in X such that for all (tn ) E Co and for all kEN,
(1 - .ok) sup Itnl ::; II L tnxnll ::; sup Itnl· n~k n~k n~k
Proof. As in the preceding theorem, we shall show the equivalence of (a) and (b). Their equivalence to condition (c) is then immediate.
Suppose that condition (a) holds. Then there is a null sequence (en) in (0,1) and a sequence (xn) in X such that
s~p (1 - en) Itnl ::; IlL tnxnll ::; s~p Itnl n
Renormings of £1 and Co and fixed point properties 281
for all (in) E co. Let m = infn(1 - cn). Then 0 < m < 1 and, for all (in) E CO,
Also, since 1 - cn :S IIxnll :S 1, we have limn-->oo Ilxnll = 1.
Conversely, suppose that condition (b) holds and let (cn) be a null sequence in (0,1). By considering xn/M rather than Xn, we can assume that M = 1. In particular, m:S Ilxnll :S 1 for all n EN and limn-->oo IIxnll = 1. Hence, by passing to subsequences if necessary, we can assume that 1 - cn :S Ilxnll :S 1, for all n E N.
By passing to subsequences again, if necessary, we can assume that cn < m/4 for all n E N. Define 81 = m and 8n = (4/m)cn for all n;::: 2.
Consider the expression II E~=1 tkxkll for scalars tl, t2, ... , tn. By assumption we have
By scaling, assume that max Itkl = 1. Then l::;k::;n
To show that X contains an asymptotically isometric copy of Co, it suffices to show that
Since we already have the right-hand inequality, it remains only to show the left-hand inequality. First, note that if Itjl < m, then
(1- 8j)ltjl :S (1 - cj)ltjl < m:S lit tkXkll· k=1
Secondly, if Itjl ;::: m, choose Cj with Cjtj = Itjl. By convexity we have
1 - Cj :S Ilxjll
:S ~ IIXj + 1::;k~k;ii CjtkXkl1 + ~ IIXj - 1::;k~k;ii CjtkXkl1
:S ~ IIXj + L Cjtkxkll + ~. 1::;k::;n,k;ii
Hence IIXj + L Cjtkxkll;::: 1 - 2cj. By convexity again we have l::;k::;n,k;ii
1- 2cj :S IIXj + L CjtkXkl1 l::;k::;n,k;ii
:S ~ II t Cjtkxkll + ~ II (2 - Cjtj)Xj + L Cjtkxk II k=1 l::;k::;n,k;ii
282
Thus 11i:>kXkll ~ Itjl- 4ej. Therefore, since Itjl ~ m, we have k=l
Putting the two pieces together we have
Thus (b) implies (a) and the proof of Theorem 2.13 is complete. • We can now give a proof of Theorem 2.10. The proof given below is for real Banach spaces. The theorem also holds for complex Banach spaces and the proof of (b) implies (a) in the complex case may be found in [l1J.
Proof of Theorem 2.10. Suppose that (a) holds. Then, by Theorem 2.12, there is a null sequence (en) in (0,1) and a sequence (xn) in X such that, for all (tn) E £1 and for all kEN,
~ Itnl :S II~ tnxnll :S (1 + ek) ~ Itnl·
Let Y = span(xn) and let (zn) be a dense sequence in the unit sphere of C[O, IJ. Define a linear operator S from Y into C[O, IJ so that S(xn) = Zn for each n E N. Using the left-hand inequality above, it is easy to check that IISII = 1. It is also easy to check that S maps Y onto C[O, IJ. (This is similar to the proof that every separable Banach space is a quotient of £1; see e.g., [lOJ p.73.) Identifying C[O, IJ with its canonical image in C[O, IJ** and using the fact that C[O, IJ** has the Hahn-Banach extension property (i.e., C[O, IJ** is a P=-space), there exists a linear mapping T : X --t C[O, IJ** such that IITII = IISII = 1. The adjoint T* maps C[O, IJ*** into X' and IIT*II = 1. Identifying C[O, IJ* with its canonical image in C[O, IJ***, if Ji, E C[O, IJ*, then
IIT*(Ji,)1I = sup{/-i(Tx) : x E X, Ilxll = I}
~ sup{Ji,(Ty) : y E Y, Ilyll = I}
= sup{Ji,(Sy) : y E Y, Ilyll = I}
= sup{Ji,(f) : f E C[O, 1], Ilfll = I}
=1IJi,1I·
Combined with IIT*II = 1, it follows that IIT*(Ji,) II = 11Ji,11. Thus T* maps C[O,IJ* isometrically into X*.
Since (c) implies (b) is obvious, we need only show that (b) implies (a) in order to complete the proof. So assume that T : L1 [0, IJ -t X* is a linear isometry. Then T* : X** --t LOO[O, IJ is a weak*-weak* continuous, norm-one map onto LOO[O, IJ. By Goldstine's Theorem, W = T*(Bx) is dense in the unit ball B L =[O,lj of LOO[O, IJ.
The idea is now easy to explain. Since the span of the Rademacher functions (Tn) in LOO[O,IJ is easily seen to be isometric to 1'1, we shall get a sequence (T*(xn)) in W
Renormings of £1 and Co and fixed point properties 283
which mimics the action of the Rademacher functions closely enough so that the closed span of the sequence (T*(xn)) is an asymptotically isometric copy of £1 in Loo[O, 1]. It will then be easy to check that the closed span of (xnl is an asymptotically isometric copy of £1 in X.
Let (en) be a null sequence in (0,1). Consider the weak* neighborhood Noo in Loo[O, 1] about the first Rademacher function
rl = X[o,~] - X(~,l]
determined by the functions X[o,~] and X(~,I] in Ll[O, 1] and the number eI/2. Since W is dense in B Loo[O,l], there exists II E W n Noo . It is easy to check that both At = {t E [O,~]: 1-e1 < lI(t):::; I} and A~ = {t E (~,1]: -1:::; lI(t) < -1+c1} have positive measure. Thus there exist pairwise disjoint sets DJ where j = 1,2,3,4
such that each DJ has positive measure and D~ U D~ = A~ and D! U Di = A~. Consider next the "Rademacher-like" function
7'2 = XDt - XD~ + XDA - XDi'
Let NE be the weak* neighborhood in Loo[O, 1] about 7'2 determined by the L1-functions XD!, XD!, XD!, and XD! and the number 62 = min {c2 . 1-£( DJ
1) /2 : j = 1,2,3,4}. Again 1 2 3 4
the density of W in B Loo[O,l] implies that there exists a function f2 E W nNE' As before, it is then easy to check that the sets
Ai = {t E Ai: 1- e2 < f2(t) :::; I} n Dl
A~ = {t E Ai: -1 :::; f2(t) < -1 + e2} n D~ A~ = {t E A~ : 1- c2 < f2(t) :::; I} n D~ A~={tEA~:-I:::;Mt)<-1+e2}nDl
are pairwise disjoint sets with positive measure.
Continuing as above, we obtain subsets
{A~} nEN k=l, ... ,2n
of [0,1] with positive measure and functions (In) in W such that
A~t!l U A~t1 C A~ A~t!l n A~t1 = 0
1 - en < fn(t) :::; 1 if t E A~, k odd
-1:::; fn(t) < -1 + cn if t E A~, k even
Let Xn E Bx such that T*(xn) = fn. For each finite sequence (an)n=l, ... ,N, choose tEA ~ n A2 n··· n AN where jk is odd if ak > a and jk is even if ak < O. Then, since
J1 32 3N -liT" II = 1,
~ II~anxnll ~ IIT*(~anxn)11 = II~anfnll
284
~ itanfn(t)i-N
= I:anfn(t) n=l
n=l
N
~ I: lanl(1 - en) n=l
Thus (xn) spans an asymptotically isometric copy of £1 in X. This completes the proof of Theorem 2.10. •
The reformulations of the definitions of a Banach space having an asymptotically isometric copy of £1 or CO are also useful in identifying other Banach spaces which fail to have the fixed point property. In fact, as we shall see momentarily, several "large" Banach spaces can not even be renormed to have the fixed point property.
Theorem 2.14 ([15, 16]) Let r be an uncountable set. Then the following hold:
(a) Every renorming of £1 (r) contains an asymptotically isometric copy of £1.
(b) Every rcnorming of co(r) contains an asymptotically isometric copy of co.
Consequently, if r is uncountable, neither £1 (r) nor co(r) can be renormed to have the fixed point property_
Proof. Let e-r be the element in £l(r) with e-r(-y) = 1 and e-r(a) = 0 if a i= ,. Let III . III be an equivalent norm on £1 (r). Then there exists constants 0 < m :So M < 00
such that
m I: la-rI:So IIII: a-re-rIII:So MI: la-rl , -rEF -rEF -rEF
for all finite subsets F of r and for all scalars a-r" E F. Define
mA = inf {/III: a-re-rlll : I: la-rl = 1, F is a finite subset of A} -rEF -rEF
where A is an uncountable subset of r. Note that m :So mA :So M for all uncountable subsets of A of rand mA increases as A decreases. Let W1 be the first uncountable ordinal and let (Au)u<w, be a decreasing chain of uncountable subsets of r with nU<Wl Au = 0. Then (mAo )U<Wl is a nondecreasing transfinite sequence of real numbers and hence eventually constant_ Thus there exists aD such that if a ~ aD, then
mAu = mAUD == mO·
Consider Aao. There exists n1 E N, and real numbers a} and elements 'f E Aao for j = 1, ... , n1 such that
n, I:la}1 = 1 j=l
Renormings of C1 and Co and fixed point properties 285
Since na<w,Aa = 0, there exists 0!1 ~ o!o such that 'YJ ¢: An" for j = 1, ... ,n1. Since mAQ , = mo, there exists n2 E N, and real numbers a; and elements 'Y; E An, for j = 1, ... , n2 such that
n2
Lla;1 = 1 j=l
Continuing in this manner we obtain a block basic sequence (Xk) of (e')') where
nk
Xk = Laje,),k and mo::; Illxklll ::; mo + Tk. J
j=l
Then, for all scalars a1, ... , an, we have
and limn~oo Illxklll = mo. Hence, by Theorem 2.11, (C1(r), III· liD contain an asymptotically isometric copy of C1 . This completes the proof of (a).
Since the proof of (b) is analogous using Theorem 2.13, we leave the rest of the proof ~fure~. •
One application of the above theorem is to improve Corollary 2.11 which stated that the dual of a Banach space containing an asymptotically isometric copy of C1 fails the fixed point property. In fact, much more can be said. Note the weakened hypothesis and the stronger conclusion in the next corollary.
Corollary 2.15 If X is a Banach space containing an isomorphic copy of C1 , then X' cannot be renormed to have the fixed point property. In particular, Coo cannot be renormed to have the fixed point property.
Proof. Following the statement of the result of Pelczynski and Hagler (Theorem 2.9), we noted that if a Banach space X contains an isomorphic copy of C1 , then X' contains an isomorphic copy of C1 (r) for some uncountable set r independent of the separability of X. This and an application of Theorem 2.14 completes the proof. •
Before continuing with further applications of these theorems, we pause for a somewhat unexpected example. As just noted, every renorming of Coo contains an asymptotically isometric copy of C1 . One might expect that every renorming of Coo would also contain an asymptotically isometric copy of co. This turns out to be false.
Example 2.16 (COO, III· III) does not contain an asymptotically isometric copy of co, where the III . III norm is defined on Coo as follows: if x = (ej )~1 E Coo then
00
Note that 111·111 is an equivalent norm on COO and Ilxll oo ::; Illxlll ::; 211xll 00 for all x E Coo.
286
Proof. In order to obtain a contradiction, assume that (£00, III· III) does contain an asymptotically isometric copy of co. That is, there is a null sequence (en) in (0,1) and a sequence (xn) in £00 such that
for all scalars tl, t2, ... , tn and for all n E N.
Without loss of generality, we can assume that the sequence (xn) converges pointwise to O. For each n E N, let Xn = (ej)~l' Since IIlxI111 ~ 1-el > 0, there exists j EN
such that eJ =fi O. Let k = min{j : eJ =fi O} and 0 = 3.1. I~l;l. Choose NI ~ k so that L~N'+l2-j < ~. Choose N2 E N so that en < 0, for all n ~ N2. Since (xn) converges pointwise to 0, choose N ~ N2 such that lejl < ~ for j = 1,2, ... , NI and for all n ~ N. Hence, for each n ~ N,
00
j=l N, 00
= IIXnlloo + 2: Tjlejl + 2: Tjle}'1 j=l j=N,+l
N, 00
:S IIXnlloo + 2: 2-j~ + 2: Tj j=l j=N,+l o
:S IIxnlloo + 2
By the convexity of II . 1100' we have IIxnlloo :S ! (IIXI + xnlloo + IIXI - xnlloo). Thus, either Ilxl + xnlloo ~ IIxnlloo or Ilxl - xnlloo ~ IIxnlloo. If IIXI + xnlloo ~ IIxnlloo, it follows that:
which is clearly impossible.
00
j=l ~ IIxnlloo + 2-k I e~ + e;: I ~ Illxnlll- ~ + Tk (le~I-le;:l)
~ Illxnlll- ~ + 2-k (Idl- ~) ~ Illxnlll- 0 + Tk Idl ~ 1 - en - 0 + 2-k lek I ~ 1 - 0 - 0 + 2-k le~ I = 1+0,
A similar contradiction occurs if we assume that IIXI -xnlloo ~ IIxnlloo. This completes the proof that (£00,111'111) contains no asymptotically isometric copy of co. •
Renormings of C1 and Co and fixed point properties 287
The usefulness of the above results in relating the fixed point property with reflexivity can further be illustrated in the context of Orlicz spaces. The basic definitions and facts about Orlicz spaces can be found in [31] or [41]. For now, let us recall a few specific results that we shall need. Hopefully these few comments will make it possible to follow the proof of Theorem 2.17 even for those readers unfamiliar with Orlicz spaces.
First, if (n, I:, f-L) is a finite non atomic measure space, the Orlicz space LiP(f-L) it; reflexive if and only if the Young function <I> and its conjugate function l]i both satisfy a .6.2-
condition for large values. (This is a result due to W. A. J. Luxemburg ([32],[41]).) Recall also that an Orlicz space LiP(f-L) can be endowed with either of two equivalent norms: the Luxemburg norm IlflliP = inf{r > ° : I <I>(f Ir) df-L :s: I} or the Orlicz norm NiP(-) = supU fgdf-L : I l]i(g)df-L :s: I}. (Please note that notation in Orlicz spaces is not standard and our notation is the opposite of the notation in [41].) We shall also use that, if the measure space is finite and not purely atomic, the Young function <I> satisfies a .6. 2-condition for large values of its argument if and only if the Orlicz space LiP (f-L) endowed with the Luxemburg norm contains no isometric copy of Coo ([45]). Since the exact nature of the isometry will prove useful in the following, let us briefly describe how Coo embeds in (LiP(f-L) , 1I·lIiP) when <I> fails to satisfy a .6.2-condition for large values. In this setting, one can construct a sequence of norm-one functions (gn) such that the gn's have disjoint support, J <I> (gn) df-L decreases to 0, and L:::"=1 gn is also norm-one. The map which sends a bounded sequence (an) to L: angn is then an isometry of £00 into LiP(jJ,). Finally, it is useful to note that IlflliP :s: NiP(f) for every f E LiP(f-L), and that if f i' 0, the inequality is strict. ([31], p. 78, [41], p. 73).
A connection between reflexivity and the fixed point property in Orlicz spaces can now be stated.
Theorem 2.17 Let (n, I:, f-L) be a finite measure space that is not purely atomic. The Orlicz space LiP(f-L) endowed with the Orlicz norm has the fixed point property if and only if it is reflexive.
Proof of necessity. Suppose that (LiP (f.1) , NiP(-)) has the fixed point property but that LiP(ll) is not reflexive. Then, by Luxemburg's result mentioned above, either the Young function <I> or its conjugate function l]i fails to satisfy a .6.2-condition for large values.
Suppose <I> fails to satisfy the .6.2-condition for large values. Then Coo embeds isometrically into LiP(f-L) with the Luxemburg norm and, since the Luxemburg and the Orlicz norms are equivalent, Coo embeds isomorphic ally into LiP(f-L) with the Orlicz norm. But this would give a renorming of Coo with the fixed point property, a contradiction to Corollary 2.15.
The only other possibility is that l]i fails to satisfy a .6.2-condition for large values while <I> satisfies a .6.2-condition for large values. Let (gn) denote the functions in (Lo/, 11·110/) = (LiP(f-L),NiPU)* giving rise to the isometry of Coo in (Lo/, 11·110/). Choose functions (fn) in LiP(f-L) such that NiP(fn) = 1, the support of in is contained in the support of gn, and I gnfn df-L > 1 - En where (En) is a sequence decreasing to 0. Then, sinee II L:n(sgnan)gnllo/ = 1,
NiP (~anfn) ~ I (~(sgnan)gn) (~anfn) df.1
= l)anl I gnfn df-L n
288
n
Thus, since NiP(Jn) = 1,
Thus, if W fails the ll.2-condition for large values, (LiP(I-£),NiP (·» contains an asymptotically isometric copy of £1 and thus fails to have the fixed point property. This contradicts the hypothesis and the proof that the fixed point property implies reflexivity is complete. •
Since the proof of the sufficiency in Theorem 2.17 uses techniques unrelated to asymptotically isometric copies of [lor CO, we refer the reader to [15] for the remaining half of the proof.
One last note concerning Orlicz spaces: Although the sufficiency in Theorem 2.17 holds for Orlicz spaces endowed with either the Orlicz norm or the Luxemburg norm, it is unclear if the necessity in Theorem 2.17 remains true if the Orlicz norm is replaced by the Luxemburg norm. The problem is in the case that the Young function <I> satisfies a ll.2-condition for large values and its conjugate W does not. If LiP(l-£) is endowed with the Luxemburg norm, although the method of proof used above still shows that [1 goes into (LiP (1-£) , II . lIiP) almost isometrically, these copies of [1 need not be asymptotically isometric copies of [1. Indeed, if they were, the space (LW, Nw(-) = (LiP (1-£) , II . lIiP)' would, by Theorem 2.10, contain an isometric copy of L1[0, 1]. It is known however (Theorem 3.35, [6]) that, for classical Orlicz spaces, this never occurs. Thus we can ask the following question:
Question 2.18 If (0, E, 1-£) is a finite measure space that is not purely atomic and the Orlicz space LiP(l-£) endowed with the Luxemburg norm has the fixed point property, is LiP(l-£) reflexive?
There are other Banach spaces which fail to have the fixed point property because of the occurrence of asymptotically isometric copies of £1 or CO. We state several theorems without proofs.
Theorem 2.19 ([7]) With mild restrictions on the weight function w, the Lorentz space L w ,l (0,00) contains asymptotically isometric copies of [1. With such weight functions w, a subspace of the Lorentz space Lw ,l(O, OO) has the fixed point property if and only if it is reflexive.
Theorem 2.20 ([12,32]) Nonreflexive subspaces of the tmce classC1 contain asymptotically isometric copies of £1. A subspace of the tmce class C1 has the fixed point property if and only if it is reflexive.
Theorem 2.21 ([12]) Nonreflexive subspaces of the predualpredual(s) M, of a von Neumann algebm M with a faithful, normal and finite tmce T contain asymptotically isometric copies of [1. Thus, nonreflexive subspaces of the predual M, of a von Neumann algebm M with a faithful, normal, finite tmce T fail the fixed point property.
Renormings of £1 and Co and .fixed point properties 289
Theorem 2.22 ([4, 19)} Nonreflexive subspaces of the space K(H) of compact operators on a Hilbert space H contain asymptotically isometric copies of co. A subspace of the K(H) has the fixed point property if and only if it is reflexive.
Since there do exist Banach spaces which contain copies of £1 or Co but do not contain asymptotically isometric copies of £1 or cO, the above methods will not completely answer the general question:'
Question 2.23 Can £1 or Co be renormed to have the fixed point property?
Given the usefulness of reformulating the definitions of a Banach space containing an asymptotically isometric copy of £1 or an asymptotically isometric copy of co, it seems tempting to see if similar reformulations of a Banach space containing an isomorphic copy of £1 or an isomorphic copy of Co are useful in getting new fixed point theorems in the more general setting. Compare the following theorems to Theorems 2.12 and 2.13.
Theorem 2.24 Let X be a Banach space. The following conditions are equivalent:
(a) X contains an isomorphic copy of £1.
(b) There is a sequence (xn) in X and constants ° < m ::; !VI < 00 such that, for all (tn ) E £1 ,
and
lim Ilxnll = M. n-->oo
(c) There is a null sequence (en) in (0,1) and a sequence (xn) in X such that, for all (in) E £1 and for all kEN,
Of course, the equivalence of (a) and (c) is just the stronger version of James's distortion theorem for £1 stated earlier. We state it again here for comparison to Theorem 2.12. The equivalence of (b) to the other two conditions is easy.
Theorem 2.25 Let X be a Banach space. The following conditions are equivalent:
(a) X contains an isomorphic copy of co.
(b) There is a sequence (xn) in X and constants ° < m ::; M < 00 such that, for all (in) E £1 ,
and
lim Ilxnll = m. n-->oo
290
(c) There is a null sequence (en) in (0,1) and a sequence (xn) in X such that for all (tn ) E CO and for all kEN,
sup Itnl ~ II L tnxn II ~ (1 + ek) sup Itnl· n~k n~k n~k
Unlike Theorem 2.24, note that the equivalence of conditions (a) and (c) in Theorem 2.25 is an improvement over the corresponding statement in the stronger version of James's distortion theorem for co. Once we have shown the equivalence of (a) and (c), their equivalence to condition (b) is immediate.
In order to prove the equivalence of conditions (a) and (c) in Theorem 2.25, we employ a beautiful result due to B. J. Cole, T. W. Gamelin, and W. B. Johnson.
Lemma 2.26 ([8]) Let (Xn) be a basic sequence in an infinite-dimensional Banach space X. Then there is a block basic sequence (Yn) of (xn) and a sequence of functionals (y~) in X' which form a unit biorthogonal system of X. That is, for each n E N,
IIYnl1 = IIY~II = Y~(Yn) = 1 and Y~(Ym) = 0 for all m =I n.
Proof of Theorem 2.25. Suppose that (a) holds; i.e., assume that X contains an isomorphic copy of co. Then, by the stronger version of James's distortion theorem, there is a decreasing null sequence (8n ) in (0,1) and a sequence (Xn) in X so that
(1 - 15k) ~~~ Itnl ~ II~ tnxnll ~ (1 + 15k) ~~~ Itnl,
for all (in) E CO and for all kEN.
Since (xn) is a basic sequence in X, there is a block basic sequence (Yn) of (xn) and a sequence of functionals (y~) in X' which form a unit biorthogonal system of X by Lemma 2.26. Thus there is a strictly increasing sequence of integers (kn)~=o, with ko = 0, and scalars t,], where kn - 1 < j ~ kn and n E N, so that
k n
Yn = L t']Xj. j=kn _ 1+l
Since IIYnl1 = 1, we have
Thus Itjl ~ (1 - 8kn _l+d-1 for all kn-l + 1 ~ j ~ kn and for all n E N.
Let (an) E Co and e E N. Then, for each m ~ e,
00
Hence II L anYnl1 ~ sup laml· Also n=l m?::i
Renormings of £1 and Co and fixed point properties
::; (1 + 8"'e_1+1) sup lant}'1 kn-l +l~j~kn
n?:e
::; (1 + 8"'e_1 +1) (1 - 8"'e_1+1) -1 sup lanl n?:l
= (1 + ce) sup lanl , n?:l
where Ci = (1 + 8"'e_1+1) (1 - 8"'e_1+1r1 -1 .
291
Since 8n --> 0 as n --+ 00, ce --> 0 as £ --> 00. Hence the proof of (a) implies (c) is complete. Since it is obvious that (c) implies (a), this also completes the proof of Theorem 2.25. •
With this "improved" version of James's distortion theorem for co, we are now in a position to improve on the result mentioned after Proposition 2.1 that CO cannot be renormed to have the fixed point property for uniformly lipschitzian mappings. Recall that a mapping T : C --> C is said to be asymptotically nonexpansive if IITnx - Tnyll ::; knllx - yll for all X,y E C and for all n E N, where (kn) is a sequence of real numbers converging to 1.
Theorem 2.27 ([17)) If a Banach space X contains an isomorphic copy of co, then there exists a closed, bounded, convex subset C of X and an affine, asymptotically nonexpansive mapping T: C --> C without a fixed point. In particular, Co cannot be renormed to have the fixed point property for asymptotically nonexpansive mappings.
Proof. If X contains an isomorphic copy of co, then, by Theorem 2.25, there is a null sequence (cn) in (0,1) and a sequence (xn) in X so that
for all (tn) E Co and for all n E N.
Define C = n=~=l tnxn : 0 ::; tn ::; 1 and (tn) E co}. Clearly, C is a closed bounded convex subset of X. Define T : C --> C by T(L~=l tnxn) = Xl + L~=l tnXn+1, for all L~=l tnxn E C. It is easily seen that T is affine and has no fixed point in C. Let z = L~=l tnxn and w = L::1 SnXn be elements of C and let kEN. Then
IIT"'z - T"'wll = "~(tn - sn)xn+k II
::; (1 + ck+1) sup Itn - snl n
::; (1 + ck+1) II ~(tn - Sn)xn II
= (1 + ck+1) liz - wll· Since 1 + ck+1 --+ 1 as k --> 00, this shows that T is an asymptotically nonexpansive mapping on C and the proof is complete. •
Whether the analogous result for £1 holds is still unknown.
292
Question 2.28 If a Banach space X contains an isomorphic copy of [1, does X fail the fixed point property for asymptotically nonexpansive mappings or for affine asymptotically nonexpansive mappings on closed bounded convex subsets of X?
Theorem 2.25 can also be applied to relate the occurrence of isomorphic copies of CO in a Banach space and the absence of a geometric condition which implies weak normal structure. A Banach space X has the generalized Gossez-Lami Dozo property (GGLD) if, whenever (xn) is a weakly null sequence in X that is not norm null, then lim infn Ilxnll < lim supn lim supm Ilxn - Xm II . A Banach space X has property asymptotic (P) if, whenever (xn) is a weakly null sequence in X that is not norm null, then liminfn Ilxnll < diama{xn}, where diama{xn} = limndiam{xk : k 2: n} is the asymptotic diameter of the sequence (xn). B. Sims and M. Smyth [43] have shown that the generalized Gossez-Lami Dozo property and property asymptotic (P) are equivalent. The next theorems appear in [20].
Theorem 2.29 If X contains an isomorphic copy of co, then X fails to have the generalized Gossez-Lami Dozo property.
Proof. Suppose that X contains an isomorphic copy of CO. By Theorem 2.25, there is a null sequence (en) in (0,1) and a sequence (xn) in X so that
(**)
for all (tn ) E Co and for all kEN. By passing to subsequences if necessary, we assume that the sequence (en) is decreasing. Since (xn) is equivalent to the unit vector basis of Co, (xn) is weakly null. Clearly, from (**), the sequence (xn) is not norm null.
Also 1 :::; II Xk - Xn II :::; 1 + ek for n 2: k implies that 1 :::; diam {xn : n 2: k} :::; 1 + ek for all kEN which, in turn, implies that diama{xn} = 1. Thus X fails to have property asymptotic (P), and thus fails to have the GGLD. This completes the proof of Theorem 2.29. •
Although Banach spaces containing isomorphic copies of Co fail the generalized GossezLami Dozo property, they may have weak normal structure. In fact, since every separable Banach space can be renormed to be uniformly rotund in every direction and Banach spaces that are uniformly rotund in every direction have weak normal structure [46], there exists an equivalent norm on CO which has weak normal structure and fails the generalized Gossez-Lami Dozo property. The next result shows that such renormings of Co also fail to contain asymptotically isometric copies of Co.
Theorem 2.30 If a Banach space X contains an asymptotically isometric copy of co, then X fails to have weak normal structure.
Proof. Suppose that X contains an asymptotically isometric copy of CO. Then there is null sequence (en) in (0,1), which we may assume is decreasing, and a sequence (xn) in X such that
Renormings of £1 and CO and fixed point properties 293
Since the sequence (xn) is equivalent to the unit vector basis of Co and the unit vector basis of CO is weakly null, (xn) is weakly null. Hence the set {xn : n E N} is a relatively weakly compact set in X and so, by the Krein-Smulian Theorem, K = co( {xn : n E N}), the closed convex hull of {xn : n EN}, is a weakly compact convex subset of X.
Since Ilxn - xmll :::; 1 for all n,m E N, diamK :::; 1. Since Ilxn - xmll ~ 1 - en for all n,m E N with n > m, and (en) is a null sequence in (0,1), diamK ~ 1. Thus diamK = 1.
Consider an element x E co({xn : n EN}). Then x = L:f=I tjXj where tj ~ 0 for all
1 :::; j :::; Nand L:f=I tj = 1. For each n > N, Ilx - xnll = II L:f=I tjXj - xnll ~ 1 - en· Since (en) is a null sequence, this implies that sup{lIx - yll : y E K} ~ 1. Since diamK = 1, we get that sup{llx - yll : y E K} = 1. Hence each x in CO({Xn: n EN}), and each x in the weakly compact set K, is a diametral point of K. Therefore X fails to have weak normal structure. This completes the proof of Theorem 2.30. •
Combining this with previous results yields:
Corollary 2.31 Every infinite-dimensional subspace of (co, 11·1100) and every renorming of co(r), where r is uncountable, fails to have weak normal structure.
Note that the second part of this corollary is a generalization of the result of M. M. Day, R. C. James, and S. Swaminathan [9] that co(r), r uncountable, cannot be renormed to be uniformly rotund in every direction.
Although there are many other results of a geometric nature concerning Banach spaces containing asymptotically isometric copies of £1 or CO, we end with a result concerning the duality of these conditions. Statements and references of other results will be given in the Notes and Remarks section to follow.
Theorem 2.32 Let X be a Banach space containing an asymptotically isometric copy of co. Then X', with its dual norm, contains an asymptotically isometric copy of £1 .
Proof. Since X contains an asymptotically isometric copy of Co, there is a null sequence (en) in (0,1) and a sequence (xn) in X such that
max (1 - en)lanl :::; II../!-- anxnll:::; max lanl, 1 <n<N ~ 1 <n<N - - n=l - -
for all scalars aI, ... , aN and all N E N. Let (x~) be the Hahn-Banach extensions to elements of X' of the linear functionals on the span of (xn) that are biorthogonal to (xn).
N Fix mEN. Then, for all vectors x of the form L: anxn with N ~ m, we have
n=I
IX;" (x) I = laml = (1- cm)-l(l - cm)laml
:::; (1 - cm)-l max (1 - cn)lanl :::; (1 - cm)-Illxli. I:<O;n:<O;N
Therefore Ilx;"11 :::; (1- cm)-I.
294
Define y~ = x~lIx~II-1 for each n E N. Fix scalars aI, a2, ... , aN and let bn = sign an for
all 1 ::; n ::; N. Then, since II I;;;=1 bnxnll ::; l~,;;xN Ibnl = 1, we have
~Ianl ~ lI~anY~1I ~ (~anY~) (~bnxn)
N
= L Ilx~II-1Ianl n=l
N
~ L(1 - cn)lanl· n=l
Thus X* contains an asymptotically isometric copy of £1. • The converse of Theorem 2.32 does not hold as is easily seen by considering X = £1 with its usual norm.
Finally, as an immediate corollary of Theorem 2.32 and previous results, we have:
Corollary 2.33 If a Banach space X contains an asymptotically isometric copy of co, then neither X nor X* has the fixed point property, and X** cannot even be renormed to have the fixed point property.
3. Notes and Remarks
The notion of an asymptotically isometric copy of £1 in a Banach space was introduced in [14] in order to show that the nonreflexive subspaces of L1[0, 1] fail to have the fixed point property. Earlier, J. Hagler, in his Ph.D. thesis [27], Remark 2.3, had noted that a real Banach space contains an asymptotically isometric copy of £1 if and only if its dual space contains an isometric copy of L1 [0,1]. Hagler did not publish this result from his thesis until recently when he, S. Dilworth, and M. Girardi [11] showed Theorem 2.10 (and more) for both real and complex Banach spaces. Unfortunately, until then, this particular result in Hagler's thesis was not widely known.
The corresponding notion of a Banach space containing an asymptotically isometric copy of Co was introduced in [15] where Theorem 2.4 was shown and in [13] as a condition for a Banach space X to be in partial duality with the dual space X* containing an asymptotically isometric copy of £1.
Theorem 2.5, which appearcd in [16], can be viewed as optimal. D. Alspach [2] and E. Behrends [3] have noted that there exist closed subspaces of CO which do not contain subspaces isometric to CO and, in fact, do not even contain isometric copies of twodimensional £'2. V. P. Fonf and M. 1. Kadec [23] have constructed subspaces of £1 which are rotund and thus contain no further subspaces isometric to £1.
Corollary 2.15 appeared as Corollary 3 in [15] where the Banach space was assumed to be separable. Since, in Theorem 2.9, condition (a) always implies condition (d) independent of the separability of the Banach space X, this extra assumption in [15] was unnecessary.
REFERENCES 295
Theorems 2.17, 2.19, 2.20, 2.21, and 2.22 all consider connections between Banach spaces with the fixed point property and reflexivity. One natural inclination would be to hope to connect weakly compact convex sets C with the existence of fixed points for all nonexpansive self-maps of C. However Alspach's [1] example of an isometry on a weakly compact convex subset of Ll[O, 1] without a fixed point shows that weak compactness is not sufficient for a set to have the fixed point property. For examples of non-weakly compact sets such that all nonexpansive self-maps of those sets have fixed points, see Soardi [42] where it is shown that the unit ball of Loo has the fixed point property or the book of Goebel and Kirk [24], page 31, where a certain non-weak' compact subset of £1 is shown to have the fixed point property. In fact, the example in [24] comes from [26] where a decreasing sequence of closed convex subsets of £1 is constructed such that the even-indexed terms have the fixed point property and the odd-indexed terms fail the fixed point property. Inspired by their example, Dowling, Lennard, and Thrett [18] showed that, if a closed bounded convex subset K of a Banach space contains an asymptotically isometric £1-basis, then there exists a nonempty closed bounded convex subset C of K and an affine nonexpansive self-map of C without a fixed point. One corollary of this theorem is that a closed bounded convex subset C of £1 is norm compact if and only if every closed bounded subset of C has the fixed point property for nonexpansive maps. The theorem also combines with a result of Dowling'S [21] to show that a closed bounded convex subset K of Ll[O, 1] is weakly compact if and only if every closed bounded convex subset of K has the fixed point property for nonexpansive affine self-maps. Obviously, given Alspach's example, the word "affine" cannot be omitted.
Finally, recent articles [37, 38, 39] by H. Pfitzner show the existence of asymptotically isometric copies of £1 in nonreflexive subspaces of L-embedded Banach spaces and asymptotically isometric copies of Co in nonreflexive subspaces of M-embedded Banach spaces and consider asymptotically isometric copies of £1 in preduals of von Neumann algebras. Moreover, a recent paper of N. Randrianantoanina [40] constructs asymptotically isometric copies of £1 in any nonreflexive subspace of the dual of an arbitrary CO-algebra or Jordan triple.
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[40J N. Randrianantoanina, Kadec-Pelczyriski decomposition for Haagerup LP-spaces, preprint.
[41J M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Marcel-Dekker, New York, 1991.
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[42J P. M. Soardi, Existence of fixed points On nonexpansive mappings in certain Banach lattices, Proc. Amer. Math. Soc. 73(1979), 25-29.
[43J B. Sims and M. Smyth, On nop.-uniform conditions giving weak normal structure, Quaestiones Mathematicae 18(1995), 9-19.
[44J M. A. Smith and B. Turett, Some examples concerning normal and uniform normal structure in Banach spaces, J. Austral. Math. Soc. Ser. A, 48 (1990), 223-234.
[45J B. Turett, Rotundity of Orlicz spaces, Proc. Acad. Amsterdam A 79(1976), 462-469.
[46J V. Zizler, On Some rotundity and smoothness properties of Banach spaces, Dissertationes Math. (Rozprawy Mat.) 87(1971), 1-33.
Chapter 10
NONEXPANSIVE MAPPINGS: BOUNDARY/INWARDNESS CONDITIONS AND LOCAL THEORY
W. A. Kirk
Department of Mathematics
The University of Iowa
Iowa City, IA 52242-1419 USA
C. H. Morales
Department of Mathematics
University of Alabama in Huntsville
Huntsville, AL 35899 USA
1. Inwardness conditions
1.1. Introduction
Boundary and inwardness conditions have been particularly useful in extending fixed point theory for nonexpansive mappings to broader classes of mappings, particularly to mappings satisfying local contractive and pseudocontractive assumptions. At the same time these conditions often enable one to relax the assumption that the mapping takes values in its own domain.
The principal inwardness conditions in metric fixed point theory are analogues of one found in the topological theory. For a convex set with nonempty interior the usual inwardness and boundary conditions are stronger than the Leray-Schauder boundary condition, yet weaker than the assumption that the mapping send the boundary of the domain back into the domain. In a variety of circumstances they assure the viability of continuation arguments in much the same way the Leray-Scauhder condition does.
We use standard notation. Throughout X will denote a Banach space and B(x;r) will denote the closed ball centered at x E X with radius r > O. We use aD to denote the boundary of a set D c X.
299
WA. Kirk and B. Sims (eds.). Handbook a/Metric Fixed Point Theory. 299-321. © 200! Kluwer Academic Publishers.
300
Condition 1.1 A mapping f : D --4 X is said to satisfy the Nagumo boundary condition if
lim h-1 dist (x - hf(x), D) = o. h~O+
For a subset D of X and xED define the inward set ID (x) as follows:
I D (x) = {x + c (u - x) : u E K and c 2 I} .
Similarly we set
ID (x) = X + {Y EX: lim inf h-1 dist (x + hy,D) = o}. h~O+
If D is closed and convex then it is easy to see that
I D (x) = x + {c (u - x) : u E K and c 2 I}.
Condition 1.2 A mapping T : D -> X is said to be inward if T (x) E ID (x) for each xED.
Condition 1.3 A mapping T : D --4 X is said to be weakly inward if T (x) E ID (x) for each xED.
Condition 1.4 A mapping T: D -> X is said to be Nagumo inward if T (x) E ID (x) for each xED.
Conditions 1.3 and 1.1 are also related.
Remark 1.5 Let D be a convex subset of X and let f : D --4 X. Then f satisfies the Nagumo boundary condition for each xED if and only if the mapping 1- f is weakly inward on D.
The following theorem is due to Caristi [3] and implicit in the work of Martin [21].
Theorem 1.6 Let K be a closed convex subset of a Banach space X and suppose T: K --4 X is a weakly inward contraction mapping. Then T has a (unique) fixed point inK.
Proof. Suppose T has Lipschitz constant k < 1 and let f = I - T. Choose e: > 0 so that k < (1 - e:) / (1 + e:). By Remark 1.5 f satisfies the Nagumo boundary condition so if x E K with x 1= T (x) there exists hE (0,1) such that
dist ((1 - h) x + hT (x), K) < he: Ilx - T (x)ll_
Therefore it is possible to select y E K so that
This gives
11(1 - h) x + hT (x) - yll < he: Ilx - T (x)lI_
( I-e:) Ily-T(y)11 < Ilx-T(x)ll+ k-- Ilx-YII-1+e:
Boundary/inwardness conditions and local theory 301
A simple calculation now leads to
( 1 )-1 Ilx-yll::; l~:-k [lIx-T(x)II-lIy-T(y)II]·
If xi T (x) for each x E K it is now possible to define 'P : K --+ ffi.+ by taking
'P(x)= ~-k Ilx-T(x)lI· ( 1 )-1 l+c
Upon setting y = 9 (x) this results in a mapping 9 : K --+ K satisfying
IIx-g(x)11 ::;'P(x)-'P(g(x)), xEK.
By Caristi's Theorem [3] 9 has a fixed point, contradicting the assumption x i T(x) for each x E K. •
In the following variant of the above one seeks the weakest condition which will assure the existence of a fixed point. This theorem is due to Martinez Yanez [22].
Theorem 1. 7 Let D be a closed subset of a Banach space X and let T : D --+ X be a contraction mapping which satisfies for each xED:
T (x) E x + { c (y - x) : y E D and c 2: I}.
Then T has a unique fixed point.
For an extension of the above result to the multivalued case see Deimling [5], Chapter 5, §11; also Yi and Zhao [33] for compact-valued T, Xu [32] for T satisfying the condition that each set Tx is proximinal relative to x, and Lim [20] for multivalued contractions merely having closed values.
2. Boundary Conditions
2.1. Introduction
There are three common boundary conditions which frequently arise in metric fixed point theory. The first and most well-known is the so-called Leray-Schauder boundary condition (L-S). This condition is used extensively in conjunction with continuation methods in a wide variety of settings. The remaining two have a more geometric character.
Condition 2.1 A mapping T : D --+ X satisfies (L-S) if there exists Zo E int (D) such that
T(zo) - Zo i A(T(z) - z) for all z E aD.
Condition 2.2 A mapping T : D --+ X is inwardly directed if T (aD) c D.
Condition 2.3 A mapping T : D --+ X satisfies the boundary condition D. if there exists Zo E D such that
Ilzo - T (zo)11 < liz - T (z)11 for all z E aD.
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Condition 2.3 has two obvious variants, although condition fj. in conjunction with inf {liz - T (z) II : ZED} = 0 automatically gives the following stronger version.
Condition 2.4 A mapping T : D -> X satisfies the boundary condition fj.. if there exists Zo E D such that
IIzo-T(zo)lI< inf Ilz-T(z)ll. zEfJD
Condition 2.5 A mapping T : D -> X satisfies the boundary condition boundary condition,fj.w fj.w if there exists Zo E D such that
IIzo - T (zo) II ::; liz - T (z)11 for all z E BD.
Two observations are worth noting at the outset.
Remark 2.6 Suppose D is convex and suppose T : D -> X is a nonexpansive mapping which satisfies either Condition 2.2 or Condition 2.3. Then T satisfies Condition 2.1.
Remark 2.7 Suppose D is convex and suppose T : D -> X is a nonexpansive mapping which satisfies Condition 2.2. Then there exists 1-£ E (0,1) such that the mapping f/' defined by
f/' := (1 - 1-£) 1+ I-£T
maps D into D (and has the same fixed points as T).
Remark 2.6 is a simple calculation. Remark 2.7 is an observation of F. Browder. See [19J or [29J for more details.
Theorem 2.8 Suppose X is a uniformly convex Banach space and suppose K is a bounded closed convex subset of X. Suppose int (K) =I 0, and suppose T : K -> X is nonexpansive and satisfies L-S on K. Then T has a fixed point.
An analogue of the above theorem holds for nonconvex K under the assumption that T satisfies fj. on K. However this is a special case of a corresponding result which holds for locally nonexpansive mappings (see Theorem 3.2 below).
Theorem 2.9 Suppose X is a uniformly convex Banach space and suppose D is a bounded open subset of X. Suppose T : K -> X is nonexpansive and satisfies fj. on K. Then T has a fixed point.
2.2. Preliminary results
The remark below is a straightforward and quite useful observation.
Remark 2.10 Let X and Y be topological spaces, G an open subset of X and let f : X -> Y be a mapping for which f(G) is open while f(G) is closed. Then Bf(G) C
f(BG). Moreover, if f is one-to-one, then Bf(G) = f(BG).
Let G be an open subset of a Banach space and let T : G -> X be a nonexpansive mapping. We begin with the following basic facts (see e.g., Gatica and Kirk [7]).
Boundary/inwardness conditions and local theory 303
Proposition 2.11 (I - tTl (G) is open for t E [0,1).
Proof. Fix t E (0,1) and set ft = (1 - tTl . Now let Xo E G, choose p > 0 so that B (xo; p) c G, and let r = (1 - t) p. Fix z E B (ft (xo); r) . We only need to show that for such z there exists wEB (xo; p) such that it (w) = w - tT (w) = z. To this end define Tz : G --+ x by setting
Since
Tz(x) = tT(x) +z, x E G.
IITz (x) - xoll = IltT (x) + z - xoll
= IltT (x) + tT (xo) - tT (xo) + z - xoll
::; tIIT(x) +T(xo)11 + liz - (xo -tT(xo))11
::; t Ilx + xoll + liz - it (xo)11
::; tp+ r = p,
it follows that Tz : B (xo; p) --+ B (xo; p), and since Tz is a contraction mapping it has a unique fixed point, say wEB (xo; p) . Thus tT (w) + z = w from which
ft (w) = w - tT (w) = z.
• Proposition 2.12 If G is bounded and if 0 E G then 0 E (1 - tTl (G) for sufficiently small t E (0,1).
Proof. Choose p > 0 so that B (0; p) c G. Since T is nonexpansive, G bounded =? T (G) bounded =? tT (G) c B (0; p) for t > 0 sufficiently small. For such t, we have tT : B (0; p) --+ B (0; p) . Since tT is a contraction mapping, Banach's Theorem implies that there exists a unique Xt E B (0; p) such that tT (Xt) = Xt. •
It is obvious that if G is bounded and if 0 E (1 - tTl (G) for all t E [0,1), then
inf{llx-T(x)ll: x E G} = O.
Therefore it is important to study the set
S = {t E (0,1) : 0 E (1 - tT)(G)}
and seek conditions which imply S = (0,1). The fact that 0 E int (G) assures that Sis nonempty. The rest follows from he boundary condition
o rt (1 - tTl (oG)
for any t E (0,1). The key observations needed to see this are the fact that for any t E (0,1):
(1) (1 - tTl (G) is an open set. (Proposition 2.11).
(2) 0[(1 - tTl (G)] c (I - tTl (oG). (Remark 2.10).
The conclusion then follows upon showing that
[ = sup {t : t E S} = 1.
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The details are routine. Assume i < 1 and choose tn E S with tn r l. For each n there exists Xn E G such that
This gives
from which
IIXn - xmll = II tnT (xn) - tmT (xm) II ::; tn liT (xn) - T (xm) II + Itn - tmlllT (X'm) II ::; tn Ilxn - xmll + Itn - tmlliT (Xm)ll
Letting m, n --> 00 we see that (since G is bounded) {xn} is a Cauchy sequence. Thus Xn --> x E G from which
tnT (xn) --> iT (x) ,
whence i E S, i.e., 0 E (I -iT) (G) . Since by assumption 0 1:. (I -iT) (aG) it follows from (2) that 0 E (I -iT) (G) . Since (I -iT) (G) is open this in turn implies that o E (I - tiT) (G) for t' > i sufficiently near i, contradicting maximality of l.
The same argument proves that if T is actually a contraction mapping then 1 E S. Thus we have:
Proposition 2.13 Suppose G is a bounded open subset of a Banach space X with o E G, and suppose T : G --> X is a nonexpansive mapping which satisfies
T (x) "# AX for all x E aG and A> 1.
Then inf {llx - T(x)11 : x E G} = o. Moreover ifT is a contraction mapping, then T has a fixed point in G.
In fact, boundedness of G is not needed for the final conclusion of Proposition 2.13. Also a simple translation allows us to restate the proposition as follows.
Proposition 2.14 Suppose G is a bounded open subset of a Banach space X, and suppose T : G --> X is a non expansive mapping which satisfies for some Zo E G :
T (x) - Zo "# A (x - zo) for all x E aG and A > 1. (L-S)
Then inf {llx - T (x) II : x E G} = o. Moreover if T is a contraction mapping, then T has a fixed point in G.
2.3. Nonexpansive and pseudocontractive mappings
An obvious question is whether Proposition 2.14 holds for a wider class of mappings. The key lies in identifying a class of mappings for which Propositions 2.11 and 2.12 hold. We now describe such a class.
Let (., -) denote the usual pairing between elements of X and its dual X* and let J : X --> 2x ' denote the normalized duality mapping
J(X) = {j E X*: IIjll2 = IIxI12 = (x,j)}.
Boundary/inwardness conditions and local theory 305
Definition 2.15 A mapping f : D ~ X -+ X is said to be accretive if for each u, v E D
(f (u) - f (v) ,j) 20 (2.1)
for each j E J (u - v) .
Definition 2.16 A mapping f : D ~ X -+ X is said to be strongly accretive if there exists a constant c > 0 such that for each u, v E D
(f (u) - f (v) ,j) 2 c lIu - vll 2 (2.2)
for each j E J (u - v).
The relationship between accretive mappings and nonexpansive mappings is synergistic. It is known that condition (2.1) has the following equivalent form.
lIu-VlIsllu-V+A(f(u)-f(v))1I forallu,vED,A>O. (2.3)
Therefore if 1>.. = (I - Af), then the mapping f is accretive if and only if the resolvent J). = (f>.)-l is nonexpansive on the range of f>.. Replacing f with 1- T in (2.3) we have
lIu-VlIsll(l+A)u-V-A(T(u)-T(V))1I forallu,vED,A>O. (2.4)
Mappings satisfying (2.4) are said to be pseudocontractive.
Thus we have the following: A mapping f : D c X -+ X is accretive if and only if the mapping T = I - f is pseudocontractive.
The study of accretive operators seems partly motivated by its close connection to the theory of partial differential equations, in addition to the fact all nonexpansive mappings are pseudocontractive, while all contraction mappings are strongly pseudocontractive.
At this point two things are clear. First, if one wishes to study the existence of zeros of accretive mappings, then the focus should be on the fixed point theory for pseudocontractive mappings rather than nonexpansive mappings. Second, the nonexpansive theory is intimately intertwined with the pseudo contractive theory via nonexpansiveness of the resolvent operators. Our next two results amply illustrate this fact.
We begin with the basic existence result for pseudo-contractive mappings which was discovered independently by Caristi [3J and Vidossich [29J using entirely different methods.
Theorem 2.17 Suppose K is a closed convex subset of a Banach space X and suppose K has the fixed point property for nonexpansive mappings. Suppose T : K -+ X is a continuous pseudocontractive mapping which is and weakly inward on K. Then T has a fixed point.
Proof. We consider only the lipschitzian case here. This is Caristi's original approach. The general case may be deduced from the approach in [8], Chapter 13.
If T is lipschitzian it is possible to choose r > 0 so small that kT is a contraction mapping for k = r/ (1 + r). Set H = [I + r(I - T)J-1 . Since T is pseudocontractive H
306
is a nonexpansive mapping on its domain D = [I + r (I - T)J (K). Also, H (D) = K. We will show that D :::J K and thus conclude that H IK: K -> K.
Fix z E K and _define T : K -> X by setting T (x) = kT (x) + (1- k) z. The choice of k assures that T is a contraction mapping. Also, for each x E K, 1K (x) is a convex set which contains K, and since T (x) is a convex combination of elements of the closure of this set it follows that T is also weakly inward on K. By Theorem 1.6 T has a fixed point, say x* E K. Thus
x* = kT (x*) + (1 - k) z = r (r + 1)-1 T (x*) + (r + 1)-1 z.
This implies z = (r + 1) x* - rT (x*) = x* + r (I - T) (x*). Therefore z is in the range of 1+ r (I - T) and thus the domain of H contains K. We now conclude that there exists x E K such that H (x) = x = x + r (x - T (x)) from which T (x) = x. •
We now replace the inwardness assumption with a boundary condition. This will pave the way for the development of the local theory. Indeed, a complete local analog of Theorem 2.19 below is given later in the paper (see Theorem 4.18). A key tool in the development here and elsewhere is the following Domain Invariance Theorem due to Deimling [4J.
Theorem 2.18 If D is an open subset of a Banach space X, and if T : D -> X is continuous and strongly accretive, then T (D) is open.
Theorem 2.19 Suppose D is a bounded open subset of a Banach space X and suppose T : 15 -7 X is a continuous pseudocontractive mapping. Suppose also that there exists zED such that
liz - T (z)11 < IIx - T (x)11 for all x E aD.
Then inf {II x - T( x) II : xED} = o. Moreover, if it is the case that 15 has the fixed point property with respect to nonexpansive mappings, then T has a fixed point in 15.
Proof. The proof of Theorem 2.19 can be broken into three steps.
Step (I). inf {lix - T (x)1I : xED} = O.
PROOF. Since T is pseudocontractive, for fixed r E (0,1) and v, v ED:
(1- r) IIv - vII ::; II (I - rT) (v) - (I - rT) (v)ll·
Thus the mapping U = (1 - r) (I - rT)-l is defined and nonexpansive on
B = (I - rT) (D).
(2.5)
Using the duality between pseudocontractivity and accretivity, there exists j E J (v - v) such that
((I - rT) (v) - (I - rT) (v) ,j) :::: (1 - r) Ilv - v11 2 .
It follows that the mapping I - rT is strongly accretive and one may apply Theorem 2.18 to conclude that (I - rT) (D) is open. On the other hand (2.5) implies that (I - rT) (D) is closed. Thus aB c (I - rT) (aD).
Next let Tr (x) = x - rT (x), xED. Then Tr (x) E aB '* x E aD, and
IITr (z) - U (Tr (z))11 < IITr (x) - U (Tr (x))II·
Boundary/inwardness conditions and local theory 307
Straightforward calculations in conjunction with the fact that U is nonexpansive lead to the conclusion:
U (Tr (x)) - Tr (z) f= A (Tr (x) - Tr (z)) for all Tr (x) E aE and A> 1.
It follows from this that for each t E (0,1) the contraction mapping Ut : E --> X defined by
Ut(Tr (x)) = (1 - t)Tr (z) + tU (Tr (x))
satisfies the Leray-Schauder boundary condition
Ut (Tr (x)) - Tr (z) f= A (Tr (x) - Tr (z)) for all Tr (x) E aE and A > 1.
By Proposition 2.13, Ut has a fixed point Tr (xt) for Xt ED (i.e., for Tr (Xt) E E). Thus
IITr (Xt) - U (Tr (Xt))11 = 11(1 - t) Tr (z) + tU (Tr (Xt)) - U(Tr (xt))11
~ (1 - t) [IITr (z)11 + IIU (Tr (xt})II)·
But U maps E into (l-r)(D) with D bounded, so it must be the case that {U(Tr(xt))} is bounded. Letting t --> 1,
inf {IITr(x) - U(Tr(x))II: Tr (x) E E} = inf {rllx- T(x)11 : xED} = 0.
This proves Step (I).
In view of Step (I) there exists zED such that
IIz-T(z)11 <inf{llx-T(x)11 :xEaD}. (2.6)
Since D is bounded (2.6) implies that a E (0,1) may be chosen so near 1 that for all y E 15,
a liz - T (z)1I + (1 - a) liz - yll < inf {a IIx - T (x)ll- (1 - a) Ilx - yll : x E aD}.
Now define Ua,y : 15 --> X by setting
Ua,y(x)=(I-a)y+aT(x), xED.
Step (II). Ua,y has a fixed point Fa (y) for each y ED.
PROOF. By the choice of a
liz - Ua,y (z)11 < inf {llx - U""y (x)11 : x E aD}. (2.7)
We now fix y E 15 and r E (0,1) , and let 5 = 1- rUr,,,,. Then for u, v E 15 and suitable j E J (u - v) we have by (2.2)
(5 (u) - 5 (v) ,j) = Ilu - vl1 2 - m(T (u) - T (v) ,j)
2: (1 - ar) Ilu - vl1 2 .
This proves that 5 is strongly accretive, so again by domain invariance 5 (D) is open. Hence S (z) ED, and since 5 (D) is closed we have 05 (D) c 5 (aD) .
Now let H = (1 - r) 5-1 . It is now possible to show that:
308
(i) H is a contraction mapping.
(ii) H satisfies the Leray-Schauder boundary condition:
H (x) - z :f. A (x - z) for all x E 8G and A > 1
where G = S(D) and z = S(z).
Having established (i) and (ii) it follows that the mapping H has a fixed point w E G by Proposition 2.13. From the definition of H,
(1 - r) (I - rU""y)-l (w) = Wj
hence
u. (w)_ W "',y 1 - r - 1 - r'
This proves Step (II).
It is now possible to define a mapping F", : 15 -> 15 by taking F", (y) to be the fixed point of U""y. Thus
F",(y) = (l-a)y+aT(F",(y)), YED.
Step (III). F", is nonexpansive.
PROOF. Let u, v E D. Then
Fa (u) - F", (v) = a (T (F", (u)) - T (F", (v))) + (1 - a) (u - v)
and so for some j E J (F", (u) - F", (v)),
(F", (u) - F", (v) ,j) = a(T(F", (u)) - T(Fa (v)) ,j) + (1- a) ((u - v) ,j).
Therefore
IIF", (u) - F", (v) 112 ::; a IIF", (u) - F", (v) II + (1- a) IIu - vll 2
from which
IIF", (u) - F", (v) II ::; IIu - vII .
It is now possible to invoke the assumption that 15 has the fixed point property for nonexpansive mappings and complete the proof of the theorem:
F", (y) = y => y = (1 - a) y + aT (y) => T (y) = y.
• 3. Locally nonexpansive mappings
Boundary conditions are particularly suited to the study of the local theory. A mapping T : G -> X, where G is an open subset of X, is said to be locally k-lipschitzian if given any x E G there exists rx > 0 such that T restricted to B(xj rx) n G satisfies
IIT(u) -T(v)1I S kllu-vll for each u,v E B(xjrx ) nG.
Boundary/inwardness conditions and local theory 309
If k < 1, then T is called a local contraction, and if k = 1 T is said to be locally non expansive. T is said to be a local radial contraction if for each x E G there exists Q (x) E (0,1) and ro; > 0 such that
I/T(u) - T(x) 1/ ::; Q (x) Ilu - xl/ for each u E B(x; r",) n G.
The study of locally lipschitzian mappings is motivated by the well-known fact that if T : G -> X has a continuous Gateaux derivative T~ at each x E G and if I/T~ II ::; k, then the restriction of T to any convex neighborhood of a point in G is lipschitzian with Lipschitz constant k. (See, for example, the discussion in [1].)
Remark 3.1 Suppose D is an open subset of a Banach space and suppose T : D -> X is a continuous mapping which is a local contraction on D. Suppose also that T satisfies the boundary condition Aw on D. Then T has a fixed point.
The above is a trivial consequence of the fact that if
I/zo - T (zo) 1/ ::; I/x - T (x) 1/ for all x E aD then {T" (zo)} is a Cauchy sequence which necessarily converges to a fixed point of T.
Fixed point theory for locally nonexpansive mappings and, more generally, locally pseudocontractive mappings has been studied rather extensively. See, [15] for a survey of early results and, e.g., [25], [27], [30], for more recent developments. The standard underlying assumption in the nonexpansive case is that G is an open subset of a Banach space X with T: G -> X continuous on G and locally nonexpansive on G. Various conditions on X (e.g., uniform convexity) in conjunction with certain boundary conditions assure that such a mapping always has at least one fixed point.
3.1. Existence theorems
The following result is proved in [13].
Theorem 3.2 Let G be a bounded open subset of a uniformly convex Banach space X. Suppose T : G -> X is continuous on G and locally nonexpansive on G. Suppose also that there exists z E G such that
I/z - T(z) 1/ < I/x - T(x) 1/ for all x E aGo
Then T has a fixed point in G.
Another fact is found in Morales [25].
(3.1)
Theorem 3.3 Let X be a Banach space which is uniformly smooth, let G be a bounded open subset of X, and let T : G -> X be continuous and locally nonexpansive on G. Let y E G satisfy
I/y - T(y) 1/ < I/x - T(x) 1/ for all x E aGo
Then for each y E G there is a unique path t ....... Yt E G, 0::; t < 1, satisfying
Yt = (1 - t)y + tT(Yt)
and for which limt-->l- Yt E F(T).
As noted in [13] and [25] respectively, Theorems 2.19 and 3.2 actually hold for the more general class of locally pseudocontractive mappings. Also, Theorem 3.3 extends a result of [2] where it is assumed that X is both uniformly convex and uniformly smooth.
310
4. Locally Pseudocontractive Mappings
4.1. Introduction
Using the Leray-Schauder boundary condition as well as the Conditions ~ and ~w it is possible to extend some of the previous results to mappings characterized by their local behavior. The continuity of the path induced by the eigenvectors of pseudocontractive mappings is a useful tool in this approach. However, these local extensions require a few global results, which we discuss first.
As noted earlier, the accretive operator theory is intimately related to the pseudocontractive one. This fact is one of the motivations for the study of more general classes of operators as can be seen below.
Let a : [0,00) -> [0,00) be a function for which a(O) = ° and the lim infr~ro a( r) > ° for every 1'0 > 0.
Definition 4.1 A mapping A : D c X -> X is said to be a-strongly accretive if for 1L, v E D there exists j E J (u - v) such that
(A(u) - A(v),j) 2' a(11 u - v II) II u - v II· (4.1)
Analogously, a mapping T is said to be a-strongly pseudocontractive if I - T is astrongly accretive.
A mapping A satisfying (4.1) locally, i.e., if each xED has a neighborhood U such that the restriction of A to U is globally a-strongly accretive, is said to be locally a-strongly accretive. The notion of locally a-strongly pseudocontractive can be defined similarly.
If a(r) = cr for some c > 0, then A is strongly accretive(with constant c) since in this case (4.1) becomes
(A(u) - A(v),j) 2' c II u - v 11 2 ,
while for T = I - A and c E (0,1), T is strongly pseudocontractive (with constant 1 - c) and (4.1) takes the form of
(T(u) - T(v),j) ::; (1- c) II u - v 112 . (4.2)
These definitions have obvious local formulations. Of course if c = 0, then the operator A is accretive, while T is pseudocontractive. In the latter case a result of Kato [12] shows that (4.2) and (2.4) are actually equivalent.
The observation below can be easily derived from the definition.
Remark 4.2 If T is pseudocontractive, then tT is strongly pseudocontractive (with CfJIlstant t) for every t E (0,1). On the other hand, if A is accretive, then tI + A is strougly accretive for t > 0.
4.2. Preliminary Results
Proposition 4.3 Let K be a closed convex subset oJ a Banach space X. Suppose J1 : f( ~ X is IL continuous and a-strongly accretive mapping, where
liminfa(r) >11 A(xo) II r~oo
Boundary/inwardness conditions and local theory 311
for some Xo E K. Suppose also that
lim -hI dist(x - hA(x); K) = 0 h-+O
(4.3)
for every x E K. Then A has a unique zero in K.
Proof. As usual we may assume that Xo = O. By a result of Martin [21], K c (1 + A)(K), and since 1 + A is invertible, the mapping 9 = (1 + A)-l is a nonexpansive self-mapping on K. Because the fixed points of 9 are precisely the zeros of A, it is sufficient to show that 9 has a fixed point in K. To this end, one may first show that the set
E = {x E K: Ax = tx for some t < O}
is bounded. To see this, let x E E. Then Ax = tx for some t < 0 and,
(A (x) - A(O),j} ~ a(1I x II) II x II,
for some j E J(x). This implies,
a(1I x II) II x II::; t II x 112 + II A(O) 1111 x II .
Since t < 0, aCllxl1) ::; IIA(O)II, and this implies that E is bounded. As a consequence of this one may easily show that the set
F = {y E K: g(y) = AY for some A> I}
is also bounded. Next it is shown that (1 -g)(K) is a closed set of X (this fact in general does not hold for arbitrary nonexpansive mappings g). Suppose {Yn} is a sequence in K so that Yn - g(Yn) --+ u, for some u EX. Let g(Yn) = xn. Then Yn - g(Yn) = AXn --+ u. On the other hand,
which implies that a(llxn -xmll) ::; IIAxn -Axmll, and thus {xn} is a Cauchy sequence. This means Xn --+ x for some x E K, and since 1 + A is continuous, Yn --+ Y for some Y E K. This implies u = (1 - g)(y).
Finally, let tn E (0,1) so that tn --+ 1- as n --+ 00. Then tng(Yn) = Yn for some Yn E K, which implies that
Yn - g(Yn) = (1 - t:;;l)Yn.
Since {Yn} C F and F is bounded, Yn - g(Yn) ...... 0 E (1 - g)(K). • Proposition 4.3 and Remark 1.5 yield the following facts.
Corollary 4.4 Let K be a closed convex subset of a Banach space X. Suppose that T : K --+ X is a continuous and a-strongly pseudocontractive mapping which is weakly inward on K. Then T has a unique fixed point.
Corollary 4.5 Let K and X be as in Proposition 4.3. Suppose T : K --+ K is a continuous and a-strongly pseudocontractive mapping. Then T has a unique fixed point.
312
Proposition 4.6 Let T : D c X ----> X be a-strongly pseudocontractive mapping. Suppose there exists a bounded sequence {xn} in D and a convergent sequence {tn} in (O,IJ such that Xn - tnT(xn) = Un for a Cauchy sequence {un}. Then {xn} is a Cauchy sequence.
Proof. Since T is strongly pseudocontractive (with constant k) there exists j m J(xn - xm) such that
(t;;:lXn - t-;;,lxm,j) ~ [1- a(rn,m)] . r;,m + (un - um,j), rn,m
where rn,m = Ilxn - xmll. This implies
Therefore,
which completes the proof. • 4.3. Global Results
The fact that pseudo contractive mappings are a much wider family than the nonexpansive mappings leads to the question of whether some of the results obtained earlier for nonexpansive mappings actually extend to pseudocontractive ones. The same question can be posed for the strong pseudocontractions relative to contractions.
Theorem 4.7 Let X be a Banach space and let D be an open subset of X. Suppose T : 75 ----> X is a continuous 00- strongly pseudocontractive mapping which satisfies for some Zo E D, lim infr~(X) a(r) > Ilzo - T(zo) II and
T(x) - Zo i- .A(x - zo) for all x E aD and .A > 1. (L-S)
Then T has a unique fixed point in 75.
Proof. By replacing T( x) with T( x + zo) - Zo and D by D - zo, one may select Zo = ° in (L-S). Consider now the set
E = {x ED: tT(x) = x for some t E [0, I]}.
Then, as in the proof of Proposition 4.3, it can be shown that this set E is also bounded. On the other hand, let
E = {t E [O,lJ : tT(x) = x for some XED}.
Then E i- ¢ (since ° E E). The proof is complete by establishing 1 E E. To accomplish this, we first observe that due to the continuity of Tat 0, we may choose t E (0,1) and 6> ° such that tT maps the closed ball B(Oj 6) into itself for each t E (0, t]. Hence, as a consequence of Corollary 4.5, to := sup E > 0.
Next, let {tn} be a sequence in [; such that tn ----> t as n ----> 00. Then tnT(xn) = Xn for some Xn E D. Since T is a-strongly pseudocontractive and {xn} is bounded, Proposition 4.6 implies that {xn} is a Cauchy sequence, and hence it converges, say to
Boundary/inwardness conditions and local theory 313
x. Consequently, the continuity of T implies that tT(x) = x, and because of the (L-S) condition, xED. Therefore, supE E E.
To complete the proof, suppose supE < 1. Then we may choose a sequence {tn } E [0,1) such that tn -+ tt. Let toT(xo) = Xo for some Xo E D, and let B be an open ball centered at Xo (in D). For t E [0,1], define ht(x) = x - tT(x). Then for each n E N,
Yn := ht" (xo) E htn (B),
while ° ~ ht,,(B). Choose Un E seg[O,Yn) n Bht,,(B). Since ht" is strongly accretive, ht,,(B) is open (by Theorem 2.18), while htJE) is closed. Hence, by Remark 2.10, Bht,,(B) C ht,,(BB). This means, there exists Xn E BB such that ht,,(xn) = Un. Since Yn -+ ° as n -+ 00 and Un E seg[O,Yn), Un -+ 0. Therefore, by Proposition 4.6 again, {xn } is a Cauchy sequence, which converges to Xl EBB. Continuity of T implies that hto(XI) = 0, and since Xo i= Xl, this contradicts the one-to-oneness of hto. •
The connection between the boundary conditions Ll and L-S for nonexpansive mappings was noted earlier. A similar connection for a-strongly pseudocontractive mapping gives the following result.
Corollary 4.8 Let X be a Banach space and let D be an open subset of X. Suppose T : 15 -+ X is a continuous a-strongly pseudocontractive mapping which satisfies for some Zo E D, liminfr-+oo a(r) > Ilzo - T(zo)11 and
Ilzo - T(zo)11 ~ Ilx - T(x)11 for all x E BD.
Then T has a unique fixed point in D.
Proof. In view of Theorem 4.7 it is sufficient to show that (Llw) implies (L-S). To see this, suppose there exists X E BD and ,\ > 1 such that T(x) - Zo = ,\(x - zo). Then there exists j E J(x - zo) such that
('\(x - zo) + Zo - T(zo),j) ~ Ilx - zoW - a(llx - zolDllx - zoll,
which implies
(,\ -l)llx - zoll ~llzo - T(zo)ll- a(lIx - zolD
~llx - T(x)ll- a(llx - zoll)
=(,\ - l)1lx - zoll- a(llx - zolD·
Hence, X = Zo which is a contradiction.
Theorem 4.7 yields an extension of the first conclusion of Proposition 2.14.
•
Corollary 4.9 Let X be a Banach space and let D be a bounded open subset of X. Suppose T : 15 -+ X is a continuous pseudocontractive mapping which satisfies the Leray-Schauder condition. Then inf{llx - T(x)11 : XED} = 0.
Proof. For each t E [0,1), let Tt(x) = (1 - t)zo + tT(x). Then Tt satisfies the Condition (L-S), and hence, by Theorem 11, 'It has a fixed point Xt E 15, such that tT(Xt) + (1 - t)zo = Xt. Since D is bounded, it follows that Xt - T(Xt) -+ ° as t -+ 1-.
•
314
In view of the proof of Corollary 4.8, one might wonder whether the Leray-Schauder condition would imply the Condition llw. Indeed, the proposition below answers this question.
Proposition 4.10 Suppose D,X and T are as in Corollary 4.8. Suppose also that T satisfies the boundary condition (L-8) and is fixed-point free on aD. Then T satisfies the boundary condition llw.
Proof. We first prove that
0:= inf{llx - T(x)11 : x E aD} > O.
Suppose the contrary. Then there exists a sequence {xn } in aD such that Xn -T(xn ) -t
o as n -t 00. However, according to Proposition 4.6 {xn } is Cauchy, and consequently, it converges to some x E aD. This contradicts one of the assumptions on T. On the other hand, Corollary 4.9 implies that inf{l/x - T(x)11 : XED} = O. This means that there exists Wo E D such that Ilwo - T(wo)11 < 0, and hence, the condition 6 w holds .
• It is unknown whether Theorem 2.8 holds for pseudo contractive mappings. In fact, this seems to be one of the few results for nonexpansive mappings for which the continuous pseudocontractive case remains in doubt. At the same time, the following is true.
Theorem 4.11 Let X be a reflexive Banach space which admits a weakly sequentially continuous duality mapping J. Let D be a bounded open subset of X. Suppose T : D -t X is a continuous pseudocontractive mapping which satisfies the Leray-Schauder condition with respect to some Zo ED. Then T has a fixed point in D.
Proof. By the proof of Corollary 4.9, there exists Xn := Xtn E D such that
(4.4)
for each tn E [0,1] with tn -t 1-. Since X is reflexive, we may assume without loss of generality that {xn } converges weakly to some x E X. Then from (4.4) and the fact that T is pseudocontractive, we get
By letting m -t 00, we obtain
and thus
Ilxn - xll2 S (zo - x, J(xn - x)).
Therefore, Xn -t x as n -t 00, and hence the continuity of T implies that T(x) = x •
4.4. Local Existence Results
The boundary conditions introduced in Section 2.1 will now be used for locally kpseudocontractive operators to prove the existence of fixed points. The next result is proven in [23].
Boundary/inwardness conditions and local theory 315
Theorem 4.12 Let X be a Banach space and let D be an open subset of X. Suppose T : 15 -+ X is a continuous and locally strongly pseudocontractive mapping on D which satisfies for some Za ED:
T(x) - Za # ,\(x - za) for all x E aD and ,\ > 1.
Then T has a fixed point in D.
The following facts will be used in the proof of Theorem 4.12.
Proposition 4.13 Let X be a Banach space, D an open subset of X, and T : 15 -+ X a continuous mapping which is locally k-pseudocontractive on D (with k ~ 1). Suppose taT(xa) = Xa for some Xa E D and to E (0,1], and suppose for lTa > 0, B(xo; lTa) C D. Then:
(a) If k < 1 and for all t E (0,1] satisfying
(4.5)
there exists a unique point Xt E B(xa; lTo) such that tT(xt) = Xt. Moreover, this point satisfies
Ilxall IIXt - xall ~ ta(1- tk) It - tal·
1 + to) . .. (b) If k = 1 and t E (0, -2-] whzch satzsfies
to(1 - to) It-tal ~ 211xall lTo,
(4.6)
there exists a unique point Xt E B(xo; lTo) such that tT(xt) = Xt. Moreover, this point satisfies
IIXtl1 Ilxt - xoll ~ ( ) It - tol· t 1 - to
Proof. To prove (a), we use Theorem 4.7. Since T is globally strongly pseudo contractive on B := B(xo; lTo), it is enough to show that tT satisfies the Leray-Schauder condition on aB for t that satisfies (4.5). To see this, suppose for some ,\ > ° and x E aB,
tT(x) - Xo = ,\(x - xo).
Then by the strong pseudocontractiveness of T on B, there exists j E J(x - xa) such that
It follows that
It - tal (,\ - tk)llx - xoll ~ -to-llxoll,
316
which implies that A ::; 1. Therefore, tT has a fixed point Xt E B. The uniqueness follows directly from the fact that tT satisfies (4.2) on B. To complete the proof of (a), observe that for some j E J(Xt - xo),
(C1Xt - t(jlxo,j) ::; kllXt - x01l2,
which implies
Thus
(1- tk)IIXt - xOll2 ::; (tt(jl - 1)(xo,j),
and hence
The proof of (b) follows the same path as in the proof of (a) with k = 1. • Proposition 4.14 Let X, D and T be as in Proposition 4.13. For H c D, set
CH = {t E [O,IJ : tT(x) = x for some x E H}
and let E = {x ED: tT(x) = x for some t E [0, I]}.
Then
(i) the set E is the union of disjoint nontrivial components, each of which is a continuous image of a subinterval of [0, IJ.In addition, if F is any component of E, then
(ii) the function h : CF -+ lR defined by h(t) = IIXt - T(Xt)11 where Xt E F and tT(xt) = Xt is nondecreasing;
(iii) if k < 1 and to E CF, then the set S = {x E F : tT(x) = x for some t E cFn[to, I]} is bounded, and
(iv) iftnT(xtn) = Xtn with tn -+ t- E (O,IJ and {Xtn} C F (with k < 1), then {Xtn} is a Cauchy sequence.
Proof. (i) is an immediate consequence of Proposition 4.13.
(ii) Let t E CF and Xt E F such that tT(xt) = Xt. Then, by assumption, there exists U> ° such that T is k-pseudocontractive on B(XtiU) C D. Suppose sT(xs) = Xs with Xs E B(XtiU) n F and 0::; t < 8::; 1. then by (4.6) we have
Ilx - x II < IlxLiI It - sl s t - t(1- sk)
(4.7)
which yields
l-s 1-8 Ilxs - T(xs)11 =-llxsll ::; - [llxs - Xtll + IIxtll]
8 S
Boundary/inwardness conditions and local theory 317
(I-S) [S-t ] ~ -s- t(1- sk) + 1 IIXtl1
I-t ~-t-IIxtil = IIXt - T(Xt) II.
(iii) Suppose Xto E F with toT(xto) = Xto and select s, x. as in (ii). If to > 0, then by (4.7)
IIX.II ~ [to(l~t~k) +1] IIXtoil
s(1 - tok) IIxto II to(1- sk) 1- tok
< to(l- k) IIXtoll·
If to = 0, then Xto = 0 and IIx.II ~ l~kllT(O)II for x. E B(OjD) with an appropriate.D. Therefore, S is bounded.
(iv) For this part of the proof, we follow the argument given by Kirk and Morales [16]. If tm < tn, then by Proposition 4.13, the segment Itm - tnl can be covered by finitely many overlapping subintervals {Ii}i=l which have the property that for each i and s, t E Ii, the eigenvectors Xs, Xt corresponding to s and t satisfy
M II x s - Xtll ::; to(1- k) It - sl (4.8)
where M = sup{IIXtll : Xt E F, to ~ t ~ I} with to = inf{tn}.
We may now select Si E Ii n Ii+! such that tm = So < Si < ... < Sr = tn. Then by (4.8),
and thus
Therefore, {Xtn } is a Cauchy sequence. • Proof of Theorem 4.12 By replacing T(x) with T(x+zo) - zo one may take zo = 0 in (L-S). Define e as in Proposition 4.14. Since 0 E D and T is continuous, there exists to E (0,1) and r> > 0 such that tT maps the closed ball B(Oj r» into itself for each t E (0, to). Hence, by Corollary 4.5, tT(x) = x for some xED. This means, [0, to) C e. Now consider the t = supe. Then, by Proposition 4.14 (iv), tEe, and according to Proposition 4.13 (a), t = 1. Therefore, T has a fixed point in D. •
Corollary 4.15 Let D be an open subset of a Banach space X and let T : 15 ---+ X be continuous and locally strongly pseudocontractive on D, which satisfies for some zo E D
IIzo - T(zo)11 ::; IIx - T(x) II for x E aD (4.9)
Then T has a fixed point in D.
318
Proof. As before, without loss of generality, one may take Zo = 0 in (4.9). Let E be as in Proposition 4.14, and let Fo be the component of E for which 0 E Fa. Let B be a closed ball centered at 0 such that T restricted to B is strongly pseudocontractive, and let x E B n Fo with tT(x) = x for some t E (0,1). Then by (4.2) there exists j E J(x) such that
(t-Ix - T(O),j) :::; kllxl12,
which implies that (t- I - k)llxll :::; IIT(O)II, and thus Ilx - T(x)11 :::; IIT(O)II. However, Propositions 4.13 and 4.14 (ii) imply that Ilx - T(x)11 :::; IIT(O)II for all x E Fo. Hence, Fo c D, and since this set is a component of E, there exists an open set G in X such that G n E = Fo, which implies that T satisfies the IrS condition on aG. The conclusion now follows from Theorem 4.12. •
We are now in the position of obtaining a local version of Theorem 2.19. We begin with some preliminaries.
Remark 4.16 (cf. Kirk[13]). Let D be an open subset of a Banach space X, and suppose T : D -+ X is a continuous and locally strongly accretive mapping. Let u = T(x) for xED, and let S = seg[u,v] C T(D) for some vEX. Then there exists a unique path (up to parameterization) whose image r begins at x, ends at wED with T(w) = v, and T(r) = S. Moreover, the inverse of the restriction of T to r is a Lipschitz mapping of S onto r.
Remark 4.16 leads to a proof of the existence of an inverse for a locally strongly accretive operator.
Proposition 4.17 (Morales-Mutangadura [26]) Let X be a Banach space, let D be a connected open subset of X, and let T : D -+ X be continuous and locally strongly accretive. Then T is globally one-to-one on D.
We now give a result which is identical to Theorem 2.19, except that the mapping T is assumed to be locally pseudocontractive rather than globally pseudocontractive. However the proofs in the two cases are markedly different.
Theorem 4.18 Let X be a Banach space, D a bounded open subset of X, and T : 15 -+ X a continuous mapping which is locally pseudocontmctive on D. Suppose there exists zED such that
liz - T(z)11 < Ilx - T(x)11 for all x E aD. (4.10)
Then inf{llx - T(x)11 : XED} = O. Moreover, if it is the case that 15 has the fixed point property with respect to nonexpansive seljmappings, then T has a fixed point in D.
Proof. As before we may take z = 0 in (4.10). Since T is continuous at 0, there exists t E (0,1) such that t E CD. Let to = SUpCFo, where Fo is the component of E (as in Proposition 4.14) which contains O. If to < 1, then by Proposition 4.14 (iv) , to E c15' This means, toT(xo) = Xo for some xo E D. However, as we observed earlier, Ilxo - T(xo)11 :::; IIT(O)II, which due to (4.10) implies that Xo E D. This means,
Boundary/inwardness conditions and local theory 319
according to Proposition 4.13, there exists t > to with t E EFo' which contradicts the maximality of to, and hence, to = 1. Therefore, inf{llx - T(x)11 : XED} = O.
To prove the existence of a fixed point for T under the additional assumption on 15, we use the first part of this theorem to redefine the element z in (4.10) so that
liz - T(z)11 < inf {llx - T(x)11 : x E aDI}·
Since D is bounded, we may choose p > 0 large enough and l' E (0,1) such that Dc B(z;p) and
r(llz - yll + Ily - xii) + liz - T(z)11 < Ilx - T(x)11
for all y E B(z; p) and for all x E aD.
For each fixed y E B(z;p), define U: 15 -> X by setting
U(x) = T(x) + r(y - x), x ED.
If x E aD, then (4.11) implies
liz - U(z)11 =llz - T(z) + r(z - y)11
:Sllz - T(z)11 + rllz - yll
<llx - T(x)ll- rilY - xii
:Sllx-U(x)ll·
(4.11)
Since U is continuous on 15 and locally strongly pseudocontractive on D, Corollary 4.15 implies that U has a fixed point xED. Hence (1 + r)x - T(x) = ry, and thus, f(x) := (1 + s)x - sT(x) = y for s = 1'-1. Therefore, B(z;p) C f(D). Since, without loss of generality, D may be assumed to be connected, Proposition 4.17 implies that f is globally one-to-one, and hence the mapping g := f- 1 : B(z;p) -> D is locally nonexpansive, and consequently, is also globally nonexpansive. Therefore, the restriction of g to 15 has a fixed point and the proof is complete. •
As a consequence of the proof of Theorem 4.18, the following result can be obtained. See [25] for the details.
Theorem 4.19 Let X be a Banach space for which the closed unit ball has the fixed point property for nonexpansive selfmappings, let D be a bounded open subset of X, and suppose T : 15 -> X is continuous mapping which is locally pseudocontractive on D. Suppose there exists zED such that
Ilz-T(z)11 < Ilx-T(x)11 for all x E aD. (4.12)
Then T has a fixed point in D.
As a corollary of Theorem 4.19 we obtain an extension of Theorem 3.2 for locally pseudocontractive mappings.
Corollary 4.20 Let X be a uniformly convex Banach space, let D be a bounded open subset of X, and let T : 15 -> X be a continuous mapping which is locally pseudocontractive on D. Suppose there exists zED such that
liz - T(z)11 < Ilx - T(x)11 for all x E aD.
320
Then T has a fixed point in D.
As mentioned earlier, Condition Ll provides a way to approximate fixed points for locally pseudocontractive mappings. The following result is proved in [25]. See also [2], [24], and [26] for related results.
Theorem 4.21 Let X be a uniformly smooth Banach space, D a bounded open subset of X, and T : D -> X a continuous mapping which is locally pseudocontmctive on D. Suppose there exists zED such that
liz - T(z)11 < Ilx - T(x)11 for all x E aD.
Then there exists a unique path t -> Zt E D, t E [0,1), satisfying
Zt = tT(zt) + (1 - t)z,
where the strong lim Zt exists, and this limit is a fixed point T. t--> 1 -
5. Remarks
( 4.13)
As we have remarked earlier, the locally lipschitzian mappings mapping(s),locally lipschitzianarise in a natural way from differentiability assumptions. Whether locally pseudocontractive mappings arise in a similar natural way is less clear. Thus many of the results of the previous section might seem to be more interesting when viewed as results about locally nonexpansive mappings. On the other hand, it seems to be the case that it is just as easy (or difficult) to obtain results for locally pseudocontractive mappings as it is for locally nonexpansive mappings.
There are also two observations in [18] that might be worth mentioning in connection with our discussion of the local theory. If D is an open subset of a metric space X and if T is a mapping of D into a convex metric space Y, then T is said to be locally expansive on D if for each Uo E D there exists r > 0 such that B (uo; r) <:;; D and d(T(u),T(v» 2': d(u,v) for all u,v E D. In [18] it is shown that if X and Yare complete with Y metrically convex, and if T is a closed mapping which maps open subsets of D onto open subsets of Y, then an element y E Y belongs to T (D) if and only if there exists Xo E D such that d (T (xo), y) ::; d (T (x), y) for all x E X\D.
It is also shown in [18] that if a continuous locally accretive mapping defined on a convex open subset of a Banach space is actually accretive. In view of this, for a convex domain D Theorem 4.18 coincides with Theorem 2.19. Also, in both cases, the domain D is assumed to have the fixed point property for nonexpansive mappings. While the class of such sets has not been completely classified, it is known to include nonconvex sets (e.g, [9]).
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[13] W. A. Kirk, A fixed point theorem for local pseudo-contractions in uniformly convex spaces, Manuscripta Math. 30(1979), 89-102.
[14] W. A. Kirk, On zeros of accretive operators, Bull.U. M. I. 17-A(1980), 249-253.
[15] W. A. Kirk, Locally nonexpansive mappings in Banach spaces, in Fixed Point Theory (E. Fadell and G. Fournier, eds.),Springer-Verlag, Berlin, Heidelberg, New York, 1981, pp. 178-198.
[16] W. A. Kirk and C. H. Morales, Fixed point theorems for local strong pseudo-contractions, Nonlinear Anal. 4(1980), 363-368.
[17] W. A. Kirk and R. Sehoneberg, Some results on pseudo-contractive mappings, Pacific J. Math. 71(1977), 89-100.
[18] W. A. Kirk and R. Schoneberg, Mapping theorems for local expansions in metric Banach spaces, J. Math. Anal. Appl. 72(1979), 114-12l.
[19] W. A. Kirk and S. S. Shin, Fixed point theorems in hyperconvex spaces, Houston J. Math. 23(1997),175-187.
[20] T. C. Lim, A fixed point theorem for weakly inward multivalued contractions, J. Math. Anal. Appl. 247(2000), 323-327.
[21] R. H. Martin, Differential equations on closed subsets of a Banach space, Trans. Amer. Math. Soc. 179(1973), 399-414.
[22] C. Martinez Yanez, A remark on weakly inward contractions, Nonlinear Anal. 16(1991), 847-848.
[23] C. H. Morales, On the fixied-point theory for locally k-pseudocontractions, Proc. Amer. Math. Soc. 81(1981), 71-74.
[24] C. H. Morales, Strong convergence theorems for psuedo-contractive mappings in Banach space, Houston J. Math. 16(1990), 549-557.
[25] C. H. Morales, Locally accretive mappings in Banaeh spaces, Bull. London Math. Soc. 28(1996), 627-633.
[26] C. H. Morales and S. A. Mutangadura, On the approximation of fixed points for locally pseudocontractive mappings, Proc. Amer. Math. Soc. 123 (1995), 417-423.
[27] C. H. Morales and S. A. Mutangadura, On a fixed point theorem of Kirk, Proc. Amer. Math. Soc. 123(1995), 3397-340l.
[28] W. Takahashi and G-E. Kim, Strong convergence of approximants to fixed points of nonexpansive nonself-mappings in Banach spaces, Nonlinear Anal. 32(1998),447-454.
[29] G. Vidossieh, Applications of topology to analysis: On the topological properties of the set of fixed points of nonlinear operators, Conf. Sem. Mat. Univ. Bari 126(1971), 1-62.
[30] C. Waters, A fixed point theorem for locally nonexpansive mappings in normed spaces, Proc. Amer. Math. Soc. 97(1986), 695-699.
[31] H. K. Xu, Approximating curves of nonexpansive nonself-mappings in Banach spaces, C. R. Acad. Sci. Paris Ser. I Math. 325(1997), 151-156.
[32] H. K. Xu, Multivalued nonexpansive mappings in Banach spaces, Nonlinear Anal. 43(2001), 693-706.
[33] H. W. Yi and Y. C. Zhao, Fixed point theorems for weakly inward multivalued mappings and their randomizations, J. Math. Anal. Appl. 183(1994), 613-619.
Chapter 11
ROTATIVE MAPPINGS AND MAPPINGS WITH CONSTANT DISPLACEMENT
Wieslawa Kaczor
wkaczor<ilgolem.umcs.lublin.pl
Malgorzata Koter-M6rgowska
mkoterCl!golem.umcs.lublin.pl
Institute of Mathematics
Maria Curie-Sklodowska University 20-031 Lublin, Poland
1. Introduction
This part is a continuation of the chapter by K. Goebel, so we adopt all his notations and definitions.
In general, to assure the fixed point property for nonexpansive mappings some assumptions concerning the geometry of the space are added. Another way is to put some additional restrictions on the mapping itself. One of them is the so-called rotativeness. This property seems to be quite natural in the case of nonexpansive mappings. Moreover, it assures the existence of fixed points in the case of k-lipschitzian mappings with k slightly greater than l.
In what follows mappings with constant displacement will also be discussed.
Let C be a nonempty subset of a Banach space X and T : C -+ C. We say that T is the mapping with constant displacement if it moves each point in C the same distance, that is, if IIx - Txll = const for each x E C. Recall that the mapping T is said to be nonexpansive if IITx - Tyll :::: IIx - yll for all x, y E C.
Note that if T is nonexpansive or with constant displacement, then for any x E C and any n E N,
IIx -T'xll :::: nllx - Txll· (1 )
2. Rotative mappings
Rotativeness is a certain generalization of the well known notion of periodicity. Recall that a mapping T : C -+ C is said to be n-periodic if rn = I d.
323 W.A. Kirk and B. Sims (etis.), Handbook of Metric Fixed Point Theory, 323-337. © 2001 Kluwer Academic Publishers.
324
Definition 2.1 A mapping T: C --4 C is said to be (a,n)-mtative for 0::; a < n, n:::: 2, if
(2)
for each x E C.
We will simply say that the mapping is n-mtative if it is (a, n)-rotative with some a < n, and mtative if it is n-rotative for some n :::: 2. The term "rotative" originates from the fact that all rotations in the euclidean plane satisfy this condition. Clearly, n-periodic mappings are (0, n)-rotative. Note also that all contractions are rotative for all n :::: 2. Since each nonexpansive mapping satisfies (1), rotativeness of such a mapping is caused either by a certain "turning" of a sequence of successive iterations {x,Tx,T2x, ... }, meaning that they do not lie on the same "metrical line", or by a "shortening" of the distances IITix - Ti-1xlI between consecutive terms of this sequence. In the case of mappings with constant displacementrotativeness is resulted only by a turning of the sequence of successive iterations.
It is worth noting here that if a nonexpansive mapping is (a, n)-rotative, then it is also (m - n + a, m)-rotative for all m :::: n.
The following theorem shows that the condition of rotativeness is actually quite strong; it assures the existence of fixed points of nonexpansive mappings even without weak compactness, or another special geometric structure of the set C.
Theorem 2.2 If C is a nonempty, closed and convex subset of a Banach space X, then any nonexpansive mtative mapping T : C --4 C has a fixed point.
Proof. For x E C and a E [0,1) consider the mapping Tx,a : C --4 C defined by
Tx,az = (1- a)x + aTz, z E C.
The mapping Tx,a is a-lipschitzian and therefore it is a contraction. By Banach's Contraction Mapping Principle there exists exactly one fixed point of Tx,a in C, i.e. a point Fax E C such that Tx,aFax = FaX. So we can define a mapping Fa : C --4 C whose set of fixed points coincides with the set of fixed points of T and which is also nonexpansive. Indeed, we have
and
From (3) we get
Fax = (1 - a) x + aTFax, x E C,
l!Fax - FaYIl = 11(1- a) (x - y) + a (TFax - TFaY) II ::; (1 - a) Ilx - YII + a l!Fax - FaYII·
(1 - a) (x - Fax) = a (Fax - TFax)
for any X E C. Moreover, by iterating Fa we obtain
F~x = (1 - a) F~-lx + aTF~x, k = 1,2, ....
Now suppose T is (a, n )-rotative. Then
IIFax - F~xll = IIFax - (1 - a) FaX - aTF~xll
(3)
Rotative mappings
It then follows that
= a IIFax - TF~xll ::; a IlFax - Tn FaxJI + a 11m Fax - T F~xll ::; aa IlFax - T Fax II + a Ilm-1 Fax - F~xll
= (1 - a) a IIx - Faxll + a Ilm-1 Fax - F~xll.
IIFax - F~xll ::; (1 - a) a Ilx - Faxll + a IITn- 1 Fax - F~xll
325
(4)
for all X E C. Now, using only the fact that Tis nonexpansive, we will use induction on m to establish the following:
For m = 2 we have
a IITFax - F~xll ::; a (1 - a) IITFax - Fax II + a2 IITFax - TF~xll
::; (1 - a)2l1x - Faxll + a2 IIFax - F;xll .
Assuming (5) to hold for m we will prove it for m + 1.
a II1""Fax -F~xll ::; a(l- a) IITmFax - Faxll +a2I1TmF"x - TF~xll ::; ma (1 - a) IITFax - Faxll + a211Tm-1 F"x - F;xll = m (1 - a)2l1x - Faxll + a2 111""-1 FaX - F~xll
::; m (1 - a)211x - Faxll + a [em - 1) - ma + am]llx - Faxll + am+! IIFax - F~xll
= [m - (m + 1) a + am+1]lIx - Faxll + am+! IIFax - F~xll .
Finally, using (4) and (5) we conclude
IIFax - F~xll ::; (1 - an)-1 [(1 - a) a + (n - 1) - na + an]llx - Faxll
~ [(a+n) (~"r -l]llx-F.XII (6)
=g(a)lIx-Faxll, xEC.
Note that g is continuous, decreasing for a E (0,1] and g(l) = a/n < 1. So there exists ao E (0,1) such that g(a) < 1 for a E (ao,I). For such a, the iterates {F;x} converge for any X E C to a fixed point of T. •
As we mentioned, in the class of nonexpansive mappings rotativeness appears in a very natural way. Obviously, this condition is independent of nonexpansiveness. Therefore we can also consider k-lipschitzian rotative mappings with k > 1. In this case the condition (2) puts a less natural restriction on the mapping T. However, it assures (provided k is not too large) the fixed point property for closed convex sets in an arbitrary Banach space. Recall that even in a Hilbert space one can construct a selfmapping of the unit ball which is fixed point free and (1 +e )-lipschitzian with ° < e < 1 arbitrarily small (see, e.g., Example 1 in the chapter by K Goebel). Of course this mapping is not rotative, which can be concluded from the following
326
Theorem 2.3 If C is a nonempty, closed and convex subset of a Banach space X, then for any n E N and a < n there exists 'Y > 1 such that any (a, n)-rotative and k-lipschitzian mapping T : C -+ C has a fixed point provided k < 'Y.
Proof. Reasoning similar to that in the proof of Theorem 2.2 can be applied. If T is a k-lipschitzian self-mapping of C, then the mapping Fo : C -+ C given by (3) is well defined for Q E [0, 11k) and it is ll~:k-lipschitzian. In this case one can prove by induction that for all mEN,
( km - 1 1 - OImkm) OIk IITm- 1 Fax - F~xll ~ (1 - 01) k""="l - 1 _ OIk IIx - Foxll (5')
+ OImkm IIFax - F~xll '
and then obtain the following counterpart of (6) :
[ kn -1 1 a+--
IIFax - F~xll ~ 11~ OIk n-l k - 1 - 1 01 ~ (OIk)'
;=0
(6')
= g(OI,k) IIx - F",xll·
It is easy to see that g(OI,k) approaches g(OI), defined in (6), as k --+ 1. Thus, since g(OI) < 1 for Q sufficiently close to 1, there exists 01 E (0, i) such that g(OI, k) < 1 if k is sufficiently close to 1 (say, if 1 < k < 'Y) . For such 01 and k the iterates {F~x} converge for any X E C to a fixed point of F"" which is also the fixed point of T. •
Clearly, 'Y which appears in the proof of the above theorem depends on a, n and the space in which the set C is contained. Thus it is convenient to define the function 'Y: (a) as follows
'Y; (a) = inf {k: there is a closed and convex set C C X and a
fixed point free k-lipschitzian (a, n) -rotative selfmapping of C}.
Now we can reformulate Theorem 2.3 in following way
For any Banach space X, n 2: 2 and a < n, we have 'Y; (a) > 1.
In general, precise values of 'Y: (a) are not known. However, in some cases we can give certain estimates.
For an arbitrary positive integer n, only estimates of the value of the function 'Y: at 0 are known. Namely,
'Y; (0) 2: {2 . ,----;--_-;::::=:==~ n-J n:2 (-1 + In (n -1) - n:l)
Some better results can be obtained for n = 2.
Theorem 2.4 In an arbitrary Banach space X,
'Yi (a) 2: max {~ (2 -a + J(2 - a)2 + a2) ,
for n = 2,
for n> 2.
(7)
Rotative mappings 327
Proof. Let C be a closed and convex subset of X. Suppose T : C -+ C is k-lipschitzian and (a,2)-rotative. For arbitrarily chosen a E (0,1) and x E C, put
Then
z = (1 - a) x + aTx,
u = (1 - a) T 2x + aTx.
liz - Tzil ::; liz - ull + Ilu - Tzil
::; (1 - a) Ilx - T2xll + (1 - a) IIT2x - Tzil + a IITx - Tzil
::; (1 - a) a Ilx - Txll + (1 - a) k IITx - zll + ak Ilx - zll
::; [(1-a)a+(1-a)2 k + a2k] Ilx-Txll
= h1 (a) Ilx - Txll.
On the other hand,
Ilu - Tull = 11(1- a) T 2x + aTx - aTu - (1- a) Tull
::; (1 - a) k IITx - ull + ak Ilx - ull
= (1 - a)2 k IITx - T2xll + ak Ilax + (1 - a) x - (1 - a) T 2x - aTxl1
::; (1 - a)2 k211x - Txll + a2k Ilx - Txll + a (1 - a) k Ilx - T2xll
::; [(1-a)2 k2+ a2k+a(1-a)ka] Ilx-Txll
=h2(a)llx-Txll·
An easy calculation shows that h1 attains its value 4k2±~kk-a2 at a1 = a!;k, while h2 attains its minimum value (k(4k-a2))j(4(1+k-a) at a2 = (2k-a)j(2(1+k-a)). Setting
x6 = X5 = x, x~±1 = (1 - ad Xn + a1Txn, x~±1 = (1 - (2) T 2xn + a2Txn,
we see that the sequences {xh}' i = 1,2, satisfy
and if
n = 0,1, ... ,
{ 2 - a + /(2 - a)2 + a2 a2 + 4 + /(a2 + 4)2 - 64a + 64}
k < max 2 ' 8 '
then either h1(a1) < 1 or h2(a2) < 1, which implies that at least one of the sequences {xh}, i = 1,2, is convergent to a fixed point of T. •
It is interesting that the first term in (7) gives a better evaluation for a E [0, 2( J2 -1)], while the second for a E [2( J2 -1), 2). In particular, for a = ° we get "of (0) 2: 2. This
328
evaluation is not the best possible. For example in the case of Hilbert space H one can prove that If (0) 2: J5 ~ 2.24, which follows from the inequality
H/5 12 (a) 2: V ~. (8)
The above evaluation can be obtained using arguments similar to that in the proof of Theorem 2.4 and additionally applying the fact that for all u, v E H and a E [0,1],
11(1 - a) u + avll2 = (1 - a) lIul1 2 + a IIvl12 - a (1 - a) Ilu - vl1 2 .
The estimate (8) is also not sharp as the following example shows.
Example 2.5 We will show that in the case of Hilbert space
If! (0) 2: Jrr2 - 3 ~ 2.62 .
To this aim let T : C -t C, where C cHis nonempty, closed and convex, be a klipschitzian involution. We will prove that if k < vrr2 - 3, then there is z E [x, Tx) such that IITz - zll :s: (1 - 1':)lIx - Txll with some I': E (0,1). So for x E C and t E (0,1) put z(t) = (1 - t)x + tTx and suppose, contrary to our claim, that
IITz (t) - z (t)11 > (1 - 1':) IIx - Txll
for all t,1': E (0,1). Let get) = Tz(t) - z(t). Then the curve g: t t-+ get) is lipschitzian, g(O) = -g(l) and IIg(t)1I > (1 - 1':) IIg(O) II = (1 - 1':)llx - Txll. Using the fact that the length of a curve laying on S and joining antipodal points exceeds the so-called girth of the sphere, which is rr in Hilbert space, we get that the length of 9 is not less than rr(l - 1':)lIx - Txll. Let 0 = to < tl < ... < tn = 1, ti - ti-l = ~. Then for sufficiently large n the length of the piecewise linear curve U~=l[g(ti-l),g(ti)) is also not less than rr(l - 1':)lIx - Txll. It then follows that
rr2 (1 - 1':)211x - Txll2 :s: (t IIg (ti) _ 9 (ti-1)1I) 2
n
:s: n L IIg (ti) - 9 (ti_l)1I 2
i=l n
= n L IITz (ti) - Tz (ti-l) - [z (ti) - Z (ti_l)JII 2 i=l
= n t [IITZ (ti) - Tz (ti-l)112 + liz (ti) - Z (ti_l)1I 2
- 2 (Tz (ti) - Tz (ti-I), Z (ti) - z (ti-l)) ]
:s: n [ (k2 + 1) t liz (ti) - z (ti_l)1I2
- 2 t (Tz (ti) - Tz (ti-d , z (ti) - z (ti-l)) ] i=l
Rotative mappings
- 2 (~[TZ (ti) - Tz (ti-d] ,Tx - X)
= (k2 + 3) Ilx - Txl12 .
329
Hence 11"(1 - c) :::; ,,)k2 + 3. Since c can be arbitrarily chosen, we get the desired result.
It turns out, however, that the above estimate is also not sharp. Namely, if T is a k-lipschitzian involution which maps the whole Hilbert space H into itself, then it has a fixed point provided k < 1(11"+ ")11"2 - 4) ~ 2.78. To see this one can apply arguments similar to that of the preceding example to the curve t ,..... get) = Faz(t) - z(t), where z(t) = (1- t)x +ty, Fa : H -> H is defined by (3) and y E H is such that FaY = TFax.
We now give two examples providing upper bounds for 'Yr (a).
Example 2.6 Let R : B -> S be a k-lipschitzian retraction of the unit ball B onto the unit sphere S. For c E (0,1) put
{ -R(~) Tx= E
-fxlr for IIxll :::; c,
for c < IIxll :::; 1.
Clearly, T maps B into S and is ~-lipschitzian. Moreover, since T2 = -T, we see that T is fixed point free. It is also easy to show that for x E B,
Ilx- T2X II:::; ~~: Ilx-Txll·
So if c-< 1, then T is (~, 2)-rotative. Therefore in any Banach space X,
X ko (X) (a + 1) "12 (a):::; a-I for a E (1,2) , (9)
where ko(X) is the minimal Lipschitz constant of the retraction of the unit ball onto the unit sphere in X.
In some particular cases better evaluations can be obtained.
Example 2.7 Let X = e[o, 1] and let e c X be defined as follows
e = {x: x is nondecreasing and x (0) = 0, x (1) = I}.
Fork>lset
Tx(t) = kmax{x(t) - (1-~) ,o}. It is easy to see that T : e -> e is k-lipschitzian and moves each point the same constant distance 1- ~. Moreover,
Ilx - T2xll = 1 - :2 = (1 + ~) IIx - Txll .
So T is (1 + ~,2)-rotative. Therefore for 1 < a < 2,
C[O,l) ( ) < 1 "12 a _ --. a-I
(10)
330
It follows from Example 5 in the chapter by K. Goebel that (10) holds also for L1(0, 1).
The above results can be illustrated pictorially in Fig. 1.
k
2
a 2
Figure 1
Set
2 - a + V(2 - a)2 + a2 a2 + 4 + v(a2 + 4)2 - 64a + 64 1" (a) = 2 ,'Y"(a)= 8
and
D1 = {(a, k) E [0,2) x [0,00) : k < max [1" (a), 1''' (a)]),
{ ko (X) (a + 1) } D2 = (a,k)E[0,2)x[0,00):k> , a-I
D3 = {(a,k) E [0,2) x [0,00): k > _1_}, a-I
D4 = ([0,2) x [0,00)) \ (D1 UD3).
It follows from (7), (9) and (10) that the graph of the function 'Yf always lies in the region D4 U D3 and for some spaces X in D4. The "lower" boundary of D4 can be slightly moved up in the case of Hilbert space (see (8)).
3. Firmly lipschitz ian mappings
In this section we show that the estimate (10) is sharp for a certain subclass of the family of k-lipschitzian mappings.
Rotative mappings 331
Definition 3.1 A mapping T : C --> C is said to be firmly k-lipschitzian if for each t E [0,1] and for any X,y E C,
IITx - TYII ::; Ilk (1 - t) (x - y) + t (Tx - Ty)lI· (11)
Of course each firmly k-lipschitzian mapping is k-lipschitzian. We will now consider firmly k-lipschitzian and (a, 2)-rotative mappings. Observe first that for such a mapping T, putting in (11) t = k!l and y = Tx, we get
Next, note that
a Ilx - Txll 2': Ilx - T2xll 2': IIx - Txll -IITx - T2xll
2': (1 - k ~ 1) Ilx - Txll .
Consequently, if T is firmly k-lipschitzian and (a, 2)-rotative, then
k+1 a>--. - 2k+ 1
(12)
On the other hand, it follows from (12) that if k < a~l' then T has a fixed point. So if we define an analogue of ,; by setting
i;; (a) = inf {k: there is a closed and convex set C C X and a fixed
point free firmly k-lipschitzian (a, n) -rotative selfmapping of C}
we get
-x ( ) 1 12 a 2': a-I·
Moreover, since the mapping defined in Example 2.7 is firmly k-lipschitzian and (1 + k, 2)-rotative,
-e[O,ll ( ) _ 1 12 a - a-I·
The same equality holds for Ll(O, 1) (see Example 5 in the chapter by K. Goebel).
332
We can summarize the results of this section pictorially in Fig. 2.
k
-----_ .... _-,
.!. 2
a 2
Figure 2
{ I-a} El = (a,k) E [0,2) x [0,00): k < 2a-l '
{ I-a 1 } E2 = (a, k) : -- ~ k < -- , 2a-l a-I
E3 = { (a, k) : k 2: a ~ I} .
The graph of the function if always lies in the region E3 and for some spaces X it coincides with the lower boundary of E3.
The shapes of E; are a bit different if we deal with a Hilbert space. In this case we have the following
Proposition 3.2 The mapping T : C --+ C, C c H, is firmly k-lipschitzian if and only if for any x,y E C,
IITx - Tyll2 ~ k (x - y, Tx - Ty) . (13)
Proof. Assume that (11) holds. Then
IITx - Tyll2 ~ k2 (1 - t)211x - yll2 + 2kt (1 - t) (x - y, Tx - Ty) + t2 11Tx - Tyll2
and consequently,
2 k2 (1 - t) 2 2kt IITx-Tyll ~ l+t IIx-yll +1+t(x- y,Tx-Ty), tE[O,I).
Rotative mappings
Upon passage to the limit as t -> L we get (13). Conversely, if (13) holds, then
Ilk (1 - t) (x - y) + t (Tx - Ty)11 2
= k2 (1 - t)211x - Yl12 + 2kt (1 - t) (x - y, Tx - Ty) + t211Tx - Tyll2
2: (1 - t)2l1Tx - Tyl12 + 2t (1 - t) IITx - Tyll2 + t211Tx - Tyl12
= IITx - Tyll2 ,
which completes the proof.
333
• Note that (13) implies that T is monotone, i.e. (x - y,Tx - Ty) 2: 0. Consequently, if T is also (a,2)-rotative, then
a211Tx - xl1 2 2: IIT2x _ xl1 2
= IIT2x - Txl12 + 2 (T2x - TX,Tx - x) + IITx - xl1 2 2: IITx - x11 2, which gives a 2: 1. In other words, in a Hilbert space there are no firmly lipschitzian and (a,2)-rotative mappings with a < 1.
Proposition 3.3
-H ( ) 2 12 a 2: a2 _ 2·
Proof. Let T : C -> C, C - closed and convex subset of H, be firmly k-lipschitzian and (a,2)-rotative. Then
a211Tx - xl1 2 2: IIT2x _ xl1 2
= IIT2x - Txl12 + 2 (T2x - Tx,Tx - x) + IITx - xl12
2: IITx - xl12 + G + 1) IIT2x - Txl12 ,
which yields
2 a2 1 IIT2x - Txll ::; k k; 2 IITx - xll 2 .
Hence if a ::; v'2 or if a > v'2 and k < i-2' then T has a fixed point. • It follows from the above considerations that the region E2 in Fig. 2 in the case of Hilbert space is equal to {(a, k) E [1, 2) x [0,00) : k < 2/( a2 - 2)}. The sets El and E3 are changed respectively. However, we do not know whether the lower boundary of E3 coincides with the graph of ;:Y:¥.
4. Mappings with constant displacement
The mapping T considered in Example 2.7 is k-lipschitzian, rotative and moves each point of C the same constant distance (for all x E C, Ilx - Txll = 1- i). We will show later that for any noncom pact set C and any k > 1 there is a k-lipschitzian mapping T : C -> C satisfying IIx - Txll = const > o. So it seems to be natural to discuss k-lipschitzian mappings with constant displacement. It follows from (1) that if
d (T) = inf IIx - Txll , xEC
334
then for any mapping T with constant displacement we have d(T") S nd(T). As we mentioned in Section 2, rotativeness of such a mapping is resulted by a turning of the sequence of its successive iterations. We can introduce an analogue of 'Y: for mappings with constant displacement by setting
1; (a) = inf {k: there is a closed and convex set C C X
and a k-lipschitzian (a, n) -rotative selfmapping
of C with constant positive displacement} .
Of course, 'Y: (a) S 1: (a). Moreover, for an arbitrary Banach space X,
_ 4 'Yf(a) 2 a+2· (14)
To see this, it suffices to take u = !(Tx + T 2x) and to verify that if IIx - Txll = d(T) for all x E C, then
Thus
(15)
which implies that for (a, 2)-rotative mapping T it must be either d(T) = 0 or k 2 a!2. Note that (14) gives a better evaluation of 1;' (a) than the general estimate (7). On the other hand, it follows from Example 2.7 that
-C[O,ll( ) < _1_ 1 < a < 2. 'Y2 a - a-I '
Let us give another example of mappings with constant displacement.
Example 4.1 Assume R: B ..... S is a k-lipschitzian retraction. For 12k, define T by setting
Rx Tx = x - -1-.
Then T is a (1 + f)-lipschitzian mapping of B onto B(O, 1- t-). Clearly,
1 IIx-Txll = y.
Considering mappings with constant displacement we can formulate some equivalents to Schauder's Fixed Point Theorem. They might appear weaker than the Schauder Theorem itself but they are still its equivalents.
Theorem 4.2 Let C be a nonempty, convex and compact subset of a Banach space X. If a continuous self-mapping T of C moves each point the same constant distance d, then d = O.
Rotative mappings 335
Proof. Clearly, this result follows immediately from Schauder's Theorem. For the reverse implication, suppose that a continuous mapping Tl : C -> C is fixed point free. By compactness of C, min"'Ec Ilx - Tlxll = d > 0 and therefore T defined by
X-T1X Tx = x + d II II' x E C, X-T1X
is the mapping with constant positive displacement d. • Recall that lipschitzian mappings form a dense subset in the set of all continuous selfmappings of a compact set C. Consequently, Schauder's Theorem has an equivalent formulation: If C is a nonempty, convex and compact subset of a Banach space X, then every lipschitzian mapping T : C -> C has a fixed point. Similar approximation argument allows us to formulate an analogue of Theorem 4.2.
Theorem 4.3 If C is noncompact and convex, then for each k > 1 there is a klipschitzian mapping T : C -> C which moves all points the same constant distance.
Proof. It is known that for any noncompact set C and any k > 1 there is a klipschitzian mapping Tl : C -> C such that d1 = inf{llx-Tlxll : x E C} > o. Therefore
X-T1X T2X = X + dl II II' x E C, X-T1X
is a lipschitzian mapping with constant displacement. Next, setting T = (I-A)Id+AT2 with an appropriate A, we get a k-lipschitzian mapping such that Ilx - Txll = Ad1 .•
It was shown at the beginning of this section that mappings with constant displacement behave in a special way. Namely, we have (15), which in turn leads to the following
Proposition 4.4 Given 10 E (0,2], if C is noncompact and convex, then there is a continuous mapping T : C -> C such that d(T) = infxEc IIx - Txll > 0 and
d(T2) ~ (2 - ~)d(T).
As such a mapping T we can take any k-lipschitzian mapping with k sufficiently close to 1, which displaces each point by the same fixed distance.
This consideration leads to a subsequent equivalent to Schauder's Theorem.
Theorem 4.5 Given 10 E (0,2], if C is compact and convex, then for any continuous mapping T : C -> C there is x E C such that
In particular, for 10= 2 we have the following, seemingly weaker, equivalent to Schauder's Theorem.
Theorem 4.6 If C is compact and convex, then any continuous mapping T : C -> C has a periodic point with period 2, i. e. T2 has a fixed point.
Using Theorem 2.3 we can modify Proposition 4.4 in the following way: Given 10 E (0, n], n ~ 2, if C is noncompact, then there is a continuous mapping T : C -> C
336
such that d(T) > ° and d(T") 2: (n - c)d(T). Therefore we get another equivalent formulation of Schauder's Theorem:
Theorem 4.7 Given c > 0, if C is compact and convex, and T : C -+ C is a continuous mapping, mapping(s),continuousthen there exists x E C such that IIx -T"xll ~ (n - c)lIx - Txll.
Theorem 4.8 If C is compact and convex, then any continuous mapping T : C -+ C has a periodic point with period n, i.e. T" has a fixed point.
Finally, let us add another theorem to our list of equivalent formulations of Schauder's Theorem.
Theorem 4.9 Given M > 0, if C is compact and convex, then for any continuous mappings Tl, T2 : C -+ C there exists x E C such that
Proof. Clearly, Schauder's Theorem implies our statement. For the converse, suppose that Schauder's Theorem does not hold. Then by Theorem 4.7 there is a continuous mapping T2 such that IIx-T2'xll > (n-c)lIx-T2XIl for all x. Now we can take Tl = 12' .
• 5. Notes and remarks
The notion of rotativeness was introduced by K. Goebel and M. Koter in [7). Basic results about rotative mappings, which we present in Section 2, originate from [7), [8) and [15). The estimate (8) and a part of (7) are due to T. Komorowski [14). In [11) one can also find some evaluations of "f(a) and "f(a). In the book by K. Goebel and W. A. Kirk [6) one chapter is devoted to rotative mappings. Moreover, in the survey [9) not only the current state of knowledge concerning such mappings is presented, but also several interesting open problems are posed.
It is worth noting here that A. T. Plant and S. Reich [18) extended the notion of rotativeness onto nonlinear semigroups. If S is a semigroup generated by -A, then S is said to be rotative if for some p > ° and a < p, IIx - S(P)xll ~ all Ax II for all x in the domain of A. It is shown in [18) that if C is a closed subset of Banach space and a nonexpansive mapping T : C -+ C is such that A = I d - T satisfies the so-called range condition, then the semigroup S generated by -A is rotative if and only if T is. The fixed point result for nonexpansive rotative semigroups is also established.
As we mentioned, rotative mappings include periodic mappings and in particular involutions. First fixed point theorems for involutions are due to K. Goebel and E. Zlotkiewicz [3), [10). It is shown that if T : C -+ C, where C is a closed and convex subset of a Banach space, is a k-lipschitzian involution with k < 2, then T has at least one fixed point. The proof of this fact is a straightforward verification that starting from any x E C, the sequence of iterates {Fnx} for F = ! (I + T) always converges to a fixed point of T. W. A. Kirk [13) extended this result by proving that the same is true if Tis n-periodic and such that IlTix - Tiyll $ kllx - yll for x, y E C, i = 1, ... , n - 1, where n-2[(n - 1)(n - 2)k2 + 2(n - l)k] < 1. This result leads to the estimate (*).
REFERENCES 337
The notion of firmly lipschitzian mappings was introduced in [15] and is analogous to that of firmly nonexpansive mappings originating from Bruck's paper [2]. Basic results of Section 3 are taken from [15]. Certain properties of firmly lipschitzian mappings and some facts regarding their asymptotic behavior can be found in [12].
Mappings with constant displacement are studied in [5]. This paper contains also a number of other (not presented here) equivalent formulations of Schauder's Fixed Point Theorem [19].
Other basic facts used here come from [1] and [17] (existence of a lipschitzian retraction of the unit ball in an infinite dimensional Banach space onto its unit sphere), [20] (girth of sphere) and [16] (existence of a lipschitzian self-mapping of a noncompact set with positive minimal displacement).
References
[1] Benyamini Y, Sternfeld Y. Spheres in infinite dimensional normed spaces are Lipschitz contractible, Proc. Amer. Math. Soc. 88 (1983), 439-445.
[2] Bruck R. E. Nonexpansive projections on subsets of Banach spaces, Pacific J. Math 48 (1973), 341-357.
[3] Goebel K. Convexity of balls and fixed point theorems for mappings with a nonexpansive square, Compostio Math. 22 (1970),269-274.
[4] Goebel K. On the minimal displacement problem, Pacific J. Math. 48 (1973), 151-163.
[5] Goebel K., Kaczor W. Remarks on failure of Schauder's Theorem in noncompact settings, Ann. Univ. Mariae Curie-Sklodowska 51 (1997),99-108.
[6] Goebel K., Kirk W. A. Topics in metric fixed point theory, Cambridge University Press, 1990
[7] Goebel K., Koter M. A remark on nonexpansive mappings, Canadian Math. Bull. 24, (1981) 113-115.
[8] Goebel K., Koter M. Fixed points of rotative lipschitzian mappings, Rend. Sem. Mat. Fis. Milano 51 (1981), 145-156.
[9] Goebel K., Koter-M6rgowska M. Rotative mappings in metric fixed point theory, Proc. NACA98, World Scientific (1999), 150-156.
[10] Goebel K., Zlotkiewicz E. Some fixed point theorems in Banach spaces, Colloquium Math. 23 (1971), 103-106.
[11] G6rnicki J. Remarks on fixed points of rotative Lipschitzian mappings, Comment. Math. Univ. Carolinae 40 (1999), 495-510.
[12] Kaczor W., Koter-M6rgowska M. Firmly lipschitzian mappings, Ann. Univ. Mariae CurieSklodowska 50 (1996), 77-85.
[13] Kirk W. A. A fixed point theorem for mapping8 with a nonexpansive iterate, Proc. Amer. Math. Soc. 29 (1971), 294-298.
[14] Komorowski T. Selected topics on lipschitzian mappings, (in Polish) Thesis, Univ. Maria CurieSklodowska, 1987.
[15] Koter M. Fixed points of lipschitzian 2-rotative mappings, Boll. Un. Mat. Ital. Sel". VI, 5 (1986), 321-339.
[16] Lin P. K., Sternfeld Y. Convex sets with the Lipschitz fixed point property are compact, Proc. Amer. Math. Soc. 93 (1985), 633-639.
[17] Nowak B. On the Lipschitz retraction of the unit ball in infinite dimensional Banach spaces onto boundary, Bull. Polish Acad. Sci. 27 (1979), 861-864.
[18] Plant A. T., Reich S. Nonlinear rotative semigroups, Proc. Japan Acad. 58 (1982), 398-40l.
[19] Schauder J. Der Fixpunktsatz in Punktionalriiumen, Studia Math. 2 (1930), 171-180.
[20] Schaffer J. J. Geometry of spheres in normed spaces, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, Inc., New York-Basel, 1976.
Chapter 12
GEOMETRIC PROPERTIES RELATED TO FIXED POINT THEORY IN SOME BANACH FUNCTION LATTICES
S. Chen
Harbin University of Science and Technology
Harbin 150080, P.R. China
schen<ilpublic.hr.hl.cn
Y. Cui
Harbin University of Science and Technology
Harbin 150080, P.R. China
H. Hudzik
Institute of Mathematics
Poznan University of Technology
Piotrowo 3a 60-965 Poznan, Poland
hudzik<ilamu.edu.pl
B. Sims
Mathematics, School of Mathematical and Physical Sciences
The University of Newcastle
NSW 2308, Australia
1. Introduction
The aim of this chapter is to present criteria for the most important geometric properties related to the metric fixed point theory in some classes of Banach function lattices, mainly in Orlicz spaces and Cesaro sequence spaces. We also give some informations about respective results for Musielak-Orlicz spaces, Orlicz-Lorentz spaces and CalderonLozanovsky spaces.
339
W.A. Kirk and B. Sims (eds.), Handbook of Metric Fixed Point Theory, 339-389. © 2001 Kluwer Academic Publishers.
340
1.1. Orlicz Spaces
Some general facts. We denote by R, R+ and N the sets of real numbers, nonnegative real numbers and natural numbers respectively.
A mapping iP: R --> R+ is said to be an Orlicz function if:
(i) iP is even, continuous, convex and vanishing only at zero,
(ii) limu-+o(¥) = 0 and limu -+oo(¥) = 00.
An Orlicz function iP is said to satisfy the ~2-condition at zero (iP E ~2 for short) if there are constants K 2:: 2 and a > 0 such that iP(a) > 0 and iP(2u) :'S KiP(u) for all real u with lui < a.
It is well known (see [Lu 55], [Mal 89), [Mu 83] and [Ra-Re 91)) that iP is an Orlicz function if and only if iP(u) = Iciul p(t)dt, where p is the right derivative of iP satisfying the following conditions:
(iii) p is right-continuous and nondecreasing on R+,
(iv) pet) > 0 whenever t > 0,
(v) p(O) = 0 and limt-+oop(t) = 00.
Hence it follows immediately that
1 (u) iP(u) 2 P 2 :'S ----:;;- :'Sp(u)
By the convexity of iP and iP(O) = 0, we get
(u > 0).
iP(au) < aiP(u) (0 < a < 1, u > 0),
which yields
iP(u) < iP(v) (0 < u < v). u v
For the function p satisfying conditions (iii), (iv) and (v), we define
q(s) = sup{t > 0: p(t):'S s} = inf{t > 0: pet) > s},
(Ll)
(1.2)
which we call the right-inverse function of p. It is easy to show that q also satisfies conditions (iii), (iv) and (v). If iP is an Orlicz function with the right derivative p and q is the right-inverse function of p, then the function
(Ivl \lI(v) = 10 q(s)ds
is called the complementary function of iP (or the Young conjugate of iP). It is well known (see [Ch 96], [Lu 55], [Mal 89], [Mu 83] and [Ra-Re 91)) that we have the Young inequality
uv :'S iP(u) + \lI(v) (u, v 2:: 0)
and that the equality uv = iP(u) + \lI(v) (1.3)
holds for u 2:: 0 if and only if v E [p_(u),p(u)], where p_ is the left derivative of iP.
Geometrical properties in function lattices 341
Sometimes Orlicz functions are defined only by condition (i). It is easy to see that (if>(u)/u) -+ 0 as u -+ 0 is equivalent to the fact that W vanishes only at zero and (if>(u)/u) -+ 00 as u -+ 00 is equivalent to the fact that W has only finite values.
Example 1.1 Let if> be an Orlicz function. If W1 is the Young conjugate of the function if>1 defined on R by if>1(U) = aif>(bu), where a,b are fixed positive numbers and p is the right derivative of if> on ~, then the right derivative of if>1, is Pl(t) = abp(bt) and. so its right-inverse function is
ql(S) = ~qCbs), where q is the right derivative of W, and W is the Young conjugate of if>. Hence
{Ivl (W (Ivl) Wl(V) = Jo ql(s)ds = a Jo q(s)ds = aW ab .
Example 1.2 Let Wi, W2 be the Young conjugates of Orlicz functions if>1 and if>2, respectively. Suppose that
Consider the relationship between W1 and W2. By the Young inequality and equality (1.3), we have
Hence by
we obtain
Let (T, E, j.t) denote a nonatomic, complete and finite measure space and denote by LO = LO(T, E, j.t) the space of all (equivalence classes of) E-measurable real functions defined on T.
Given an Orlicz function if>, we define on LO a convex modular Iif! by
Iif!(x) = l if> (x(t))dj.t.
The Orlicz space Lif! generated by if> is the set of those x E LO that Iif!(AX) < 00 for some A > O. If 1° is the space of all real sequences x = (x(i))~1' then the modular Iif! is defined on 1° by
00
Iif!(x) = Lif>(x(i)) i=l
and the corresponding space Iif! = {x E 1°: Iif!(AX) < 00 for some A > O} is called the Orlicz sequence space. We also define
Eif! = {x E LO: Iif!(AX) < 00 for any A> O}, and
hif! = {x E 1°: Iif!(AX) < 00 for any A> O}.
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Lemma 1.3 Let cJ> be an Orlicz function and W be its Young conjugate. Then the following are equivalent:
(i) cJ> E .6.2.
(ii) There exist I> 1,11.0 > 0 and K > 1 such that
cJ>(lu) ~ KcJ>(u) (11. > 11.0). (1.4)
(iii) For any II > 1 and 11.1 > 0 there exists K' > 0 such that (1.4) holds for I =
11,11.0 = 11.1 and K = K'.
(iv) For any 12 > 1 and 11.2 > 0 there exists e in the interval (0,1) such that
(1.5)
(v) For any 13 > 1 there exist Vo > 0 and «5 > 0 such that
W(13v) 2: (13 + 8)W(v) (v 2: va). (1.6)
(vi) There exist 13 > 1, Vo > 0 and «5 > 0 such that (1.6) holds.
Proof.
The implication (i) =} (ii) is obvious.
(ii) =} (iii). Given II > 1, choose an integer Q such that I" > II' Then by (1.4),
cJ>(l)u) ~ cJ>(I"u) ~ K"cJ>(u) (11. 2: 11.0).
Hence, if 11.1 2: 11.0, then K" is a candidate for K'. If 11.1 < 11.0, then, we choose K' = max(K", Ko), where
Ko = max{cJ>(IIU)/cJ>(U): 11. E [U1,Uo]}·
(iii) =} (iv). For 12 > 1 and 11.2 > 0, by (iii), there exists K' > 12 such that
cJ>(2u) ~ K'cJ>(u) (11. 2: 11.2).
Take e = (12 - 1)/(K' - 1). Then 0 < e < 1 and by the convexity of cJ>,
cJ>«1 + e)u) = cJ>«1 - e)u + 2eu) ~ (1 - e)cJ>(U) + ecJ>(2u),
~ (1 - e)cJ>(u) + eK'cJ>(u) = 12cJ>(U) (11. 2: 11.2).
(iv) => (v). For any 13 > 1 and Vo > 0, choose 11.2 E (0, q(vo)]. Then inequality (1.5) and Examples 1.1 and 1.2 imply
1 (13 ) W(v) ~ z;cJ> 1 + e v (v 2: vo)·
Hence it follows that
13W(V)~WC~eV) ~ l:eW(13(v)) (v 2: vo).
By setting «5 = 13e, we get (1.6).
The implication (v) => (vi) is trivial.
Geometrical properties in function lattices
(vi) => (i). Let (3 = (/3 + 8)/13. Then (l.6) can be written in the form
1 (313 w(13v ) 2 w(v) (v 2 va).
343
Choose n E N such that (3n 2 2 and set K = (3nl'3. Then by Examples 1.1 and 1.2, we get
2. Normal structure, weak normal structure, weak sum property, sum property and uniform normal structure
•
Let X be a Banach space and {xn} a sequence in X such that Xi # Xj whenever i # j. If for any X E co {xn}, the convex hull of {xn}, the limit 6(x) = limn Ilx - xnll > 0 exists and 6(x) is affine on co {xn}, then {xn} is called a limit affine sequence. If, in particular, 6(x) is equal to a constant on co{xn}, then {xn} is called a limit constant sequence. If X contains no (weakly convergent) limit affine sequence {xn} satisfying 6(xn) t, then it is said to have the (weak) sum-property. We say that X has (weak) normal structure (NS (WNS)) if it contains no (weakly) convergent limit constant sequence.
The original definition of the (weak) normal structure is given in the following equivalent way:
X has (weak) normal structure if for any nonsingleton (weakly compact) nonempty bounded closed convex subset C of X, there exists x E C such that
rc(x) := sup{llx - YII: Y E C} < diam (C) := sup{llu - vii: u, v E C}.
Moreover, if there exists h E (0,1) such that for each nonsingleton nonempty bounded closed convex subset C, there exists x E C such that rc(x) :os: (1- h)diam (C), then X is said to have uniform normal structure (UNS).
The above concepts are introduced as a powerful tool in fixed point theory. For instance, if X has weak normal structure, then it has weak fixed point property (w-FPP), that is, any nonexpansive self-mapping defined on a weakly compact convex nonempty subset of X has a fixed point (see [Ki 65] and [Go-Ki 90]).
In this section we will consider Orlicz spaces over a finite nonatomic measure space only.
Theorem 2.1 Let X be equal to one of the space L1>, L~, 11> or l~. Then X has UNS if and only if it is reflexive.
To prove this theorem, we need the following lemmas.
Lemma 2.2 Suppose cI> E 62. Then for any (3 > 1 and c > 0, there exists K 2 2 such that for all x E L1>,
Iq,((3x) :os: Klq,(x) + c.
Proof. Let a> 0 satisfy w((3a){L(T) < c. Then since cI> E 62, there exists K 2 2 such that cI>((3u) :os: KcI>(u) for all u 2 a. For given x E Lq" set F = {t E T : Ix(t)1 2 a}. Then
344
where xlF = XXF. • Lemma 2.3 Assume <1> E 6.2 n V'2. Then for any a > 0, there exist c > 1 and 8> ° such that
<1> (u;v) :':: 1;8[<1>(u)+<1>(v)]
whenever lui ~ a and lui ~ clvl, or uv :':: 0.
Proof. By Lemma 1.3, there exist 'Y > O,and e E (0,1/2) such that
<1> (~) :':: 1; 'Y <1>(w) (Iwl ~ a)
and 2
<1>((1 + e)W) :':: -2 -<1>(w) (Iwl ~ a). -'Y
Set c = ~ 8 = 1 _ 2 - 2'Y
e' 2-'Y'
Then lui ~ a, lui ~ clul or uv:':: ° imply
(U+V) (1+c- l ) 1-'Y (( 1)) <1> -2- :':: <1> --2-u :':: -2-<1> 1 +; u
1-'Y 2 1-8 :':: -2-2 _ 'Y <1>(u) :':: -2-[<1>(u) + <1>(v)].
• Lemma 2.4 If a Banach space X does not have UNS, then for each n E Nand e > 0, there exists a family {Xi : 1 :':: i :':: n + 1} in X such that
IIxjll :':: 1, IIxi - Xjll :':: 1 (1:':: i :':: j :':: n + 1)
and
(m = 1,'" ,n).
Proof. By the assumption, there exists a bounded nonempty closed convex subset C of X such that for each z E C, there exists x E C satisfying liz - xII > (1 - e) diam C. Without loss of generality ,we may assume that ° E C and diamC = 1, that is, IIxil :':: 1 and IIx - yll :':: 1 for all X,y E C.
Pick Xl E C arbitrarily. Then, by the hypothesis, there exists X2 E C such that IIx2 - xIII> 1 - e. Since C is convex, (Xl + x2)/2 E C. Therefore, there exists Xs E C such that IIx3 - (Xl +X2)/211 > l-e, and so on, by induction, we can choose the desired system of elements. •
Proof of Theorem 2.1 We only prove the theorem for X = Lif!. The proofs for other spaces are analogous. Since all Banach spaces with UNS are reflexive, we only need to show the sufficiency. By Lemmas 2.2 and 2.3, there exist K ~ 2, b > 0, c > 1 and 8 > ° such that
(2.1)
Geometrical properties in function lattices 345
if!(b)fL(T) ::; 8~ (2.2)
and
(U+v) 1-0 if! -2- ::; -2-[if!(u) + if!(v)] (lui:::: b, lui:::: clvl, or uv::; 0). (2.3)
Select an integer p > 16c2 K2 and set n = 8p. If LiP fails to have UNS, then Lemma 2.4 and if! E 62 yield the existence of {Xi}, 1::; i ::; n + 1, such that
liP(xi) ::; 1, liP(xi - Xj) ::; 1 (1::; i::; j ::; n + 1) and (2.4)
1 m 0 1 :::: liP(xm +1 - - "" Xi) > 1 - -2-' (2.5)
m~ 4nK i=l
Now, we first introduce some notation. Set Ui(t) = Xn+1(t) - Xi(t) (i ::; n) and for each t E T, rearrange {Ui(t)}i:<;n into {Ys(t) = Uis(t)(t)} such that Yl(t) ::; ... ::; Yn(t). Then it is not difficult to check that each Ys(t) is E-measurable. Moreover, define
Then
2 n
xo(t) = - L IUi(t)l, n i=l
l(t) = {i ::; n : IUi(t)1 > clx(t)1 or CIUi(t)1 < x(t) or Ui(t)X(t)::; O}, A = {t E T : l(t) contains at least 4p elements}, B = T \ A.
Ix(t)I::; max{IYs(t) I, IY4p+s(t)l} ::; xo(t).
Moreover, (2.1), (2.4) and the convexity of if! imply
(1 n ) 1 K n 1 1
liP(xo) ::; KliP ;;: ~ IUil + 8::; -;;: ~ liP (Ui) + 8 < K + 8'
In the first step, we show that
Since (2.4) and (2.1) yield
7 0 ~-~ 1 8 < 1- 4n2K < lq,(X1 - X2) ::; KliP(-2-) < K + 8'
That is, liP(Xl"2x2) > 4~' to verify (2.8) it suffices to show that
j if!(x1(t) - x2(t))d _1 2 t < 4K'
A
For this purpose, we first check that tEA implies
IYs(t)1 > CIY4p+s(t)1 or cIYs(t) I < IY4p+.(t) I or y.(t)Y4p+s(t)::; 0
(2.6)
(2.7)
(2.8)
for each s ::; 4p. In fact, if there exist some j ::; 4p and tEA such that none of the above three inequalities holds, then we get
C-1Y4p+j(t) ::; Yj(t) ::; CY4p+j(t) or C-1Y4P+j(t):::: Yj(t) :::: CY4p+j(t).
346
Since x(t) is between Yj(t) and Y4p+j(t), we derive that
c-1x(t) :s; Ys(t) :s; cx(t) or c-1x(t) 2': Ys(t) 2': cx(t)
for all s = j, j + 1,··· ,4p + j, which contradicts the definition of A.
Hence if we define for each s :s; 4p, A(s) = {t E A : max{IYs(t)l, IY4p+s(t) I} > b}, then (2.3) and the convexity of <I> imply
It follows from (2.4) that
4p J 1 1 2:= [<I>(Ys(t)) + <I> (Y4p+s(t))]dt < 4nK < 8K· 8=1 A(s)
(2.9)
Now, we define
Di = {t E A: IUi(t)1 > b} (i = 1,2), and
Bi(S) = {t E A: Ui(t) = Ys(t) or Y4P+8(t)} (i = 1,2).
Then from (2.2), (2.9) and the fact that U!::lBi(s) = A, Di n Bi(S) c A(s) (i = 1,2), we derive
Geometrical properties in function lattices 347
1 1 1 < 8K + 8K = 4K·
This ends the proof of inequality (2.8).
In the second step, we set for each i = 3, ... ,n - 1,
G(i) = {t E B : IXs(t) - xi(t)1 :::: clx(t)l/p for some s > i and s:::: n}.
Then n-l
U G(i) =B. (2.10) i=3
In fact, for any t E B = T \ A, by the definition of A, there exist at least five Ui(t) such that their distance from each other is no more than clx(t)l/p, and thus, there exist ~,J; 3 :::: i < j :::: n, such that
IUi(t) - Uj(t)1 :::: clx(t)1 :::: clx(t)l/p,
that is t E G(i). This proves (2.10).
Now, we define
i-I
D(3) = G(3), D(i) = G(i) / U G(k) (i = 4,··· ,n - 1). k=3
Then {D(i)}?':l are pairwise disjoint and U?,:lD(i) = B. Let, for each i = 3,··· ,n-1 and each t E D(i),
i'(t) = i,il/(t) = max{k:::: n: IXk(t) - xi(t)1 :::: cIx(t)l/p}·
Then i'(t), il/(t) are well defined by the definition of G(i) and i'(t) < il/(t). Next, we construct two ~-measurable functions as follows:
n-l 1£-1
x'(t) = 2..: Xi'(t) (t)XD(i) (t), xl/(t) = 2..: Xi"(t)(t)XD(i)(t). i=3 i=3
Then by (2.6) and the definition of i'(t), il/(t),
Ix'(t) - xl/(t) I :::: clx(t)l/p:::: =o(t)/p. (2.11 )
Since (2.8) and the convexity of M imply
~ /[iI>(xl/(t) - Xl(t)) + iI>(xl/(t) - x2(t))]dt ~ / iI>(x1(t); x2(t))dt > 2~' B B
without loss of generality, we assume that
/ iI>(xl/(t) - xl(t))dt > 2~. (2.12)
B
Finally, let E = {t E B : Ixl/(t) - xl(t)1 ~ max{b, c2xo(t)/p}}. Then by (2.11), tEE implies Ixl/(t) - xl(t)1 ~ c2xo(t)/p ~ Ix'(t) - xl/(t)l. It follows from (2.3) that if tEE, then
iI> (xl/(t) - x'(t) ~ xl/(t) - X1(t)) :::: 1; (j [iI>(xl/(t) _ x'(t)) + iI>(xl/(t) _ Xl(t))]. (2.13)
348
Moreover, (2.12), (2.3), (2.1), (2.7) and the inequality c21p < (4K)-2 < 1 imply that
J iI>(x"(t) - Xl (t))dt =.1 iI>(x"(t) - xl(t))dt - .I iI>(xl/(t) - xl(t))dt (2.14)
E B B\E
> 2~ - [.I iI> (c2X;(t)) dt + iI>(b)J-L(BIE)]
B\E
~ 2~ - [% ! iI>(xo(t))dt + 8~ ] > _1 _ [~ (K + ~) + _1 ] - 2K p 8 8K 111 1
> 2K - 8K - 8K = 4K'
In light of (2.13) and the convexity of iI>, for all tEE, we have
,; iI> (m ~ 1 ~(xm(t) - Xk(t)))
= 2s~n iI> ( m ~ 1 ~ (xm(t) - Xk(t))) m;#:i"(t)
+ i" / _ 1 [ L iI>(xl/(t) - Xk(t)) + 2iI> (Xll(t) - x'(t) ~ X"(t) - X1(t))] ( ) 2:SkSill(t)-1
k¥i'(t)
n m-l ::; f2 m ~ 1 t; iI>(x"(t) - Xk(t)) - ill(t~ -1 [iI>(x"(t) - x'(t)) + iI>(x"(t) - Xl(t))].
Combining this with (2.5), (2.14) and x"(t) - 1 ::; n - 1, we deduce
Geometrical properties in Junction lattices 349
This contradiction completes the proof. • Lemma 2.5 Suppose xn E B(Lif» for n E Nand Xn """"' x in measure. Then x E
B(Lif».
Proof. Since Iif>(xn ) < 1 for all n E N, by the Fatou Lemma, we have Iif>(x) < liminfnIif>(xn) ::; 1. •
Theorem 2.6 The spaces L~ and l~ have the weak sum-property, so they have WNS.
We will prove this theorem together with the next theorem.
Lemma 2.7 If {xn} is a bounded sequence in L~, kn E K(xn) for n = 1,2, ... , and kn """"' 00, then Xn -> 0 in measure.
Proof. For each (J > 0, define Gn = {t E T: IXn(t)1 :::: (J}. Then
Ilxnllo = k1 (1 + Iif>(knxn)J :::: k1 iI>(kna)J-L(Gn). n n
Applying the fact that iI>(u)/u """"' 00 as u -> 00, we get J-L(Gn) """"' O. • Lemma 2.8 (i) If {xn} is a bounded sequence in Lif> and it converges to zero in
measure, then Xn """"' 0 EifI-weakly, where \]I is the Young conjugate ofiI>.
(ii) If {xn} 'is a bounded sequence in lif> and it converges to zero coor'd'inate-wise, then Xn -> 0 hifl-weakly,
Proof. We only prove (i) because the proof of (ii) is analogous. Suppose that IIxnll ::; K for any n E N. For any v E Eifl and c > 0, choose /5 > 0 such that E E ~ and J-L(E) < /5 imply IlxXEII~ < c. Since Xn -> 0 in measure, we can find Gn E ~ with J-L(Gn) < /5 such that IXn(t)1 < c on T \ G n for all large n. Hence, for such n,
l(v,xn)l::; }' Ixn(t)v(t)ldJ-L
T\Gn
::; cllxTllllvll~ + IlxnllllvXGn II~ ::; cllxTlillvll~ + cK. Since c > 0 is arbitrary, this shows that (v, xn) """"' O. • In the following by SAl of iI> we denote an interval [a, bJ such that iI> is affine on [a, b] but it is not affine neither on [a - c, aJ nor on [b, b + c], where c > O.
Theorem 2.9 Let X be one of the spaces L~,Eg,l~ or h~. Then the following are equivalent:
(i) X has the sum-property.
(ii) X has NS.
(iii) There exist a> 0 and c > 1 such that for any SAl [u, v] of iI>,
u>a*v::;cu(whenX=L~ or Eg),
350
o < 'U ::; a => v ::; cu (when X = l~ or h~).
Proof of Theorems 2.6 and 2.9. We only prove the theorems for X = L~ and X = E~. Let {xn} be a limit affine sequence in L~ and kij E K(Xi - Xj) (i i- j). First we show that there exists a subsequence N j of N such that for any j E N j , {kij : i E N j } is bounded. Indeed, if {kij : i E N} is bounded for all j E N, then we set N j = N. Otherwise, there exist some mEN and a subsequence I of N such that kim -t 00
as i( E 1) -t 00. This shows that Xi : i E I converges to Xm in measure according to Lemma 2.7. Since Xi i- X j for all i i- j, by the same reason, we get that N j = I \ {m} satisfies our requirement.
By the diagonal method, we can pick a subsequence N2 of Nj such that kij -t k < 00
as i(E N 2) -t 00 for each j E N j . We claim that k j -t 00 as i(E N2) -t 00. In fact, if this is not true, then N2 contains a subsequence N3 such that kj -t k < 00 as j(E N3) -t 00. Therefore, for all n,i,j E N3 with n i- i,j,
Ilxn - xillo + Ilxn - xjllO -112xn - Xi - xjllO 1 1
;::: -k 0 [1 + Iq,(kni(xn - Xi))] + -k 0 [1 + Iq,(knj(xn - Xj))] n'l nJ
_ kni + knj [1 + lq, ( kniknj (2xn - Xi - Xj))] kniknj kni + knj
(2.15)
= J {fif>(kni(X(t) - Xi(t))) + fif>(knj(Xn(t) - Xj(t))) n'l nJ
G
_ kni + knj if> ( kniknj (2xn(t) _ Xi(t) - Xj(t)))} dt. kniknj kni + knj
Denote the last integrand in (2.15) by!:!(t). Then!:! ;::: 0 for all t E T since if> is convex. Recall that 6(x) is affine on co{xd. By letting n -t 00 we get
J !:/(t)dt -t 0,
G
and thus !:! (t) -t 0 in measure. Hence, the diagonal method allows us to find a subsequence N4 of N3 such that !:! (t) -t 0 fL-a.e. on T as n( E N 4) -t 00 for all i,j E N3.
Now, for each t E T, we pick a subsequence {n-y = n-y(t)} of N4 such that
Iv(t)1 = liminf IXn(t)l, and limxn~(t) = v(t). nEN4 -y
(2.16)
Then, by the Fatou Lemma, Iv(t)1 < 00 wa.e. on T, analogously to the proof of Lemma (2.5). Let "Y -t 00. Then the convexity of if> gives
00 1 1 o = lim!~~(t) = -if>(ki(V(t) - Xi(t))) + -k if>(kj(v(t) - Xj(t))) (2.17)
-y 0_, ki j
_ ki + kj if> ( kikj (2v(t) _ Xi(t) - Xj(t))) kikj ki + k-j
fL-a.e. on T. Since for fL-a.e. t E T, (2.17) holds for all i,j E N3, by replacing j by n-y in (2.17) and letting "Y -t 00, we have for fL-a.e. t E T,
1 k + k (kk ) -koif>(kiv(t)-Xi(t)))= 'kok}if> k o' Jk(V(t)-Xi(t)) . 7. 1. J 1, + J
(2.18)
Geometrical properties in function lattices 351
Since 0 < Ki~kj < 1, by (2.7), condition (2.18) holds only for v(t) = Xi(t). This means v = Xi for all i E N3 , contradicting the assumption that Xi #- Xj whenever i #- j. This contradiction proves that k j -t 00 as j (E N3 ) -t 00.
Now, we prove (iii) ~ (i) in Theorem 2.9. If LO does not have the sum-property, then there exists a limit affine sequence {Yn} in Lt such that 1::,.(Yn) i. By the above discussion, {Yn} contains a subsequence {xn} satisfying kij -t kj < 00 as i -t 00, and kj -t 00 as j -t 00, where kij E K(Xi - xj),i #- j. Since ip(u)lu -t 00 as u -t 00, for the constant a> 0 in (iii), we can find b> a such that
..... (a + b) ip(a) + ip(b) "'i.' 2 < 2 .
Since ip is convex, by (iii),
ip(ou + (1- o)v) < oip(u) + (1 - o)ip(v) (2.19)
for all 0 < 0 < 1 and all u ::; a, v 2: b or u 2: a, v 2: cu. If we define v(t) as in (2.16), then by (2.16) and (2.19), for p,-a.e. t E T, if kilv(t) -xi(t)1 ::; 0, then kjlv(t) - xj(t)1 ::; bj if kilv(t) - xi(t)1 2: 0, then kjlv(t) - xj(t)1 ::; ekilv(t) - xi(t)l. Therefore, for p,-a.e. t E T,
kjlv(t) - xj(t)l::; max{b,ekjlv(t) - xi(t)l} =: Ui(t). (2.20)
By the Fatou Lemma,
1::,.(Xj) 2: kjl[l + lip (kj(v - Xj))] 2: lie - xjllO. (2.21)
Thus V-Xj E Lip, whence Ui E Lip. Since 1::,. > 0, liminfj IIv-xjllO =: 'Y > O. It follows from (2.20) that kj = II kj (v - xj)IIO/liv - xjllo::; lIu;lIo/llv - xjllo. Letting j -t 00, we get a contradiction: 00 = IIUilio h < 00.
Now, we turn to Theorem 2.6. If L~ does not have the weak sum-property, then by (2.15), there exists a weakly convergent (to zero) limit affine sequence (xn) with IIxnllo -t 1 and 6(xn) -t 1. By the first part of the proof, passing to a subsequence if necessary, we may assume that kij -t kj < 00 as i -t 00 and kj -t 00 as j -t 00, where kij E K(Xi - Xj). It follows from (2.21) that Xj -t v in measure (verified as in the first part of the proof). Therefore, by Lemma 2.7, v = O. We may also assume that Xj -t 0 p,-a.e. on T. We prove the theorem by showing that lim1::,.(xj) 2: ~, which contradicts the assumption 1::,.(Xj) -t 1.
For each j E N, we choose a set Gj E L: such that Xj is bounded on Gj and
1 k;t[l + lip (kjxjXG;)] > kj l[l + lip (kjxj)] - "k-.
J
Then by (2.21),
that is
(2.22)
352
It follow~ that
IIXjXG, 11 0 :2: IIxjllO - IIXjXT\Gj 110 o 2 > 211xjll - 6(xj) - k:.
J
(2.23)
Since Xj is bounded on Gj , there exists 15 = 15(j) > a such that
o 1 IIXjXEIi < - whenever E C Gj and p,(E) < 15. (2.24)
kj
Since Xi ~ a p,-a.e. on T, there exists FEE with p,(F) < 15 such that Xi -7 a uniformly on T \ F. Hence, there exists 1= I(j) E N such that for all i > I,
It follows that
Hence, by (2.22)-(2.26),
o 1 IIXiXT\F1I < k:.
J
IIXi - xjllO = kijl[l + Iq, (kij (Xi - Xj)xT\(T\F»)]
+ kijl[l + Iq,(kij(Xi - Xj)XT/F)]- kl 'J
o 1 :2: II(Xi - Xj)xT\CT\F) II + II (Xi - Xj)XT\FII- ~
'J
(2.25)
(2.26)
Finally, we prove the implication (ii)*(iii) of Theorem 2.9. If (iii) does not hold, then there exist sequences {Uj}, {Vj} such that iI>(uJ)p,(T) > 1,uj+1 > 2j uj,vj > 2j uj and p(u) is a constant on [Uj,Vj]' j E N. By the first two assumptions, we can choose disjoint sets Gj E L: such that p,(T \ UjEN Gj) > a and
(2.27)
Hence, we can find Uo large enough so that there is Go c T \ Uj Gj satisfying
1: w(p(Uj))p,(Gj) + w(p(uo))p,(Go) = 1. (2.28) JEN
Geometrical properties in function lattices 353
Define
v = L p( Uj )XGj , Xn = UOXGo + L VjXG j + L UjXG j •
j~O j~n j~n
Then by (2.28), Iq,(v) = 1, whence v E Ll, and Ilvllw = 1.
First we show that Xn E Eq, for any n E N. Given arbitrary K > 1, choose J> n such that 2J > K. Then for all j > J, we have Vj > 2juj > KUj > Uj. Therefore,
j>J j>J
j>J
j>J j>J
This implies that Iq,(Kxn) < 00. Since K > 1 is arbitrary, we have Xn E Eq,.
Let kn = Ilxnllo and Yn = xn/kn. Then Yn E Eq, and IIYnllo = 1. By (2.28),
IIYnll°?: (v,Yn) = k;;l [UOP(UO)J1(GO) + LVjp(Uj)J1(Gj) + LUjP(Uj)J1(Gj)] J~n J>n
= k;;l [/?w(v) + Iq,(knYn)] ?: IIYnll o = 1.
Moreover, since
we have kn -t 00 as n -t 00.
We complete the proof by showing that 6, = 2 on CO(Yn). Indeed, for any Y E cO(Yn), there exist Ai ?: ° with ~i<m Ai = 1 such that Y = ~i<m AiYi. Since (v, Yn) = 1, we have (v, y) = ~i<m Ai(v, yJ = 1. For any E > 0, since Y E Eq" there exists I > m
such that IIYXFllo- < E, where F = Ui<JGi. In view of xn(t) ~ max{vJ,uo} on G \ F and kn -t 00 as n -t 00, we can find ~o E Gi such that IIYnXT\Fll o < E for all n > no· Define Vo = VXT\F - VXF. Then IIvollw and for n > no,
2?: IlylIO + IIYnllo ?: Ily - Ynll o
?: (vo, Y - Yn)
= (VO,YXT\F) + (VO,YXF) - (VO,YnXT\F) - (V,YnXF)
= (V,YXT\F) - (V,YXF) - (V,YnXT\F) + (V,YnXF)
= (v,y) - 2(V,YXF) - 2(V,YnXT\F) + (V,Yn)
> 1 - 211YXFlio - 2I1YnXT\FII - 21IYnXT\Fllo + 1> 2 - 4E,
which shows that 6,(y) = 2. • Theorem 2.10 Let X be one of the spaces Lq" Eq" lq, 01' hq,. Then the following aTe equivalent:
(i) X has the sum-pmpeTty.
354
(ii) X has WNS.
(iii) ME 62.
Proof. This time, we prove the theorem for X = lif! and X = hif!.
(i) => (ii).This implication is trivial.
(ii)=> (iii). If <I> rt 62, then there exist ak ! 0 such that <I>(at) < c and <I>((1 + i)ak) > 2k <I>(ak) (k EN), where 0 < c < 1 is a given constant. For each kEN, choose an integer mk such that
and define m"
xn(i) = an L:>i+s" (n EN), i-I
where {ei} is the natural basis of CO and Sn = E~~l m;. Obviously, {xn} have mutually disjoint supports, and so, Iif!(xi-xj) ::; c/2i +c/2j < 1 (i # j). Moreover, for any v> 1, it is easy to check that Iif!(vxn) ..... 00 as n ..... 00. Therefore, for any n E N,6(xn ) = 1 and 6(x) = 1 for all x E co{xn}. Clearly, Xn ..... 0 lif!-weakly, that is, Xn ..... 0 weakly in hif!. This means that {xn} is a weakly convergent limit constant sequence, thus, hif! does not have the WNS.
(iii) => (i). Assume that lif! has a limit affine sequence {xn} with 6(xn) i 6'. By the diagonal method, we can find a subsequence of {xn}, again denoted by {xn}, such that Xn ..... x coordinate-wise. By Lemma 2.5, x E lif!o Hence, we may assume that Xn ..... 0 coordinate-wise and that 6' = lim 6(xn ) > O.
For any i,j EN, since 6 is affine on co{xn},
Hence, as <I> E 62,
Let Aij = 6(x~tg(Xj)' Then by the convexity of <I>,
(2.30)
Recall that Xn ..... 0 coordinate-wise. By letting n ..... 00, we conclude from (2.19) and (2.30) that
( xi(k) ) ( Xj) ( xi(k) + xj(k) ) Aij<I> 6(Xi) + (1 - Aij)<I> 6(xj) = <I> 6(Xi) + 6(xj) (k EN).
Geometrical properties in function lattices 355
Letting j --+ 00, this equality becomes the equality
6(xi) cI> (Xi(k) ) _ cI> ( xi(k) ) (k N) 6(xi) + 6' 6(xi) - 6(xi) + 6' E.
But C --.: 6(xi)/(6(xi)+6') < 1 by (2.7), so the above equality holds only for x;(k) = o. This means that Xi = 0 (i E N), contradicting the assumption that {xn } is a limit affine sequence. •
To end this section, we present a different sufficient condition for hiP to have the weakly fixed point property.
Lemma 2.11 The space hiP has the weak orthogonality property, that is for any sequence {xn} in hq, such that Xn --+ 0 weakly, there holds
liminfliminf IlIxnl /\ IXiPl1i = 0, n m
where (x /\y)(t) = min{x(t),y(t)} (see [8 88J and [8 92]).
Proof. The lemma results from obvious fact that the mapping y --+ IXI!\ Iyl is weak-norm continuous for every fixed x E hiP. In fact weak convergence of (Yn) to zero implies that Yn --+ 0 coordinate-wise. 80, if x E hiP then IYnl /\ Ixi :S Ixl and IYnl !\ Ixl --+ 0 coordinate-wise. By the dominated Lebesgue convergence theorem, we get IIIYnl /\ Ixlll --+ O. •
Lemma 2.12 The Riesz angle Q(liP) < 2 if and only ifcI> E \72, where
a(liP) = sup{lIlxl V IYIII : Ilxll :S 1, IIYII :S I}.
Proof. If cI> rf. \72, then there exist Un ! 0 such that
( Un) 1 2cI> 2 > (1 - ;;:)cI>(un) (n EN). (2.31)
Let mn be an integer satisfying mncI>(un) :S 1 and (mn + 1)cI>(un) > 1. Define
mn 2mn
xn=UnLei, Yn=Un Lei. i=l i=mn +1
Then it is easy to check that 1 2: liP(Yn) --+ 1 and by (2.31),
This shows that Ilxn V Ynll --+ 2.
Next we assume cI> E \72. That is, there exist 8 > 0 such that
cI>«2 - 8)u) 2: 2cI>(u) (lul:S cI>-1(1)).
Given X,Y E B(liP), we have IX(i)l, ly(i)l:s cI>-1(1), whence
( Ixl V IYI) (x) (Y) 1 liP ~ = liP 2 _ 8 + liP 2 _ 8 :S"2 [liP(x) + liP(y)J :S 1,
356
that is, l"xl V Iylll ~ 2 - 8.
• Applying a result of Borwein and Sims [Bo-S 84] stating that every weakly orthogonal Banach lattice X with Riesz angle o(X) < 2 has the weak fixed point property, from Lemmas 2.11 and 2.12 we deduce the following
Theorem 2.13 If <l? E V'2, then hiP has the weak fixed point property.
Remark 2.14 Theorems 2.10 and 2.13 furnish a natural example of a space with the weakly fixed point property but without WNS.
Remark 2.15 Dowling, Lennard and Thrett [Do-Le-T 96] investigated Orlicz spaces for which every nonexpansive self-mapping of a nonempty, closed, bounded, convex subset has a fixed point. This property is called the fixed point property (FPP). They proved that L~ has FPP if and only if it is reflexive. In fact, this can be obtained immediately from Theorem 2.1 presented above, Theorem 1.90 in [Ch 96J and the following two results given by Dowling and Lennard [Do-Le 97]:
(a) A Banach space X fails FPP if it contains an asymptotically isometric copy of 11. That is, for every positive sequence (en) decreasing to 0, there exists a sequences {xn} of norm-one elements in X such that En(l- en)lonl ~ II En onxnll for all sequences (on) of real numbers.
(b) If the dual of X contains an isometric copy of 100 , then X contains an asymptotically isometric copy of II .
It is still an open problem whether the above conclusion is true or not for the Orlicz space LiP equipped with the Luxemburg norm. The only trouble is that one cannot prove the necessity of <l? E 62 in the same way as for the Orlicz norm.
Notes. Criteria for normal structure and uniform normal structure of Musielak-Orlicz spaces were given by Katirtzoglou [Kat 97J. In Orlicz-Lorentz spaces the criteria were presented by Kaminska, Lin and Sun [Ka-L-Sun 96J.
3. Uniform rotundity in every direction
Recall that a Banach space X is said to be uniformly rotund in every direction (URED) if for any z E S(X) and every two sequences {xn} and {Yn} in S(X) such that Xn -Yn = enZ, where {en} is a sequence of reals, and Ilxn + Ynll --+ 2 we have IIxn - Ynll --+ O. If we change S(X) into B(X) in the above definition, we get the same property.
In the fixed point theory this geometric property is important because of the following well known theorem.
Theorem 3.1 Any Banach space X which is uniformly rotund in every direction has normal structure.
Now we will present criteria for URED of Orlicz spaces. We do not assume in this section that Orlicz functions cJ> satisfy condition (ii) from the definition (see page 1).
Geometrical properties in function lattices 357
Theorem 3.2 Let (T, I:, /L) be a nonatomic complete and a-finite measure space and if> be an Orlicz function. Then the Orlicz space Lq, equipped with the Luxemburg norm is uniformly rotund in every direction if and only if if> is strictly convex and if> satisfies the !::..2-condition on R+ if fL is infinite and the !::..2-condition at infinity if it fL is finite.
Proof. Sufficiency. Let {xn}, {Yn} be sequences in S(Lq,), z E S(Lq,), Xn -Yn = EnZ, where {En} is a sequence of reals and II (xn + Yn)/211 -> 1. Then by the !::..2-condition, we have lq,((xn + Yn)/2) -> 1 (see [Ch 96]). We will show that Xn - Yn -> 0 in measure. Assume for the contrary that it is not true. Then we can assume (passing to a subsequence if necessary) that for some E, a > 0 there holds fL(E)n ~ E for all n E N, where
En = {t E T: IXn(t) -Yn(t)l:::> a}.
Choose k > 1 such that F E I: and fL(F) = E/4 implies that IlxF11 = l/k and define
An = {t E T: IXn(t)1 > k}, Bn = {t E T: IYn(t)1 > k}.
Then we have 1 = Ilxnllq, ~ IlxnXAn 1Iq, > kliXAn 1Iq" whence IlxAnllq, < k and consequently, fL(An) < E/4. Similarly, fL(Bn) < E/4 (n EN). By strict convexity of if> there is 8> 0 such that if u, v E [0, k] and lu - vi :::> a, then
(u+V) 1-15 if> -2- ::; -2-{if>(u) + if> (v)}.
Denote Cn = En \ (An U Bn). Then we have fL(Cn ) :::> fL(En) - (fL(An) + fL(Bn)) = E - E/2 = E/2. Moreover for any t E Cn, we have IXn(t) - Yn(t)1 :::> a and IXn(t)1 ::; k, IYn(t) I ::; k for any kEN. Consequently
Consequently
( Xn + Yn) 1 (Xn + Yn) Of-I - lq, -2- = "2{h(xn ) + lq,(Yn)} - lq, -2-
1{ (Xn + Yn ) :::> 2 lq,(xn)XcJ + lq,(Yn)xcJ} - lq, -2-XCn
1 1- 15 ~ 2{lq,(xnXcJ + lq,(YnXcJ} - -2-{lq,(xnXcJ + lq,(YnXcJ}
= ~{lq,(XnxcJ + lq,(YnXcJ} :::> 8lq,Cn; Yn XCn )
:::> 15if>(~)fL(Cn) :::> 15if>(~)~,
a contradiction. Therefore Xn - Yn -> 0 in measure. Since Z E S(Lq,) and so z f= 0, we conclude from the equality Xn - Yn = EnZ (n E N) that En -> O. Consequently, there is no E N such that IEnl ::; 1 for n :::> no and so IXn - Ynl ::; Izi for n :::> no. By the Lebesgue dominated convergence theorem, we get (xn - Yn) -> 0 and by the suitable !::..2-condition for if>, we get lq,(,>-(xn - Yn)) -> 0 as n -> 00 for any .>- > 0, which means that Ilxn - Ynll -> 0 as n -> 00.
Necessity. Assume that if> does not satisfy the suitable !::..2-condition. Then Lq, contains an order isometric copy of 100 (see [Ch 96], [Ra-Re 91] and [T 76]). Since 100 is not URED, Lq, is not URED, either.
358
Assume now that iP is not strictly convex. We will show that L.p is not rotund and so is not URED. Since iP is not strictly convex, there exists u > v > 0 such that iP((u + v)/2) = {iP(u) + iP(v)}/2. Choose two disjoint sets A,B E ~ and a > 0 such that fL(A) > 0, fL(B) > 0 and
1 2(iP(u) + iP(V))fL(A) + iP(a)fL(B) = 1.
Let C, DCA be measurable sets such that fL(C) = fL(D) = ~fL(A) and C n D = 0. Define
x = UXC +VXD +aXB,
y = VXC + UXD + aXB·
Then I.p(x) = iP(U)fL(C)+iP(V)fL(D)+iP(a)fL(B) = ~(iP(u)+iP(V))fL(A)+iP(a)fL(B) = 1. In the same way we can prove that I.p(y) = 1. Moreover,
I.pC ;y) = iP(U; V)fL(A) + iP(a)fL(B)
1 = 2(iP(u) + iP(V))fL(A) + iP(a)fL(B) = l.
Consequently, Ilxll.p = Ilyli.p = II (x + y)/211.p = 1. Since, evidently, x =J y, L.p is not rotund. This finishes the proof. •
Notes. Kaminska [Ka 84] first gave criteria for URED of Musielak-Orlicz spaces of Bochner type. Theorem 3.2 can be easily deduced from her paper. The proof that we presented here is different.
4. B-convexity and uniform monotonicity
These properties are related to the fixed point theory by the following
Theorem 4.1 (see [Ak-K 90]) If a Kothe Junction space X is B-convex and uniformly monotone, then it has the fixed point property.
Recall that a Banach space X is said to be B-convex if no nonreflexive space Y is finitely represented in X (see [Ak-K 90] and [Ch 96]). Since UR implies nonsquareness, nonsquareness implies B-convexity and B-convexity is preserved by equivalent norms, we know that uniformly covexifiable Banach spaces are B-convex. The converse is also true.
Now, we will present criteria for B-convexity and uniform monotonicity of Orlicz spaces. We do not assume generally in this section that Orlicz functions that they must satisfy condition (ii) from the definition on page 1. First we will prove the following.
Lemma 4.2 Let iP be an Orlicz function such that its right derivative p on R+ satisfies the condition:
For any c: > 0, there exists K > 1 such that p((l + c:)t) 2: Kp(t) (t 2: 0).
Then iP is uniformly convex.
Geometrical properties in function lattices 359
Proof. Given c: E (0,1), take K > 1 such that
p((1 + c:/2)t) :c:: Kp(t) (t:C:: 0).
We shall show that for any u, v E R satisfying lu - vi :c:: c:max{lul, Ivl}, the inequality
<1>(U; V) ~ (1 _ 6) <1>(u) ; <1> (v) (4.1)
holds for 6 = c:(1 - I/K)/4 > O. We may assume without loss of generality that u - v :c:: fU > c:v > 0, that is (1 - c:)u :c:: v > O. Define
rp(t) = <1>(u) + <1>(t) _ 2<1>(U; t) (t :c:: 0).
Then for almost all t E [0, u],
rp'(t)=p(t)_p(U;t) ~O.
Hence rp(t) is nonincreasing on [0, u]. Therefore,
(u+V) rp(v) = <1>(u) + <1>(v) - 2<1> -2-
:c:: <1>(u) + <1>((1 - c:)u) - 2<1>((1 - c:/2)u) u (1-c/2)u u J p(t)dt - J p(t)dt = J [P(t) - p((1 - c:/2)t)]dt
(1-c/2)u (l-c)u (1-c/2)u u
:c:: J (1-I/K)p(t)dt:c:: (1-I/K)[<1>(u) - (1- c:/2)<1>(u)]
(1-c/2)u
c: > 4(1- I/K) [<1>(u) + <1>(v)]
for u and vas above, that is inequality (4.1) holds with 8 = (c:/4)(I-I/K) > O. •
Theorem 4.3 For any Orlicz space Lip the following are equivalent:
(i) Lip is reflexive.
(ii) <1> E ~2 and \[t E ~2.
(iii) Lip is uniformly covexifiable.
(iv) Lip is B-convex.
Proof. It is well known that (i) {o} (ii) (see [Ra-Re 91] and [T 76]). Let us prove that (ii) =?- (iii). We consider only the case of a nonatomic finite measure space, when <1> E ~2 means that <1> satisfies the ~2-condition at 00. By (ii) there exist Uo > 0, K > 2 and 8 > 0 such that
(2 + 8)<1>(u) ~ <1>(2u) ~ K<1>(u) (u:C:: uo).
Since changing the value of <1> on [0, uo] does not affect the equivalence, we may assume that the above inequalities hold for all u E R. Let
l !U! <1>(t) l!U! <1>o(t) <1>o(u) = --dt, and <1>1 = --dt.
o tot
360
We claim that <I>o ~ <I>l ~ <I>. Indeed, denoting by P the right derivative of <I> on R+, we have
<I>(u) <I> (2u) 1 p(u) ~ --;;- ~ Ku ~ KP(u) (u> 0).
Integrating each term of the last inequalities from zero to u, we get K<I>(u) ~ K<I>o(u) ~ <I>(u), that is <I> ~ <I>o. Similarly <I>o ~ <I>l' Next, we will show that <I>l is uniformly convex. Since
lul/2 lui
J <I>(uj2) dt < J <I>(t) dt luj21 - t
o lul/2
::; <I>o(u) lul/2 lui
= J <I>;t) dt + J <I>;t) dt
o lul/2
< <I>(~) + <I>(u) - 2 2
( 1 1) 1 ::; 2 + b + 2 <I>(u) = Z<I>(u),
where L = ~~2f > 1, we obtain for t > 0,
Hence we get
L < <I>(t) = t<I>o(t) < K. - <I>o(t) <I>o(t)-
~ < <I>o(t) < K. t - <I>o(t) - t
Integrating this inequality with respect to t from u to (Ju, where (J ~ 1, we have
(JL<I>O(U) ::; <I>o((Ju) ::; (JK<I>(u) ((J ~ 1, u E R).
Set PI(t) = MW) = <I>o(t)jt for t > O. We have for any c > 0 and u > 0,
((1 )) <I>o((1 + cluj (1 + c)L<I>O(u) PI + c u = > -'---;:-'----;C=-"':"
(l+c)u - (l+c)u
= (1 +c)L-l<I>O(u)ju = (l+c)L-Ipl (u),
whence, by Lemma 4.2, <I>l is uniformly convex. Finally we will show that WI is also uniformly convex, where W I denotes the function complementary to <I>l in the sense of Young. Let (WI)'(U) =: ql(U) and
ql((l + c)v) = O(V)ql(V) (v> 0).
Then o(v) > 1 (v> 0). Replacing v by Pl(U), we get
(1 + c)PI(U) = PI(O(V)U) = <I>o(~(~)u) ovu
::; aK(v)<I>o(u) = oK-I(v)PI(U). o(v)u
Hence, o(v) ~ (1 + c)I/(K-I), and so ql((l + c)v) ~ (1 + c)l/(K-l)ql(V), that is WI is uniformly convex.
Geometrical properties in junction lattices 361
Therefore, applying the result of Kaminska [Ka 82b] and the fact that a Banach space X is uniformly rotund if and only if X' is uniformly smooth, we get that both spaces Lif>1 and Lw are uniformly rotund and uniformly smooth for both the Luxemburg and Orlicz norms. Since <I> ~ <I> 1 , the space Lif> is B-convex, which finishes the proof of the implications (ii) '* (iii) '* (iv).
Let us prove that (iv) '* (i). We will show that if Lif> is not reflexive, then Lif> is not B-complex. Lif> is nonreflexive if and only if <I> (j. fl2 or \If (j. fl2. If <I> (j. fl2, then Lif> equipped with the Luxemburg norm contains an order isometric copy of leo (see [T 76]). Therefore Lif> is not B-complex. Assume now that \If (j. fl2 and <I> E fl2. Then the dual space of L~ is Lw, which contains an order isometric copy of leo and consequently L~ contains an asymptotically isometric copy of h (see [Ch-H-Sun 92]), so L~ (and consequently Lif» is not B-convex. This completes the proof. •
Let us denote by <I> E fl2 the fact that the Orlicz function <I> satisfies the suitable fl2-condition, which means the fl2-condition on R+ if p is non atomic and infinite, the fl2-condition at infinity if p is non-atomic and finite and the fl 2-condition at zero if p is the counting measure.
Theorem 4.4 An Orlicz space Lif> equipped with the Luxemburg norm is uniformly monotone if and only if <I> vanishes only at zero and <I> E fl2.
Proof. Sufficiency. Assume that a(<I» := sup{u 2: 0: <I>(u) = o} = ° and <I> E fl2. Let x E 8(Lif», ° ::; Y ::; x and lIyllif> 2: c, where c E (0,1). Then, by <I> E fl2 and Iif>(x) = 1, there is 8(c) E (0, c] such that Iif>(Y) 2: 8(c) (see [Ch 96]). Since <I> is superadditive on R+, we have
Iif>(x) = Iif>((x - y) +y) 2: Iif>(x - y) + Iif>(y),
whence Iif>(x - y) ::; 1 - Iif>(Y) ::; 1 - 8(c). Again by <I> E fl2, there is a function 0": (0,1) ...... (0,1) such that Ilxll ::; 1 - O"(c) whenever Iif>(x) ::; 1 - c (see [Ch 96]). Consequently, IIx - yll ::; 1 - 0"(8(c)), that is Lif> is uniformly monotone.
Necessity. Assume that a(<I» > 0. First assume that the measure space is infinite. Then x = XT E 8(Lif». Let A E :E be such that !l,(A) = p(T \ A) = 00 and define y = XA. Then ° ::; y ::; x and Ilx - yllif> = IIXT\AIIif> = 1, which means that Lif> is not uniformly monotone. If p is finite take y 2: ° such that Ilyilif> = 1 and f1(T\ supp y) > 0, Define x = y + a(<I»XT\suppy' Then 0::; y ::; x and IIXIIif> = 1, whence we get that Lif> is not uniformly monotone.
Assume now that <I> (j. fl2. Then there exists x E S( Lif», x 2: 0, such that Iif> (AX) = +00 for any A > 1 (see [Ch 96] and [Ra-Re 91]). Consequently, there exists A E :E such that IIXXAIIif> = IIXXT\AIIif> = IIx - XXAIIif> = 1, that is Lif> is not uniformly monotone. This finishes the proof of the theorem. •
Theorem 4.5 Let <I> be an Orlicz function such that <I>(u)ju ...... 00 as u ...... 00. The Orlicz space L~ equipped with the Orlicz norm is uniformly monotone if and only if <I> E fl2 and <I> vanishes only at zero.
The proof of this theorem is similar to that for Theorem 4.4 and so will be omitted.
Notes. B-convexity for Musielak-Orlic7: spaces was characterized in [H-Ka 85] and for Orlicz-Lorentz spaces in [H-Ka-M 96]. Uniform monotonicity of Musielak-Orlicz spaces was characterized in [H-Ka-Ku 87], [Cu-H-W] and [Ku 92]. For Orlicz-Lorentz
362
spaces it was done in [H-Ka 95]. In some Calderon-Lozanowsky spaces and Banach lattices monotonicity properties were considered in [C-H-M 95]' [F-H 97], [F-H 99] and [H-Ka-M 00].
5. Nearly uniform convexity and nearly uniform smoothness
First we introduce the notions of nearly uniform convexity, k-nearly uniform convexity and nearly uniform smoothness.
For a given c > a a sequence {xn} in a Banach space X is said to be c-separated if
A Banach space X is called nearly uniformly convex (NUC) if for any c > a there is 0 > a such that for every sequence {xn} in B(X) with sep( {xn}) > c there is an element x E co({xn}) such that [[xII < 1- o. This notion was introduced by Huff [Hu 80], where it was also proved that a Banach space X is NUC if and only if it is reflexive and it has the uniform Kadec-Klee property (UKK). Recall that a Banach space X is said to have the UKK - property if for any c > a there is 0 = 0 (c) > a such that for any sequence {xn} with sep({xn}) > c and any x E B(X), we have IIxll :::; 1- 0 whenever Xn -+ x weakly.
It is well known that NUC Banach spaces have the FPP (see [Go-Ki 90]). The property NUC has also been defined by using the measure of noncompactness by Goebel and S~kowski [Go-Se 84].
Kutzarova [Kur 30] introduced the notion of k-nearly uniform convexity of Banach spaces (k-NUC). Let k be an integer, k ::::: 2. A Banach space X is said to be k-NUC if for any c > a there exists 0 > a such that for every sequence {xn} in B(X) with sep( {xn}) > c, there are nI, n2, ... ,nk E N such that [[(xnl +xn2+· +xnk)/kll < 1-0. Clearly k-NUC Banach spaces are NUC but the opposite implication does not hold in general (see [Cu-H-Li]).
The notion of nearly uniform smoothness (NUS) has been introduced by S~kowski and Stachura [Se-St 82]. The definition uses the notion of the measure of noncompactness. Prus [Pr 89], [Pr 99] used another (equivalent) definition of this property which is easier to formulate. Namely a Banach space X is said to be NUS if for every c > a there exists 0 > a such that for each basic sequence {xn} in B(X) there is k > 1 such that
for each t E [0,0]. Prus [Pr 89] showed that a Banach space X is NUC if and only if its dual space X* is NUS.
A natural generalization of NUS is WNUS where the condition "for every c > 0" in the definition of NUS is replaced by "for some c E (0,1)". Let A be a bounded set of X. Its Kuratowski measure of non-compactness a(A) is defined as the infimum of all numbers d > a such that A may be covered by finitely many sets of diameter smaller than d.
A Banach space X is said to be nearly uniformly *-smooth provided that for every c > a there exists Ii > a such that if x E S(X), then
a(S*(x,6)) :::; E:,
Geometrical properties in function lattices 363
where S*(x, 0) = {x* E B(X*): x*(x) 2: 1- oJ. NUC and NUS have been also studied by Banas [B 87], [B 91].
Also nearly uniform smoothness and weakly nearly uniform smoothness are related to the fixed point theory as it follows from the following.
Theorem 5.1 (see [Ca 97]) If X is a WNUS Banach space, then X has the FPP. In particular, NUS Banach space have the FPP.
In order to get criteria for NUS of Orlicz spaces it is natural to present first criteria for NUC of these spaces because NUC and NUS are dual properties (see [Pr 89]).
Since k-NUC implies NUC we present first criteria for k-NUC of Orlicz spaces given in [Cu-H-Li].
Theorem 5.2 Let (T,~, /L) be a nonatomic and finite measure spaces and <I> be a Orlicz function satisfying (<I>(u)/u) -+ 00 as u -+ 00 and X be equal to Lip or L~. Then X is k-NUC if and only if <I> is a strictly convex and satisfies the i':l.2-condition at infinity and <I> is uniformly convex outside a neighbourhood of zero.
Corollary 5.3 Under the assumptions of Theorem 5.2 on /L and <I>, the spaces Lif> and L~ are NUC if and only if both <I> and W (where W is the Young conjugate of <I» satisfy the condition i':l.2 at infinity.
Proof. It follows directly from the facts that k-NUC implies NUC, NUC implies reflexivity and reflexivity of Lif> (respectively, L~) is equivalent to the fact that both <I>
and W satisfy the suitable i':l.2-condition. •
Theorem 5.4 The Orlicz sequence space lif> is k-NUC if and only if both <I> and W satisfy the i':l.2 -condition at zero, that is lif> is reflexive.
Proof. We need only to prove the sufficiency of the theorem. Suppose that the implication is not true. Let any £ > ° and {xn} C B(lif» with sep(xnJ> £ be given. By <I> E i':l.2(O), there exists 0 = 0(£) > ° such that
. {(xn-xm) } mf Iif> 2 : n =F m 2: o.
Next, we will show that for any j E N there exists nj E N such that
00 0 I: <I> (xnj (i)) 2: 3' 'L=J
Otherwise, there exists ]0 E N such that
for any] EN.
00 0 I: <I> (xnj (i)) < 3
(5.1)
Defining xn = (xn(1),xn(2), ... ,xn(jo),O,O, ... ) for n E N, we easily see that there exists a subsequence {xnk } of {xn} such that
Iif> en, ; Xnj ) < ~
364
for any i # j. Hence
This contradiction shows that (5.1) holds.
Since qr satisfies the ll.2-condition at zero, there is 8 E (0,1) such that
<1> G) ::; (1 - 8) <1>ku ) ('Va::; U ::; <1>-1(1))
(see [Ay-D-Lo 97] and [C-H-Ka-M 98]). By <1> E ll.2(0), there exists a > a such
88 II<I>(x + y) - I<I>(x) I < 6k
whenever I<I>(x) ::; 1, /;f!(Y) ::; a (see [Ay-D-Lo 97J, [Ch 96]).
Take n1 < n2 < ... < nk-li nl,n2,··· ,nk-l EN. Notice that
I (Xnl +xn2 + ... +Xnk_l ) < 1 <I> k -
and l<I> (xn.) ::; 1 for i = 1,2, ... , k - 1. There exists jo E N such that
~ (Xnl (i) + xn2 (i) + ... + Xnk_l (i)) ~ <P k <a
i=jo+l
and 00 /j L <1> (xnj(i)) <3" (j=1,2, ... ,k-l).
i=jo+l
By (5.1), there exists nk E N such that
00 8 L <1> (xnk (i)) ~ 3". i=jo+l
So, in virtue of (5.2), (5.3), (5.4) and (5.5), we get
h (Xnl + . ~. + Xnk )
(5.2)
(5.3)
(5.4)
(5.5)
Geometrical properties in function lattices 365
= f ip (Xn1 (i) + .~. + Xn.(i)) + . t ip (Xn1 (i) + .~. +Xn.(i))
,=1 '=}0+1
S ~ ~ t ip(xnj(i)) + i=tl ip (Xn~(i)) + ~: 1 k jo 1 _ 8 00 86
S kLLip(Xnj(i))+-k- L ip(xn.(i)) + 6k j=1 i=1 ;=jo+l
1 k 00 8 00 86 =kLLip(Xnj(i))-k L ip(xn.(i))+6k
j=1 i=1 i=jo+1
86 96 e6 S 1 - 3k + 6k = 1 - 3k'
which completes the proof. • Theorem 5.5 For any Orlicz function ip satisfying (ip(u)/u) -+ 00 as u -+ 00 the Orlicz sequence space l~ is k-NUC if and only if both ip and IlJ satisfy the i:l2-condition at zero, that is l~ is reflexive.
Proof. We only need to prove the sufficiency. Let any c > 0 and {xn} C B(l~) with sep( {xn}» c be given. By ip E i:l2(0), there exists 6 > 0 such that
. { (Xn - Xm) } mf Iq, 2 : n =f. m ~ 6.
By the same argument as in Theorem 5.4, we have that for any j E N there exists nj E N such that
(5.6)
Take kn ~ 1 satisfying
1 IIxnllo = kn (1 + Iq,(knxn)) (n = 1,2, ... ).
Such numbers kn ~ 1 exist by the assumption (ip(u)/u) -+ 00 as u -+ 00 (see [Ch 96]). Since IlJ satisfies the i:l2-condition at zero, the number
ko = sup{kn : n = 1,2, ... }
is finite (see [Ay-D-Lo 97]). Fix nl < n2 < ... < nlc-l; nl, n2, ... ,nlc-l E N. For any nk EN, put
k
H= nkni' ;=1
k k kk-l n n ni 0 hj = k"" h = -k-- and A = k-l .
'..J.' '-I '" h ko + 1 'rJ ,- ~ j j=1
By ip E i:l2 (0), there exists 9 E (0,1) such that
ip().u) S (1 - e).ip(u) whenever 0 SuS ip-l(ko).
366
Since <I> is convex, for any 1 E [O,.A] and U E [0, <I>-1(ko)], we have
<I>(lu) = <I> (.A±u) ~ (1 - 6).A<I> Uu)
1 ~ .A(1 - 6)~<I>(u) ~ (1- 6)l<I>(u).
h h kk-l Since --f!L = ~ ~ ~ = .A, the following holds
E hi h k+ E hi l+ko i=l i=l
hk hk -k-u ~ (1 - 8)-k-<I>(u) (5.7)
2: hi 2: hi i=l i=l
whenever 0 ~ u ~ <I>-l(ko). By <I> E ~2(0), there exists ()" > 0 such that
8kk {) 11(> (x + y) - I(>(x)1 < L1 ·-6
l+ko
if I(>(x) ~ ko and I(>(y) ~ ()" (see [Ay-D-Lo 97] and [Pr 89]). Consequently, we need only to prove that (ii) '* (iii). We will show that (ii) implies the ~2-condition at zero for <I>. If <I> does not satisfy the ~2-condition at zero, we can construct x E 8(l(» such that I(>(x) ~ 1 and 1(>((1 + ~)x) = 00 for every n E N (see [Ch 96] and [Ka 82a]). Take a sequence {id of natural numbers such that ik 1 and
ik+l
L <I> ( ( 1 + ~) x( i)) ~ 1 (k E N). i=i k +1
Put
Xk = (0,0, ... ,0, X(ik + 1), X(ik + 2), ... , X(ik+1), 0, 0, ... ) (k EN).
Then it is obvious that
Moreover, Xk -> 0 weakly. (5.8)
Indeed, for every y* E (l(»* we have y* = Yo + yi uniquely, where Yo is the order continuous part of y' and yi is the singular part of y'. That is yi(x) = 0 for any x E h(> (see [Ch 96]). The functional Yo is generated by some Yo E l\[l by the formula
00
y(;(x) = (x,Yo) = Lx(i)yo(i) (x E l(». i=1
Let .A> 0 be such that 2:~1 W(.Ayo(i)) < 00. Since Xk E h(> for any kEN, we have
ik+l
(Xk,y') = (Xk,YO) = L x(i)yo(i)
Geometrical properties in junction lattices 367
-> ° as k -> 00,
that is (5.8) holds.
Since the space lif> is nearly uniformly *-smooth, it has property A 2, that is for any c > ° there exists 8 E (0,1) such that for each weakly null sequence (zn) in B(lif» there is m > 1 such that
II Zl + tZm II ::::; 1 + tc
whenever t E [0,8J (see [Pr 89J and [Pr 99]). Take ko E N such that k!l < (1- c)8 if k ~ ko. We have for k ~ ko,
k 1+8c~ IIx + 8Xkll ~ 1I(1+8)Xkll ~ (1+8)k+l
= (1 + 8) (1 __ 1_) > 1 + 8 __ 2_ k+l k+l'
whence k!l > (1- c)8. This is a contradiction which finishes the proof of the fact that (ii) implies the n2-condition at zero for <I>.
Next, we will show that (ii) implies the 82-condition for w. By the above part of the proof, we can assume that 1'II is nearly uniformly *-smooth and <I> satisfies the n2-condition at zero. So, lif> is order continuous. Moreover, any Orlicz space lif> has the Fatou property and consequently, it is weakly sequentially complete. So, in view of Corollary 5.3, lif> is nearly uniformly smooth and consequently reflexive. This yields the n2-condition at zero for w. •
6. WORTH and uniform nonsquareness
Garcia-Falset [Ga 94J has proved that if a Banach space X has WORTH and is uniformly nonsquare, then X has the FPP.
So, we will present now criteria for uniform nonsquareness in Orlicz spaces and criteria for WORTH in Kothe sequence spaces. We say following Sims [S 88J that a Banach space X has WORTH if for any x E S(X) and any weakly null sequence (xn) in X, we have
lim IlIxn + xII - IIxn - xlIl-> 0. n~oo
Let £0 be an Orlicz sequence space. A Banach space X c eo is said to be a Kothe sequence space (or a Banach sequence lattice) if there is a sequence x = (x(i))~l EX with all x(i) =J 0 and for every x E eo and y E X with Ix(i)1 ::::; ly(i)1 for all i E N it follows that x E X and IIxll ::::; lIyll.
Theorem 6.1 (see [Cu-H-P 99]) A Kothe sequence space X has WORTH if and only if it is order continuous.
In this section we do not assume that <I> satisfies condition (ii) from the definition of an Orlicz function.
Corollary 6.2 Orlicz sequence spaces lif> equipped with the Luxemburg norm or with the Orlicz norm have WORTH if and only if <I> E n2(O).
Proof. Since order continuity of lif> and l~ is equivalent to <I> E n2(O), the corollary follows immediately from Theorem 6.1. •
368
The notion of uniform nonsquareness of a Banach space was introduced by James [J 64]. Recall that a Banach space X is said to be uniformly nons quare (UNSQ) if there is e E (0,1) such that for every x, y E B(X) there holds
Theorem 6.3 (see [H 85], [H-Ka-Mu 88] and [Su 66])
(a) In the case of a nonatomic infinite and u-finite measure space as well as in the case of the counting measure space the Orlicz space Lit> equipped with the Luxemburg norm is uniformly nons quare if and only if it is reflexive.
(b) In the case of any finite nonatomic measure space the Orlicz space Lit> equipped with the Luxemburg norm is uniformly nons quare if and only if Lit> is reflexive and <I>(b(<I»)p,(T) < 2, where b(<I» = sup{u 2 0: <I> is linear on the interval [O,u]}.
Note. Uniform nonsquareness of Musielak-Orlicz spaces was characterized in [H-Ka-Ku 87]. For Orlicz-Lorentz spaces it was done in [H-Ka-M 96], where uniform nonsquareness of some Calderon-Lozanowsky spaces was also considered.
Note. The characteristic of convexity of Orlicz function spaces equipped with the Luxemburg norm was calculated in [H-Ka-Mu 88] in the case when the measure spaces is nonatomic and infinite. Recall that this coefficient for a Banach space X is defined by
eo(X) = inf{e E (0,2]: Ox(e) > O},
where Ox denotes the modulus of convexity of X. In the case of nonatomic and finite measure space eo(Lit» was calculated in [H-W-Wa 92]. Lower and upper estimates for the characteristic of convexity of Kothe-Bochner spaces were given in [H-Lan 92].
7. Opial property and uniform Opial property in modular sequence spaces
In this section we will present some results on the uniform Opial property of modular sequence spaces. As a corollary we will obtain criteria for the Opial property and the uniform Opial property of Orlicz sequence spaces for both the Luxemburg and Orlicz norms.
Let X be a real vector space. A functional m: X -> [0,00] is called a modular if (see [Mu 83] and [Mal 89]):
(i) m(x) = 0 if and only if x = 0,
(ii) me-x) = m(x) for all x EX,
(iii) m(ax + f3y) ::; am (x) + f3m(y) for all x, y E X and a, f3 2 0 such that a + f3 = 1 (that is m is convex).
For any modular m on X, the space
Xm = {x E X: m(Ax) < 00 for some A> O}
Geometrical properties in function lattices 369
is called the modular space (generated by m). It is obvious that Xm is a vector space. The functional
Ilxll = infp > 0: m(x/>..) ~ I}
is a norm on X m , which is called the Luxemburg norm (see [Cu-H 98a] and [Cu-H 99a]). A modular m is said to satisfy the ll2-condition (m E ll2) if for any E: > 0 there exist
constants K ~ 2 and a > 0 such that
m(2x) ~ Km(x) + E:
for all x E Xm with m(x) ~ a.
If m satisfies the ll2-condition for any a > 0 with K ~ 2 dependent on a, we say that m satisfies the strong ll2-condition (m E ll~).
In this section a function <I>: (-00,00) --t [0,00) is said to be an Orlicz function if it is convex, even and <I>(O) = 0 (see [Ch 96], [Lu 55], [Mal 89], [Mu 83], [Kr-R 61] and [Ra-Re 91]). For a given Orlicz function <I> one can define on the space 1° of all real sequences x = (x(i)) the modular
00
mq;(x) = L <I>(x(i)). i=l
The modular space (CO)m", is called an Orlicz sequence space (see [Ch 96], [Kr-R 61], [Lu 55], [Mal 89), [Mu 83] and [Ra-Re 91]).
It is easy to see that if <I> vanishes only at zero, then mq; E ll2 whenever <I> E ll2(0).
Let X be a Banach sequence space (or Kothe sequence space), an element x E X is said to be absolutely continuous if
lim 11(0, ... ,0,x(n+l),x(n+2), ... )11 =0. n~oo
The set of all absolutely continuous elements in X is denoted by Xa and it is a subspace of X. X is called absolutely continuous if Xa = X.
Vie say that a Banach sequence lattice X has the Fatou property if for any x E X and a sequence {xn } in X such that 0 ~ Xn ~ x and Xn i x, there holds IIxnll r IIxll (for the theory of Kothe sequence spaces we refer to [Kan-Aki 72]). A Banach space X is said to have the Opial property (see [0 67]) if for every weakly null sequence {xn } and every x # 0 in X there holds
liminf IIxnll < liminf Ilxn + xii· n--+<X) n-tOCl
The Opial property is important because Banach spaces with this property have the weak fixed point property (see [G-La 72]).
Opial has proved in [0 67) that the Lebesgue sequence spaces Cp (1 < p < 00) have this condition but Lp[0,21l"] (p # 2, 1 < p < 00) do not have it. Franchetti [Fr 81] has shown that any infinite-dimensional Banach space admits an equivalent norm under which it has the Opial property. A Banach space X is said to be the uniform Opial property (see [Pr 92]) if for every IS> 0 there exists T > 0 such that for any weakly null sequence {xn} in S(X) and x E X with Ilxll ~ IS there holds
1 + T ~ liminf IIxn + xII. n-+oo
370
Let fJ be the ball-measure (that is the Hausdorff measure) of noncompactness in X, that is
fJ(A) = inf{c > 0: A can be covered by a finite
family of sets of diameter :s: c}
for any A c X. A Banach space X is said to have property (L) if lime_l_ ~(c) = 1, where
~(c) = inf{1 - inf{llxll: x E A}},
and the first infimum is taken over all closed sets A in the unit ball B(X) of X with fJ(A) 2: c.
The function ~ is called the modulus of noncompact convexity (see [Go-Se 84]). It has been proved in [Pr 92] that property (L) is useful to study the fixed point property and that a Banach space X has property (L) if and only if it is reflexive and has the uniform Opial property. We start with the following auxiliary lemma.
Lemma 7.1 Assume that m E ~2. Then for every L > 0 and c > 0 there exists 8 = 8(L,c) > 0 such that for all x,y in Xm with m(x) :s: Land m(y) :s: 8, there holds Im(x + y) - m(y)1 < c.
Proof. Let L > 0 and c > 0 be given. By m E ~2' we conclude that there is Ko 2: 2 such that
m(2x) :s: Kom(x) + c/8
for all x E Xm with m(x) :s: L. Set fJ = c/2KoL. Using again m E ~2' one can find Kl 2: 2 such that
m Gx) :s: K1m(x) + c/8
for all x E Xm with m(x) :s: L. Set 8 = c/2fJKl and assume that m(x) :s: Land m(y) :s: 8. Then
m(x + y) = m((1 - fJ)x + fJ(x + fJ-ly))
:s: (1 - fJ)m(x) + fJm(x + fJ-ly)
:s: m(x) + fJm(Tl(2x + 2fJ-ly))
:s: m(x) + T 1fJm(2x) + T 1fJm(2fJ-ly)
:s: m(x) + TlfJKoL + ~ + T1fJm(y) + ~ < m(x) +c.
In a similar way we can show that m(x) - c < m(x + V). Hence Im(x + y) - m(x)1 < c whenever m(x) :s: Land m(y) :s: 8, which finishes the proof. •
Corollary 7.2 If m E ~2' then for any x E X m , Ilxll = 1 if and only if m(x) = 1.
Proof. We only need to show that Ilxll = 1 implies m(x) = 1 because the opposite implication is obvious. Assume that m E ~2. We can easily get from Lemma 7.1 that the function f defined on R by f(A) = m(Ax) is continuous. Namely, it easily follows by m E .6.2 that f is finitely valued, which yields that f is continuous. Take any c > 0
Geometrical properties in function lattices 371
and An E R \ {O}, and apply Lemma 7.1 which L = m(Aox) and 8 = 8(L, c). We have Im(x + y) - m(x)1 < e whenever m(x) ::; Land m(y) ::; 8. Hence
Im(Ax) - m(Aox)1 = Im((A - AO)X + AOX) - m(Aox)1 < e
whenever IA - Aol < 8. So, we easily get that m(x) = 1 whenever Ilxll = 1. •
Lemma 7.3 If m E ~2, then for any sequence (xn ) in Xm the condition Ilxnll -+ 0 holds if and only if m(xn ) -+ o.
Proof. It is easy to see that IIxnll -+ 0 if and only if m(Axn) -+ 0 for each A > O. By m E ~2 it follows from Lemma 7.1 that the property holds for sufficiently small positive L (say L ::; Lo). Assume that m(xn) -+ O. There is mEN such that m(xn) ::; Lo for all n 2 m. So, for anye E (0,1) there is Kg > 0 such that
m Cen) ::; Kgm(xn ) + ~ for n sufficiently large. Let no E N be so large that m(xn ) ::; 1/(2Kg ) for n > no. Hence m(xn/e) ::; Kgm(xn) + ! ::; 1 for n> no sufficiently large. This yields IIxnll ::; e for n sufficiently large. The opposite implication follows from the inequality m(x) ::; Ilxll for x with IIxll ::; 1. •
Lemma 7.4 If m E ~~, then for any e > 0 there exists 8 = 8(e) > 0 such that IIxll 2 1 + 8 whenever m(x) 2 1 + e.
Proof. Suppose that there exist co> 0 and a sequence {xn } in Xm such that Ilxnll ! 1 and m(xn ) 2 1 + co. Since m E ~~, for any e > 0 there exists 8 > 0 such that
Im(x + y) - m(e)1 < e
whenever m(x) ::; 1 and m(y) ::; 8 (see Lemma 7.1). We may assume without loss of generality that 1 - l/llxnll < 8. Hence, applying the fact that m(xn/llxnll) = 1 for any n E N (see the proof of Lemma 7.1), we get
That is Im(xn) - 11 < e. This contradiction shows that Lemma 7.4 is true. •
Theorem 7.5 Suppose that a Kothe sequence space X has the Fatou property. Then X is absolutely continuous whenever it has the uniform Opial property.
Proof. Assume that X is not order continuous. Take e = 1/2 and an arbitrary T > o. Let () = 1/(1 + T /2), whence () > 1/(1 + T). By Riesz's lemma (see [Ta-Lay 80J, p. 64), for any () E (0,1) there is Xo E S(X) such that IIxo - xII > () for any x E Xa. Let Xo corresponds to () = 1/(1 +T/2). By the Fatou property of X,
Let nl = O. There is n2 E N such that
372
Since L~l xo(i)ei E Xa, it follows that
II . f xo( i)ei II ? B. ,=n2+1
So, there is 113 E N, 113 > 112, such that
One can find by induction a sequence (11j)~l in N such that 111 = 0, 111 < 112 < ... , and
Define nj+l
Xj= 2:= xo(i)ei (i=I,2, ... ). i=nj+l
It is obvious that x*(Xj) = 0, j = 1,2, ... , for any singular functional x* E X*. If x* E X' is order continuous, there is y = (y(i»~l E X' (the Kothe dual of X, see [Kan-Aki 72]) such that
00
x*(z) = Ly(i)z(i) (Vz = (Z(i»~l EX). i=l
Since L~l y(i)xo(i) is convergent it follows that
as j -t 00. Therefore, Xj -t 0 weakly as j -t 00. Moreover
(1- D 8:S; Ilxjll:S; 1 (j = 1,2, ... ). (7.1)
Define Yj = xj/llxjll. Then IIYjl1 = 1 for all j EN and
II Xo - Xj II Ilxoll 1 1 IIxo-Yjll:S; TxT :s; IIxjll:S; (1_1)8 -to·
J
Since 1/8 < 1 + T, there is jo E N such that
IIxo - Yjll < 1 + T (Vj > jo). (7.2)
Since Xj -t 0 weakly, inequalities (7.1) yield that Yj -t 0 weakly. Hence and by (7.2) it follows that X does not have the uniform Opial property. This finishes the proof. •
Theorem 7.6 Assume that a modular m E ~~ and m is countably orthogonally additive and that the modular sequence space Xm is a Banach space. Then Xm has the uniform Opial property.
Geometrical properties in junction lattices 373
Proof. Let 10 > 0 be given. There is 101 > 0 such that (see Lemma 7.3) m(x) 2 E:1
whenever Ilxll 2 E:. Since m E ,0,.~, by Lemma 7.1, there is 0 E (0,cl/4) such that
E:1 Im(x + y) - m(x)1 < "8
whenever m(x) ::; 1 and m(y) ::; o. By countable orthogonal additivity of m, there is io E N such that
m(. f X(i)ei) ::; o. ~=to+l
Let {xn} be a weakly null sequence in S(X). It is obvious that Xn --> 0 coordinate-wise. Hence, there is no E N such that
Therefore
m(Xn + x) = m(~(Xn(i) + X(i))ei ) + m(~(Xn(i) +X(i))ei )
2 m (t X(i)ei ) - E:; +m(.f Xn(i)ei ) _ E:;
t=l t=to+l
3 E:1 E:1 E:1 E:1 2 ::tE:1 - "8 - "8 + m(xn) - "8 - "8 =1+~
4
for n 2 no. By Lemma 7.4, there is E:2 > 0 that depends only on E:1 and such that Ilxn + xii > 1 + E:2 whenever n 2 no. This means that Xm has the uniform Opial property. •
In this section we write m in place of Iif! in Orlicz spaces.
Corollary 7.7 Orlicz sequence spaces £if! equipped with the Luxemburg norm have the uniform Opial property if and only if il> E ,0,.2(0).
Proof. Sufficiency. Orlicz spaces £if! are Banach spaces and they are modular spaces (£O)m .. , where
00
mif!(x) = L il>(x(i)) i=l
for x = (x(i)) E go. If il> E ,0,.2(0), then mif! is countably orthogonally additive and mif! E ,0,.2. Therefore, by Theorem 7.6, £if! have the uniform Opial property.
Necessity. If il> rf: ,0,.2(0), then£if! contains an order isometric copy of £00 (see [Ka 82aJ), so £if! is not absolutely continuous. Since £if! has the Fatou property (see [Lu 55]), by Theorem 7.5, £if! does not have the uniform Opial property. •
Corollary 7.8 Orlicz sequence spaces lif! equipped with the Luxemburg norm have the Opial property if and only if il> E ,0,.2(0).
374
Proof. If <I> E .6.2(0), then by Corollary 7.7, lif> has the uniform Opial property, and hence has the Opial property as well.
Assuming that <I> rf- .6.2 (0), one can find a sequence {xn} in S(lif» such that xm..lxn for m oF nand
00
x = L Xn E S(lif» (see [Ka 82a]). n=l
Then we easily get that Xn -> 0 weakly. However, II Xl + Xnll Consequently lif> fails to have the Opial property.
1 for any n > 2.
• Corollary 1.9 The Nakano sequence spaces e(Pi) with 1 ::; Pi < 00 for all i E N have the uniform Opial property if and only iflim sUPi~oo Pi < 00.
Proof. The Nakano space e(Pi) is a Banach space and it is generated by the modular
00
m(x) = L Ix(i)IPi i=l
defined on eo (see [Na 50] and [Mu 83]). If
limsuPPi < 00 i-H)O
then m E .6.~ and m is countably orthogonally additive. Therefore, by Theorem 7.6, e(Pi) has the uniform Opial property.
If limsuPPi = 00
i---+oo
then the Musielak-Orlicz function <I> = (<I»~l' where <l>i(U) = lul P', does not satisfy the 02-condition (for the definition of <I> E 02 see [Ka 82a] and [F-H 99]). Therefore (see rKa 82a], [F-H 99] and [H 98]) e(Pi) contains an order isometric copy of eoo, whence e(Pi} is not absolutely continuous. Moreover, e(Pi) has the Fatou property whence, by Theorem 7.5, it follows that e(P;) does not have the uniform Opial property, which finishes the proof. •
For some other properties of e(Pi) we refer to [H-Wu-Y 94].
In the following we will consider the uniform Opial property for Orlicz spaces equipped with the Amemiya norm
Ilxll~ = inf ~(1 + mif>(kx)). k>O k
We write e~ in place of (eif>, II . II~). Denote by K(x) the set of all k > 0 such that Ilxll~ = ~(1 + mif>(kx)). It is known (see [Ch 96], [Ra-Re 91] and [Wu-Sun 91]) that K(x) = [k;,k;*], where k; = inf{k > 0: mif>(pokx) 2 I} and k;* = sup{k > 0: mif>(po kx) ::; I} whenever K (x) oF 0 (that is k; < 00), where P denotes the right hand side derivative of <I> on R+ = [0,00) and po kx denotes the composition of P and kx. It is also well known that K(x) oF 0 for all x E e~ whenever (<I>(u)ju) -> 00 as u -> 00 (see [Ch 96] and [Cu-H-N-P 99]).
The following lemma from [Cu-H-N-P 99] will be useful to get criteria for the uniform Opial property of e~.
Geometrical properties in function lattices 375
Lemma 7.10 If x E £<f! and K(x) = 0, then A := limu~oo(iP(u)/u) < 00 and
00
Ilxll~ = A L Ix(i)l· i=l
Theorem 7.11 The Orlicz space £~ has the uniform Opial property if and only if <I> E b..2(0).
Proof. Since £~ is not absolutely continuous whenever <I> tt b..2 (0), by Theorem 7.5, the necessity is obvious.
Sufficiency. Take any c > 0 and x E £~ with Ilxll~ ::: c. Let (xn) be a weakly null sequence in S(£~). By <I> E b..2 (0) there is 0 E (0, c) independent of x such that mq, (~) ::: o. Take j E N such that
i,From Xn ~ 0 it follows that xn(i) --t 0 for any i E N. So, there exists no E N such that
Hence
IIX + xnll~ = Ilt(X(i) + xn(i))ei + j~l (x(i) + Xn(i))ei [
::: Ilt(X(i) + xnCi))ei + . f Xn(i)eiIIA ~
,=1 '=1+1 <f!
::: IltX(i)ei +f xn(i)eiIIA ! (7.3) ,=1 '=1+1 <f!
whenever n > no. We will consider now two cases for n > no.
I. K(~ x(i)ei + i=t1 xn(i)ei) =I 0. Then there exists kn > 0 such that
Ilxn +xll~ = :n (1 +m(kn(tX(i)ei +f Xn(i)ei))) . ,=1 '=1+1
Combining this with (7.3), we get
IIxn + xlI~ ::: :, (t <I>(knx(i)) + i~l <I>(knxn(i))) - ~
= L (~<I>(knX(i))) + kIn it1 <I>(knxn(i)) -~. (7.4)
376
Moreover, from the inequalities
m( f X(i)ei) ::; II f X(i)eiIIA
< ~ and m (~) ~ 8, '=J+1 '=J+1 4>
it follows that
.(7.5)
We may assume without loss of generality that kn ~ ~. Hence, inequalities (7.4) and (7.5) yield
Ilxn + xlI~ ~ II f xn(i)eiIIA
+ 2 t <I> (~X(i)) - ~ '=J+1 4> ,=1
8788 8 > 1 - - + - - - = 1 + -. - 8 8 4 2
lit x(i)ei + i~1 Xn(i)ei [ = At Ix(i)1 + A i~1 IXn(i)1
~ IltX(i)eiIIA + II f xn(i)eill
,=1 4> '=J+1
78 8 38 > "8 + 1 - 8 = 1 + 4·
Therefore, by inequality (7.3) we get
So, in any case, IIXn + xlI~ ~ 1 + ~ for n > no, which finishes the proof.
Corollary 7.12 Orlicz spaces 14> generated by Orlicz functions <I> satisfying
(<I>(u)/u) --t 00 as u --t 00
have normal structure if and only if <I> E Ll2(O).
•
Proof. If <I> E Ll2(O), then l~ has uniform normal structure, and so it has normal structure as well.
Assume that <I> ¢ Ll2(O). Then there exist x E S(l~) and a sequence {xn} in (l~)+ such that xm..lxn for min, xn..lx, {xn} has a majorant in (l~)+ and 14>(koxn) ::; 2-n for any n, where ko ~ 1 satisfies fo(1 + m(kox)) = Ilxll~ = 1, and Ilxnll~ --t 1 as n --t 00.
Then Xn --t 0 weakly. Moreover,
k1 (1 + m(ko(x + Xn)) = ~(1 + m(kox)) + ~m(koXn) ::; 1 + Tn. o ko ko
Therefore, lim Ilxn + xnll~ = lim Ilxnll~ = 1. That is, l~ does not have the Opial n--+oo n--+oo
property. •
Geometrical properties in function lattices 377
8. Garcia - Falset coefficient
First we need to introduce some notation and definitions. Garcia-Falset [Ga 94] defined the coefficient
R(X) = sup {liminf Ilxn - xii: x E B(X), {xn} C B(X), Xn ~ o} n-->oo
and proved in [Ga 97] that any Banach space X with R(X) < 2 has the weak fixed point property.
A Kothe sequence space X is said to have the semi-Fatou property (X E SF P) if for every sequence {xn} in X and x E X such that 0 :S Xn 1 x, we have Ilxnll -t Ilxll·
Theorem 8.1 (see [H-M 93]) If X is a Kothe sequence space with the semi-Fatou property and with the norm not being absolutely continuous, then X contains almost isometric copy of 100 • That is, for any c; > 0 the exists a closed subspace Y of X and an isomorphism P of 100 onto Y which is a (1 + c;)-isometry.
Corollary 8.2 If a Kothe sequence space with the semi-Fatou property is not absolutely continuous, then R(X) = 2.
Proof. It is easy to see that R(loo) = 2. Moreover, by Theorem 8.1, R(X) = R(loo) .
• Corollary 8.3 IfiI> does not satisfy the t::. 2 -condition at zero, then R(lq,) = R(l~) = 2.
Proof. Each of the norms 1111 and 1111 0 have the semi-Fatou property (in fact they even have the Fatou property). Moreover, if iI> rt t::.2(0), then lq, and l~ are not absolutely continuous (see [Ch 96]). So, by Corollary 8.2, we get the desired conclusion. •
Theorem 8.4 (see [Cu-H-Li 00]) For any Orlicz function iI>, the equality
R(hq,) = sup {cx : x = ~X(i)ei E S(lq,) for some mEN} holds, where Cx is positive number satisfying Iq,(x/cx ) = 1/2.
Remark 8.5 Note that R(X) = Rl(X) for any Kothe sequence space with the semiFatou property and an absolutely continuous norm.
Corollary 8.6 For any Lebesgue sequence space lp (1 < p < 00), we have R(lp) = 21/ p.
Proof. For any x E S(lp) we have Cx = 21/ p, which follows by the equalities
• To formulate the next corollary we need an equivalent definition of the Riesz angle for a Banach lattice X. It is defined by
Q(X) = sup{lIlxl V Iylll : X,y E B(X), IxlA Iyl = O}.
378
Corollary 8.7 For any Orlicz function iI>, the equality R(hq,) = a(hq,) holds.
Proof. By the equality R(hq,) = d that was obtained in the proof of Theorem 8.4, we can easily get the inequality R(hq,) :::; a(hq,). On the other hand, for any c > 0 there exist x E S(hq,) and y E S(hq,) such that Ixl II Iyl = 0 and
Illxl V Iylll > a(lq,) - c:.
For the sake of convenience, we may assume that x V y = (x(I), y(I), x(2), y(2),· .. ). By the fact that hq, has an absolutely continuous norm, there exists io E N such that
II (Ixl (1), Iyl (1),· .. ; Ixl (io), Iyl (io), 0, 0,· .. )11 2: a(hq,) - c. (io+n)th
Defining Xo = (x(I), x(2),··· , x(io), 0, 0,···) and Yn = (~, lyl (1), Iyl (2),··· , Iyl (io), 0, 0,···) for all n EN, we get Yn ~ 0 and
liminf llYn + xoll 2: a(hq,) - c. n~oo
Hence R(X) 2: a(X) - c. By the arbitrariness of c > 0, we get R(hq,) 2: a(hq,) and consequently R(hq,) = a(hq,). •
Corollary 8.8 For any Orlicz sequence space lq" R(lq,) < 2 if and only if iI> E 6dO) and W E ~2(0).
Corollary 8.9 For any Orlicz function iI>, R(hq,) < 2 if and only if W E ~2(0).
Proof. By Corollary 8.2 and Theorem 3.11 in [Ch 96], which says that if W E ~2(0), then hq, has the w - F P P, both Corollaries 8.8 and 8.9 follow. •
Notes. It is known that property (f3) which has been introduced by Rolewicz [Ro 87] is stronger than NUC and it implies normal structure of the dual space (see [Kut-Ma-Pr 92]). Property (f3) has been considered in Orlicz-Bochner spaces, Musielak-Orlicz sequence spaces of Bochner type, Orlicz-Lorentz spaces and Calder6n-Lozanovskii spaces in [Ko-a], [Ko-b] and [Ko-c]. Properties UKK and NUC in Kothe-Bochner spaces have been considered in [Ko-d].
9. Cesaro Sequence Spaces
For 1 ::; p < 00 the Cesaro sequence space cesp is defined by
{
00 1 n ]} cesp = x E 1°: Ilxll = [];(;;: 8Ix(i)IYp];; < 00
(see [Lee 84] and [Sh 70]).
Lemma 9.1 (see [Cu-H OOb]) For any £ > 0 and L > 0, there exists 8 > 0 such that
Geometrical properties in function lattices 379
Proof. It follows by the uniform continuity of the function f (u) = uP on any compact interval [0, I]. •
Theorem 9.2 For the Cesaro sequence space cesp (1 < p < 00) we have R(cesp )
21/ p .
Proof. We can apply Remark 8.5. Let c > 0 be given. For any
In h
{Xn = L Xn(i)ei} ~=2 C S(cesp ), Xn ~ 0, X = Lx(i)ei E S(cesp ), It < h < '" , i=In _ 1+1 i=l
there exists no E N such that
00
L (~y < min(c, 8) , k=ino +l
h where a = L Ix(i)1
i=1
and 8 > 0 is the number corresponding to our c > 0 and L = 1 in Lemma 9.1. Hence for any m > no there holds
> 1 - c + 1 = 2 - c.
1
That is, lim inf Ilxn - xii ~ (2 - c) P. On the other hand, for any m > no, n--;oo
That is, 1~IIl,~f Ilxn - xii::; (2 + c) ~. By the arbitrariness of c > 0 and by Remark 8.5, 1
we get R(cesp ) = 21'. •
Corollary 9.3 Cesaro sequence spaces cesp (1 < p < 00)) have the fixed point property.
Proof. For 1 < p < 00, cesp is a reflexive space and since R(cesp ) = 21/ p < 2, cesp
has the weakly fixed point property. Therefore, cesp has the fixed point property. •
380
10. WCSC, uniform Opial property, k-NUC and UNS for cesp
Our main aim in this section is to calculate the weakly convergence sequence coefficient for Cesaro sequence space cesp and to prove that for any p E (1,00), cesp is k-NUC for any integer k ~ 2 and has the uniform Opial property and property (L). The weakly convergence sequence coefficient, which is connected with normal structure, is an important geometric constant. It was introduced by Bynum [By 80].
For a sequence {xn} C X, we consider
A({xn}) = lim {sup{llxi-Xjll: i,j~n,i~j}} n ..... oo
A 1({xn }) = lim {inf{lIxi-xjll: i,j~n,i~j}}. n ..... oo
The weakly convergence sequence coefficient of X, denoted by WCS(X), is defined as follows:
WCS(X) = sup{k > 0: for each weakly convergent sequence {xn }, there is
y E col {xn}) such that k· lim sup IIxn - yll ~ A( {xn})}, n ..... oo
see [B 91].
The number M(X) = IjWCS(X) for a reflexive Banach space is called the Maluta coefficient and it is known that M(X) = 1 for every non-reflexive Banach space X (see [Ma 84]). It is also well known that a Banach space X with WCS(X) > 1 has weak normal structure (see [Cu-H-Li]). A sequence {xn} is said to be an asymptotic equidistant sequence if A({xn}) = A 1({xn }) (see [Z 92]). The formula
WCS(X) = inf{A( {xn}): {xn} C S(X) and Xn ~ O}
= inf{ A( {xn}): {xn} an asymptotic equidistant
sequence in S(X) and Xn ~ O}
was obtained in [Z 92].
A Banach space X is said to have weak uniform normal structure if WCS(X) > l. Recall that the functions a and f3 are the Kuratowski measure of noncompactness and the it Hausdorff measure of noncompactness in X, respectively. We can associate these functions with the notions of the set-contraction and the ball-contraction (see [De 85]). These notions are very useful in the study of nonlinear operator problems (see [De 85]).
The packing rate of a Banach space X is denoted by ,(X) and it is defined by the formula
,(X) = 8(X)ju(X),
where 8(X) and u(X) are defined as the supremum and the infimum, respectively, of the set
{ !~~~: A C X, A is a-minimal, alA) > O}. Recall that A C X is said to be a-minimal if alB) = alA) for any infinite subset
of A. For those definitions and for results concerning the existence of a-minimal and f3-minimal sets we refer to [Ay-D-Lo 97], Chapter X.
Theorem 10.1 If 1 < p < 00, then the space cesp is k-NUC for any integer k 2: 2.
Geometrical properties in function lattices 381
Proof. Let c> 0 be given. For every sequence {xn} C B(X) with sep({xn}) > c, we put x;:' = (0,0, ... ,O,xn(m),xn(m + 1), ... ). For each i E N, the sequence {xn(in~l is bounded. Therefore, using the diagonal method one can find a subsequence {xnk } of {xn} such that the sequence {xnk (in converges for each i EN. Therefore, for any mEN there exists km such that sep({x;::'h::::km) 2: c. Hence for each mEN there exists nm E N such that
(10.1)
Write Ip(x) = 2::::"=1 (~2:7=1 Ix(i)IY and put C1 = ;;(~=h (~r Then :3 8 > 0 such that
IIp(x + y) - Ip(x) I < C1
whenever Ip(x) ::; 1 and Ip(y) ::; 8 (see Lemma 9.1).
(10.2)
There exists m1 E N such that Ip(x'[") ::; 8. Next, there exists m2 > m1 such that Ip(x~) ::; 8. In such a way, there exists m2 < m3 < ... < mk-1 such that Ip(x';i) ::; 8 for all j = 1,2, ... ,k - 1. Define mk = mk-1 + 1. By condition (10.1), there exists nk E N such that Ip(x;::'k) 2: (c/2)P. Put ni = i for 1 ::; i ::; k - 1. Then in virtue of (10.1), (10.2) and convexity of the function f(u) = lulP , we get
382
(kP- 1 - 1) e
::; 1 + (k - l)el - kP ("2-)P
= 1 _ ~ (kP-1 - 1) (=-)P 2 kP 2'
Therefore, cesp is (k - NUC) for any integer k ~ 2. • Theorem 10.2 For any 1 < p < 00, the space cesp has the uniform Opial property.
Proof. For anye > ° we can find a positive number eo E (O,e) such that
eP 1 + 2 > (1 + eo)P.
Let x E X and IIxll ~ e. There exists nl EN such that
Hence we have
II . f X(i)eill < e: <~, t=nl+l
ith where ei = (0, ... , 1 ,0,0, ... ). Furthermore, we have
n, (1 n )P 00 (1 n )P eP
::; ~ ~ 8 Ix(i)1 + n=~+1 ~ 8 Ix(i)1
n, (1 n )P <~ ~8Ix(i)1 +(~r
n, (1 n )P P <~ ~8Ix(i)1 +e4 ,
whence 3 p n, (1 n )P +::; ~ ~~lx(i)1
Geometrical properties in junction lattices 383
For any weakly null sequence {xm} C S(X), in virtue of xm(i) ~ a for i = 1,2, ... , there exists mo E N such that
when m > mo. Therefore,
IIXm + xII = Ilt(Xm(i) + x(i))ei + .:t (xm(i) + X(i))eill .=1 ,=nl+1
2: Ilfx(i)ei +.:t Xm(i)eill-llfxm(i)eill-II.:t X(i)eill ,=1 ,=nl+1 ,=1 ,=nl+1
nl when m> mo. Moreover for a := L: Ix(i)1 there holds
i=l
IIXm+xlI2: Ilfx(i)ei+.:t Xm(i)eill- c; .=1 .=nl+1
cO cO 2: 1 + co - 2 = 1 + 2· This means that cesp has the uniform Opial property.
By the reflexivity of cesp for 1 < p < 00, we get the following.
•
Corollary 10.3 For 1 < p < 00 the space cesp has property (L) and the fixed point property.
Now, we will calculate the weakly convergence sequence coefficient of cesp .
1
Theorem 10.4 For 1 < p < 00, WCS(cesp ) = 2p.
Proof. Take any c > a and an asymptotic equidistant sequence {xn} C S(X) with Xn ~ a and put V1 = Xl. There exists i1 E N such that
11.:t V1(i)ei ll < c. t=tl+1
384
Since Xn --4 0 coordinate-wise, there exists n2 E N such that
whenever n 2: n2.
Take V2 = x n2 . Then there is i2 > i1 such that
Since xn(i) --4 0 coordinate-wise, there exists n3 EN such that
whenever n 2: n3.
Continuing this process in such a way by induction, we get a subsequence {vn} of {xn} such that
Put
for n = 2,3, . .. . Then
in
Zn = L Vn(i)ei i=in _l+1
Moreover, for any n, mEN with n oj m, we have
(10.3)
IIvn - vmll = II~ un(i)ei - ~ Vm(i)eill (10.4)
2: II. f vn(i)ei -.f Vm(i)eill-II~ vn(i)eill t=tn _l+1 t=tm_l+1 2=1
-L~l Vn(i)eill-Ili~ vm(i)eill- L~+1 Um(i)eill
2: IIZn - zml! - 4c.
This means that A({xn}) = A({vn}) 2: A({zn}) - 4c. Put Un = zn/llznli for n = 2,3, .... Then
(10.5)
Geometrical properties in junction lattices 385
(10.6)
On the other hand
for every m, n EN, m i' n. Therefore
(10.7)
By the arbitrariness of c > 0, we have from (10.5), (10.6) and (10.7) that
in WCS(ceSp)==inf{A({un}): Un = 2:= un(i)eiES(cesp),
i=in _l+1
o == io < i1 < i2 < ... , Un ~ O}. Using Lemma 2 in [Z 92], we have
in WCS(cesp) == inf{ A( {Un}): Un = 2:= 'un(i)ei E S(cesp), 0 = io < il < ... ,
i=in_l +1
Un ~ 0 and {Un} is asymptotic equidistant}.
Take mEN large enough such that
where b:= 2:::in_l+1lun(i)I. We have for n < m
Note that
1
> 1-c+ 1 = 2 - c, that is, A1({ud):::: (2- c),.
[k=i~'+1 (~(b + t IUm(i)l) rr
== [k=i~'+1 (~+ ~ t lum(i)lrr
~ [k=i~'+1 G rf + [k=i~'+1 U t 'Um(i),rr < c~ + 1.
386
Therefore
1 ::; 1+(I+cii )P
1 1
for any n,m E N, m # n. This yields A({un }) ::; [1 + (1 + cii)P)ii and, by the 1
arbitrariness of c > 0, we obtain we S( cesp) = 2ii. •
Corollary 10.5 For 1 < p < 00, cesp has the weak uniform normal structure and normal structure.
Corollary 10.6 For any 1 < p < 00, we have 'Y(cesp) = 2(P-l)/p.
Proof. By [Ay-D 93), if X is reflexive Banach space with the uniform Opial property, then 'Y(X) = 2/WeS(X). Since, by Theorem 10.1, cesp is NUe for 1 < p < 00 and property NUe implies reflexivity, Theorem 10.2 yields 'Y(cesp) = 2/21/ P = 2(P-l)p. •
Note. Banach-Saks and weak Banach-Saks properties in Cesaro sequence spaces has been characterized in [Cu-H 99b).
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Chapter 13
INTRODUCTION TO HYPERCONVEX SPACES
R. Espinola
Departamento de Analisis Matematico
Facultad dt Matematicas
Universidad de Sevilla
Aptdo 1160, Sevilla 41080, Spain
M. A. Khamsi
Department of Mathematical Sciences and Computer Science
The University of Texas at El Paso
El Paso, Texas 79968, USA
am [email protected],edu.ku
1. Preface
The notion of hyperconvexity is due to Aronszajn and Panitchpakdi [IJ (1956) who proved that a hyperconvex space is a nonexpansive absolute retract, i.e. it is a nonexpansive retract of any metric space in which it is isometrically embedded. The corresponding linear theory is well developed and associated with the names of Gleason, Goodner, Kelley and Nachbin (see for instance [19, 29, 42, 46]). The nonlinear theory is still developing. The recent interest into these spaces goes back to the results of Sine [54J and Soardi [57J who proved independently that fixed point property for nonexpansive mappings holds in bounded hyperconvex spaces. Since then many interesting results have been shown to hold in hyperconvex spaces.
Recall also that Jawhari, Misane and Pouzet [27J were able to show that Sine and Soardi's fixed point theorem is equivalent to the classical Tarski's fixed point theorem in complete ordered sets. This happens via the notion of generalized metric spaces. Therefore, the notion of hyperconvexity should be understood and appreciated in a more abstract formulation. It is not our purpose, however, to study hyperconvex spaces from this more general point of view, interested readers may consult the references [22, 27, 28, 38, 59J.
Along this chapter we will describe and study some of the most characteristic properties that hyperconvex spaces enjoy. In opposition to the lack of linearity hyperconvexity provides us with a really rich metric structure that leads to a collection of surprising and beautiful results related to different branches of mathematics as, for instance, topology,
391
WA. Kirk and B. Sims (eds.), Handbook of Metric Fixed Point Theory, 391-435. © 2001 Kluwer Academic Publishers.
392
graph theory, multivalued analysis, fixed point theory, ... It is our aim to present some of these results about hyperconvexity emphasizing their relation to fixed point theory.
This chapter has been divided into eleven sections. We begin with basic definitions and properties which will lead us to a first approach to the relation between hyperconvexity and the Hahn-Banach theorem. In Section 3 we continue with more general properties on hyperconvexity, we will learn some fundamental facts of the geometry of hyperconvex spaces, facts such as that every hyperconvex space is complete or relevant properties verified by their Chebyshev elements. In this section we will also distinguish among four different important subclasses of hyperconvex subsets of a hyperconvex space, namely hyperconvex, admissible, externally and weakly externally hyperconvex subsets. These classes of sets will be of great importance all along our treatment and more particularly when fixed point results will be stated. In Section 4 we relate hyperconvexity to injectivity and absolute retracts. We will see the role that these latter concepts played in the motivations of hyperconvexity to finish with more recent and subtle results on the existence of retractions and E-constant nonexpansive retractions. Section 5 will be devoted to the study of deeper facts of the geometry of hyperconvex spaces, we begin by studying the bad properties of these spaces with respect to the intersection, stating that hyperconvex subsets of a hyperconvex spaces do not define a closed (under the intersection of sets) class of sets. This is in part corrected by a very celebrated result due to Baillon in [3] on intersection of hyperconvex sets that allows to define the concept of "hyperconvex hull" as that one of "injective hull" given by Isbell in [24]. This section is finished with the definition of the noncompactness measures of Hausdorff and Kuratowski as well as their properties in hyperconvex spaces.
We abandon generalities on hyperconvexity in Section 6 to study some more recent and particular properties that will lead us to deep fixed point results. The existence of fixed points for nonexpansive mappings is studied as well as similar results on family of commuting mappings. This section is closed by a non-elsewhere published result on asymptotically nonexpansive mappings. Section 7 deals with another kind of fixed point theorems, those in which compactness conditions are considered. We study how different results stated in linear spaces are still true in a hyperconvex setting. Section 8 is completely devoted to the study of the "injective hull" of Isbell and extremal functions, this concept is one of the most interesting and intriguing ones in hyperconvex metric spaces and well deserves a whole section for its better understanding. Another very important characteristic of hyperconvex spaces is studied in Section 9, in this section we focus on multivalued mappings and state a surprising result on selection of multivalued mappings that implies different results on fixed point theory and existence of nonexpansive selections of the metric projection. The study on multivalued mappings will be completed in Section 10 where the KKM principle is adapted for hyperconvex spaces and new versions of classical results on fixed point theory are obtained. The concept of "lambda hyperconvexity" is studied in the last section of the chapter, Section 11. This is a recently introduced [34] idea which may be understood as an extension of the geometrical definition of hyperconvexity inspired in former studies due to Griinbaum [20].
Finally we want to point out that the material we present here goes from classical to very recent facts that will lead the reader to an updated knowledge about fixed point results on hyperconvex spaces. The reader will find, however, a large collection of items at the end of the chapter from which it is possible to continue the study of facts and related subjects that were not treated in detail in this chapter.
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2. Introduction and basic definitions
There is no doubt that the Hahn-Banach theorem has played a major role in functional analysis. In fact, it is impossible to develop the theory of Banach spaces without this theorem. So it was clear from the beginning that an extension of this theorem to metric spaces would be desirable. The first to study this question were Aronszajn and Panitchpakdi in [1]. Their investigation led to the discovery of hyperconvex metric spaces. In order to appreciate their findings, one needs to recall how the Hahn-Banach theorem is proved.
Theorem 2.1 Let X be a real vector space, Y be a linear subspace of X, and p a seminorm on X. Let f be a linear functional defined on Y such that f (y) ::=; p(y), for' all y E Y. Then there exists a linear functional 9 defined on X, which is an extension of f (i.e. g(y) = f(y), for all y E Y), which satisfies g(x) ::=; p(x), for all x EX.
Proof. The main argument behind the proof of this theorem is the structure of the real line lR. Indeed, via the Hausdorff maximality principle, it is enough to extend f to Y + lR· Xo, where Xo E X \ Y. So we need to find g(xo) such that
{ g(y + ClXo) = g(y) + Clg(XO) = f(y) + Clg(XO)
g(y + ClXo) ::=; p(y + ClXo)
for any real number Cl E lR and any y E Y. Since p is a semi norm and f is a linear functional, we may assume Cl = ±l. This means that what we need to find is a number A (which eventually will be equal to g(xo)), such that f(y) ± A ::=; p(y ± xo), which translates to f(y) - p(y - xo) ::=; A ::=; p(y* + xo) - f(y*) for any y, y* E Y. In other words, we must have
n [f(y)-p(y-xo),p(y*+xo)-f(Y*)] #0. y,y*EY
Define the interval Iy,y' = [f(y) - p(y - xo), p(y* + xo) - f(y*)], for y, y* E Y. Then using the linearity of f and the seminorm behavior of p, it is easy to check that for any Yl, Y2, yi, and Y2 in Y, Iy"yi n Iy"yi # 0. The proof will be complete if we use the following well-known fundamental property of the real line lR:
"If {Ia}aEr is a collection of intervals such that Ia n 1(3 # 0, for any Cl, f3 E r, then we have naErIa # 0". •
It is this property that is at the heart of the new concept discovered by Aronszajn and Panitchpakdi. Note that an interval may also be seen on the real line as a closed ball. Indeed, the interval [a, b] is also the closed ball centered at (a + b)/2 with radius r = (b - a)/2, i.e.
[ b] = B (a + b b - a) a, 2 ' 2
So the above intersection property may also be seen as a ball intersection property. This is quite interesting since in metric spaces it is natural to talk about balls. But keep in mind that in ordered sets for example, intervals are more natural than balls.
Throughout this chapter, the balls referred to are closed. Therefore we will routinely omit the word closed.
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Remark 2.2 Let M be a metric space. Using the triangle inequality, we have
for any Xl, X2 E M and positive numbers r}, r2. The converse is true on the real line and corresponds to Menger convexity in metric spaces.
Definition 2.3 Let M be a metric space. We say that M is metrically convex if for any points X, y E M and positive numbers a and /3 such that d(x, y) ::; a + /3, there exists z E M such that d(x, z) ::; a and d(z, y) ::; /3, or equivalently z E B(x, a) n B(y, /3).
Therefore, Mis metrically convex ifB(x,a)nB(y,/3) f= 0 if and only if d(x,y)::; a+/3 for any points X, y E M and positive numbers a and /3.
Remark 2.4 Note that some authors define metric convexity slightly differently. Indeed, (M, d) is metrically convex if and only if for any points X, y E M and any number a E [0,1], there exists z E M such that d(x, z) = ad(x, y) and dey, z) = (1- a)d(x, y). This definition is hard to extend to general structures since it uses the multiplication operation, which is, for example, hard to define in discrete sets.
The above discussion shows that the Hahn-Banach extension theorem is closely related to an intersection property of closed balls combined with some kind of metric convexity. The concept of hyperconvexity introduced by Aronszajn and Panitchpakdi captured these ideas.
Definition 2.5 The metric space M is said to be hyperconve:rif
for any collection of points {x",}",er in M and positive numbers {r",}",er such that d(x"" x(3) ::; r", + rf3 for any a and /3 in r.
3. Some basic properties of Hyperconvex spaces
Clearly from the previous section, the real line lR is hyperconvex. In fact, we can easily prove that the infinite dimensional Banach space loo is hyperconvex. One way to see this is to use the following result.
Theorem 3.1 Let (Mo, d",)",er be a collection of hyperconvex metric spaces. Consider the product space M = Il",er Ma. Fix a = (ao ) EM and consider the subset M of M defined by
Then (M, doo ) is a hyperconvex metric space where doo is defined by
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This theorem is a structural result. Its proof is an easy consequence of the fact that for any ball B((xa),r) in M, we have B((xa),r) = ITaEr Ba(xa, r) where Ba(xa,r) is the ball centered at x'" with radius r in Ma.
We will see later on that any metric space may be embedded in a "small" hyperconvex metric space. That fact, discovered by Isbell, is not immediate and somewhat hard to grasp. But, for many cases, it is worth knowing that any metric space may be embedded isometrically into a hyperconvex metric space. This is easy to see. Indeed, let (M, d) be a metric space. Set
loo(M) = {(Xm)mEM; Xm E lR. for all m and sup IXml < oo}. mEM
On loo(M) define the distance doo by doo((xm) , (Ym)) = sUPmEM IXm - Yml. The metric space (loo(M), doo) is hyperconvex. To see that M embeds isometrically into loo(M), fix a EM and consider the map I : M -->loo(M) defined by I(b) = (d(b,m)-d(a,m))mEM for b E M. It is easy to check that doo(I(b), I(c)) = d(b, c) for any b, c EM.
Next we discuss completeness of hyperconvex metric spaces. In fact, a weaker version of the binary-ball intersection property is needed to insure the completeness of the metric space. Indeed, we will say that the metric space M has the ball intersection property (BIP for short) if n",ErB", fo 0 for any collection of balls (B"')"'Er such that n",ErfBa fo 0, for any finite subset rf cr.
Proposition 3.2 Any metric space M which has the ball intersection property is complete. In particular any hyperconvex metric space is complete.
Proof. Let (xn) be a Cauchy sequence in M. For any n 2': 1, set
rn = sup d(xn' xm). m2:n
Consider the collection of balls (B(xn, rn))n>l. Since for m 2': n we have d(xn, Xm) ::; rn, then -
Xnk E B(Xnll rn1 ) n B(Xn2' rn2 ) n ... n B(Xnk' rnk )
for any nl < n2 < ... < nk, but M has the ball intersection property so we may conclude that nn:2:1B(xn, rn) fo 0. Now, since (Xn) is a Cauchy sequence, limn~oo rn = a and so the intersection nn:2:1B(xn, rn) is reduced to one point z which is the limit of the sequence (xn). •
We have just seen a property related to intersection of balls rather than hyperconvexity. Ball intersection properties have been extensively studied in connection with different problems concerning geometrical properties of Banach spaces or the extension of mappings. Interested readers may find more on this interesting subject in [6, 23, 43, 45].
At this point we introduce some notation which will be used throughout the remainder of this chapter. For a subset A of a metric space M, set:
rx(A)
r(A)
R(A) diam(A)
G(A) GA(A) cov(A)
sup{d(x,y): YEA}, x E M;
inf{rx(A) : x EM};
inf{rx(A) : x E A};
sup{d(x,y): X,y E A};
{x EM: rx(A) = r(A)};
{x E A : rx(A) = R(A)}; n{B: B is a ball and B :..2 A}.
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rCA) is called the mdius of A (relative to M), diam(A) is called the diameter of A, R(A) is called the Chebyshev mdius of A, C(A) is called the center of A (in M), CA(A) is called the Chebyshev center of A, and cov(A) is called the cover of A.
We now prove a technical lemma.
Lemma 3.3 Suppose A is a bounded subset of a hyperconvex metric space M. Then:
1 cov(A) = n{B(x,rx(A» : x EM}.
2 rx(cov(A» = rx(A), for any x E M.
3 r(cov(A» = rCA).
4 rCA) = ~ diam(A).
5 diam(cov(A» = diam(A).
6 If A = cov(A), then rCA) = R(A). In particular we have R(A) = ~ diam(A).
Proof. 1. Since B(x,rx(A» contains A for each x EMit must be the case that
cov(A) ~ n {B(x, rx(A» : x EM}.
On the other hand, if A ~ B(x, r) then rx(A) ~ r so B(x, rx(A» ~ B(x, r). Hence
n{B(z,rz(A»: z E M} ~ B(x,r).
This clearly implies
cov(A) = n {B(x, rx(A» : x EM}.
2. By 1, rx(cov(A» = sup{d(x,y) : y E nzEMB(z,rz(A»} so, in particular, y E cov(A) implies y E B(x,rx(A» for any x E M. Hence d(x,y) ~ rx(A), which proves rx(cov(A» ~ rx(A). The reverse inequality is obvious since A ~ cov(A).
3. This is immediate from the definition of r.
4. Let {) = diam(A) and consider the family {B(a,~) : a E A}. If a, b E A then d( a, b) ~ {) = {j /2 + {j /2 so by hyperconvexity
If x is any point in this intersection then d(x,a) ~ ~ so rx(A) ~ ~. On the other hand dCa, b) ~ dCa, z) + d(z, b) for any a, b E A and z E M so {j ~ 2rz(A) from which {j ~ 2r(A). Therefore {j ~ 2r(A) ~ 2rx(A) ~ {j proving rCA) = ~.
5. Using 3 and 4, diam(A) = 2r(A) = 2r(cov(A» = diam(cov(A».
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6. Note that we always have !diam(A) ~ r(A) ~ R(A). We may write A = niElBi where Bi is a closed ball for any i E I. Now, since n aEAB(a,8/2) =I 0 where 8 = diam(A), it is easy to check that any two balls drawn from the collection
{Bi, i E I} U { B (a, D ; a E A}
have nonempty intersection. Recalling now the hyperconvexity of M,
C = An { B (a, D ; a E A} = n {Bij i E I} n { B (a, D ; a E A} =I 0.
Let x E C. Then rx(A) ~ 8/2 and therefore 8/2 ~ r(A) ~ R(A) ~ rx(A) ~ 8/2, which clearly implies r(A) = R(A) = ! diam(A). •
Definition 3.4 Let M be a metric space. By A(M) we denote the collection of all subsets of M which are intersection of balls, i.e. A(M) = {A c Mj A = cov(A)}. The elements of A(M) are called admissible subset of M.
It is clear that A(M) contains all the closed balls of M and is stable by intersection, i.e. the intersection of any collection of elements from A(M) is also in A(M). From the above results, for any A E A(M), we have
C(A) = n B(a,R(A)) nA E A(M). aEA
Moreover, diam(C(A)) ~ diam(A)/2. So we have A = C(A) if and only if A E A(M) and diam(A) = 0, i.e. A is reduced to one point.
This property and the ones studied above are extremely important when we discuss the fixed point property in hyperconvex metric spaces. In fact, we will show in this chapter that admissible subsets in hyperconvex metric spaces enjoy some nice properties. But admissible subsets will not be the only class of subsets that are of interest to us, let us introduce three more classes of subsets that will be of great importance in our exposition.
Definition 3.5 A subset E of a metric space M is said to be externally hyperconvex (relative to M) if given any family {xa} of points in M and any family {ra} of real numbers satisfying d(xa,x(3) ~ ra + r(3 and dist(xa,E) ~ ro" where dist(x,E) = inf{ d(x, y): y E E}, then it follows that naB(xa, raj n E =I 0. The class of all the externally hyperconvex subsets of M will be denoted by £(M).
Definition 3.6 A subset E of a metric space M is said to be weakly externally hyperconvex (relative to M) if E is externally hyperconvex relative to E U {z} for each z E M. More precisely, given any family {xa} of points in M all but at most one of which lies in E, and any family {ra} of real numbers satisfying d(xa,x(3) ~ r", + r(3, with dist(x"" E) ~ r", if Xu 1: E, it follows that naB(xaj raj n E =10. The class of all the weakly externally hyperconvex subsets of M will be denoted by W(M).
Additionally, we will denote by 1i(M) the class of hyperconvex subsets of a metric space M.
We finish this section by studying the relation among these classes of sets. In order to do this we need to introduce the concept of proximinality.
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Definition 3.7 A subset E of a metric space M is said to be proximinal (with respect to M) if the intersection En B(x, dist(x, E)) is nonempty for each x E M.
Lemma 3.8 If E is either an admissible, externally hyperconvex or weakly externally hyperconvex subset of a hyperconvex metric space M, then E is proximinal in M.
Proof. We write the proof for the case E = A an admissible subset. Other cases are similar. Set A = niEI Bi. Then for any c > 0, there exists ao E A such that d(x, ao ) :<:::: dist(x, A) + c. Clearly this implies niEIBi n B(x, dist(x, A) + c) #- 0. Since M is hyperconvex, then we must have
A nB(x,dist(x,A)) = nBin (n B(x,dist(x,A) +c)) #- 0 iEI 0>0
which completes the proof that A is proximinal. • Remark 3.9 Notice that the assumption that M is hyperconvex is only necessary in the case when A is an admissible set. The same result for externally or weak externally hyperconvex subsets does not require the hyperconvexity of M. This is essentially due to the fact that for these other subsets the necessary hyperconvexity is implicit in the definition of these sets.
Theorem 3.10 Let M be a hyperconvex metric space, then
A(M) ~ E(M) ~ W(M) ~ H(M).
Proof. A(M) ~ E(M): Let A be an admissible subset of M and let {Xa}aEr be a family of points in M and {ra}aEr be family of real numbers satisfying d(xa, x(3) :<::::
ra + rf3 and dist(xa,A) :<:::: ra for any ex,/3 E r. Since A is proximinal, for any ex E r, there exists aa E A such that d(xa, aa) = dist(xa , A), which gives An B(xa, raj #- 0. Since M is hyperconvex, the conditions on both families imply naErB(xa, raj #- 0. Since A is admissible and An B(xa, ra) #- 0 it follows that
An ( n B(xa,ra)) #- 0 aEr
which proves the first inclusion.
e(M) ~ W(M): This follows directly from definition.
W(M) ~ H(M): This follows directly from definition. • Remark 3.11 Notice that if in the definition of weak external hyperconvexity we impose the condition that one of the balls must be out of E, then weakly externally hyperconvex subsets do not have to be hyperconvex. Think of ]R2 minus the interior of the unit square with the supremum norm and take E as the border of the unit square. E would be weakly externally hyperconvex under this new definition but not hyperconvex.
Remark 3.12 The families A(M), e(M), W(M), and H(M) do not coincide in general. Let M be the right half real plane endowed with the maximum metric, i.e.
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M = {(x,y) E]R2: x 2 O} with the maximum metric. Let E = B((O,-1),2) n M, W = {(x,x): 0::; X ::; 1}, and H = {(x,~): 0::; X ::; 1} then E is externally hyperconvex in M but not admissible, W is weakly externally hyperconvex in M but not externally hyperconvex, and, finally, H is hyperconvex but not weakly externally hyperconvex.
4. Hyperconvexity, Injectivity and Retraction
In this section, we will discuss Aronszajn and Panitchpakdi ideas on how hyperconvexity captures a Hahn-Banach extension theorem in metric spaces. Before we state Aronszajn and Panitchpakdi's main result, we recall the definition of nonexpansive mappings.
Definition 4.1 Let (MI,d!) and (M2,d2) be metric spaces. A map T: MI -+ M2 is said to be Lipschitzian if there exists a constant k 2 0 such that
for any x,y E MI. If k = 1, the map is called nonexpansive, and a strict (or Banach) contraction if k < 1.
A metric space M is said to be injective if it has the following extension property: Whenever Y is a subspace of X and f : Y -+ M is nonexpansive, then f has a nonexpansive extension j : X -> M.
Theorem 4.2 Let H be a metric space. The following statements are equivalent:
(i) H is hyperconvex;
(ii) H is injective.
Proof. First assume H is hyperconvex. Let D be a metric space and T : D -> H be nonexpansive. Let M be a metric space containing D metrically. Consider the following set
c = {(Tp,F); Tp: F -> H with D c F c M metrically}
where Tp is a nonexpansive extension of T. We have (T, D) E C. Therefore, C is not empty. On the other hand, one can partially order C by defining (Tp, F) -< (Ta, G) if and only if Fe G and the restriction of Ta to F is Tp.
It is easy to see that C satisfies the hypothesis of Zorn's lemma. Therefore, C has maximal elements. Let (TI, FI) be one maximal element of C. Let us show that PI = M. Assume not. Let z E M\FI and set F = FIU{Z}. Let us extend TI to F. The problem is to find a point Zl, which will play the role of the value of the extension at z. Since we need the extension to be nonexpansive, we must have d(TI(X),ZI)::; d(x,z) for all x E Fl. Consider the family of closed balls {B(TI(x),d(x,z))}, with x E Fl. Since d(TI(x),TI(y)) ::; d(x,y) ::; d(x,z) +d(z,y) for all x,y E FI, the hyperconvexity of H implies nxEP,B(TI(x),d(x,z)) =10. Let Zl be any point in this intersection and define T*: F -+ H by
T*(x) = {TI(X) ~f x =I z Zl If x = z.
It is easy to check that (T*, F) belongs to C, hence (TI, FI) -< (T*, F) and (Tl, FI) =I (T*, F). This contradicts the maximality of (TI' FI). Therefore, FI = M. In other words, T has a nonexpansive extension to M.
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Conversely, assume that H is injective, i.e. for every metric space D and every nonexpansive map T : D -> H, there exists a nonexpansive extension T* : M -> H of T, where M is any metric space which contains D metrically. Let us prove that H is hyperconvex. Aronszajn and Panitchpakdi's original proof is divided into two parts. First, they showed that H is metrically convex. Then they showed that it is hyperconvex. Here, we give a proof based on ideas ofIsbell's. Given {x",}",Er in H and positive numbers {r",}aer such that d(xa, xf3) ::; r", + rf3 for any a and /3 in r we want to show that naerB(xa, ra) 1= 0. Without any loss of generality, we may assume that Xa 1= xfJ whenever a 1= /3. Consider the set :F of positive real valued functions f defined on the set D = {xa; a E r} such that d(xa, xf3) ::; f(xa) + f(xfJ) and f(xa) ::; ra for all a, /3 E r. Note that the function r : D -> lR defined by r(xa) = ra belongs to this set. :F is partially ordered by the pointwise order on the real line. Obviously, any descending chain of elements of:F has a lower bound. Hence, Zorn's lemma implies the existence of a minimal element f E :F smaller than r, i.e. f(xa) ::; r(xa), for any a E r. Now, using the minimality of f, we can prove that f(xa) ::; d(xa, xfJ) + f(xf3) for any a and /3 in r. Indeed, assume this is not the case. Then there exists ao and /30 such that d(xao ,x.80) + f(x.80) < f(xao )' Set
F(x..,) = { f(x..,) if'Y 1= ao d(xao ,x.80) + f(x.80) if[ = ao·
F satisfies F ::; f and F 1= f, contradicting the minimality of f. Let w be a point not in the set H. Consider the set D* = D U {w}. The distance between the elements of D is the one inherited from H. For the new point, set d(w,xa) = f(xo,}. It is easy to check that D* is a metric space which contains D metrically. Our assumption assures us of the existence of a nonexpansive extension R : D* -> H of the identity map from D into H. It is clear that d(xa, R(w)) = d(R(xa), R(w)) ::; d(xa,w) = f(xa) ::; ra for any a E r, and so R(w) E B(xa, ra) Va E r. Hence
R(w) E n B(xa,ra)n H, aer
which completes the proof. • Remark 4.3 Clearly this answers the Hahn-Banach extension problem for nonexpansive mappings. In fact, Aronszajn and Panitchpakdi obtained a more general result involving the modulus of uniform continuity. The main ingredients behind the proof are still the same. Note that in the second part of the proof, we extended an identity map. The extension mapping is in fact a retract. Recall that a map R : M -> N is said to be a retraction if N c M and R restricted to N is the identity, i.e. R( n) = n, for any n EN. It is easy to see that this is equivalent to requiring that R is onto with R 0 R = R. The set N is therefore called a retract of M.
A similar result to Aronszajn and Panitchpakdi's main theorem may be stated in terms of retractions as follows.
Theorem 4.4 Let H be a metric space. The following statements are equivalent:
(i) H is hyperconvex;
(ii) for every metric space M which contains H metrically, there exists a nonexpansive retraction R : M -> H:
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(iii) for any point w not in H, there exists a nonexpansive retraction R : HU{w} -+ H.
The proof is similar to the one given above. The last statement needs some additional explanation. Indeed, the idea here is to consider the hyperconvex metric space loo(H) (in which H embeds isometrically). In order to show that an intersection of balls in H is not empty, embed the balls in loo(H). Then use the hyperconvexity of 100 (H) to show that the intersection is not empty. Take a point in the intersection and the retraction sends this point into the intersection of the balls in H.
Remark 4.5 Note that statement (ii) is also known as an absolute retract property. This is why hyperconvex metric spaces are sometimes called absolute nonexpansive retracts (ANR for short). The equivalence of (ii) and (iii) is somewhat surprising, in that, the global extension property captured in (ii) follows for the local version given in (iii).
Using the above remark, the second author [32] introduced the concept of I-local retract of a metric space.
Definition 4.6 Let M be a metric space. A subset N is called a i-local retract of M if for any point x E M \ N, there exists a nonexpansive retraction R : N U {x} -+ N.
If we require N to be a I-local retract of all metric spaces M we obtain the absolute i-local retract property. Note that absolute I-local retracts are absolute nonexpansive retracts, i.e. hyperconvex.
Remark 4.7 In the above definition, the number 1 stands for the nonexpansive behavior of the retract. In other words, one may define a A-local retract as well by assuming that the retraction is Lipschitzian with Lipschitz constant equal to A.
Notice that statement (ii), in the above theorem, has a nice extension. Indeed, we have the following result:
Corollary 4.8 Let H be a metric space. The following statements are equivalent:
(i) H is hyperconvex;
(ii) there exist a hyperconvex metric space H* which contains H metrically and a non expansive retraction R : H* -+ H;
In other words, a nonexpansive retract of a hyperconvex metric space is also hyperconvex. The proof is elementary.
Remark 4.9 This result was used by the second author in [31] to prove that hyperconvex metric spaces enjoy a better convexity property than the one described previously. Indeed, let H be a hyperconvex metric space. We have seen that H embeds isometrically into loo(H). Corollary 4.8 implies the existence of a nonexpansive retraction R : loo(H) -+ H. Let Xl, ... , Xn be in H and let positive numbers al, ... , an be such that L::~~ ai = 1. In loo(H), consider the convex linear combination L::~~ aiXi. Set
(i-n ) EB aiXi = R L aiXi
l~iSn i=l
EH.
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Then, for any wE H, we have
which implies
d ( ffi QiXi'W) l::;i::;n
i=n
::; LQid(Xi,W). i=l
We will rewrite the above inequality as
d ( ffi QiXi,') l:S;i:S;n
i=n
::; LQid(Xi' .). i=l
In fact we have a more general formula
One may argue that the choice of our convex combination in H depends on the retraction R and the choice of the isometric embedding. This is true, and depending on the problem, one may have to be careful about this choice.
Sine [55] began a more detailed study of the retraction property in hyperconvex spaces than we have stated so far. His results are crucial in investigating nonexpansive mappings defined on hyperconvex metric spaces. We will begin the final part of this section by focusing on admissible subsets, for which Sine proved the following relevant fact. For any positive real number r and any set A in a metric space M, we define the r-parallel set of A as
A+r=U{B(a,r); aEA}.
We have the following result.
Lemma 4.10 Let H be a hyperconvex metric space. Let J be an admissible subset of H. Set J = naErB(xa, ra). Then for any r :2: 0, we have
J+r= n B(xa,ra+ r ). aEr
In other words, the r-parallel sets of an admissible subset of a hyperconvex metric space are also admissible sets (this is not a common property of metric spaces).
Proof. Let y E J + r. Then there exists a E J such that d(y, a) ::; r. Hence d(xa, y) ::; d(xa, a) + d(a, xa) ::; ra + r for all Q E r. Thus
J +r C n B(xa,ra +r). aEr
For the reverse inclusion, let y E naaB(xa, ra +r). So we have d(xa, y) ::; ra+r for all Q E r. Now, since H is hyperconvex, naErB(xa,ra) nB(y,r) =10 which is equivalent to J n B(y, r) =10. This clearly implies that y E J + r. •
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Remark 4.11 For any positive real number r and any set A in a metric space M, we define the r-neighborhood
Nr(A) = {x E M;dist(x,A):::; r}.
If A is proximinal, i.e. for every x E M there exists a E A such that dist(x, A) = d(x, a), then the r-parallel set A + r and the r-neighborhood Nr(A) are identical. This is the case for example if A is externally hyperconvex in a hyperconvex metric space M.
Before we state Sine's result recall that a map T defined on a metric space M is said to be c:-constant if d(x, Tx) :::; c: for all x E M. We have the following result:
Theorem 4.12 Let H be a hyperconvex metric space. Let J be a nonempty admissible subset of H. Then for any c: > 0 there exists an c:-constant nonexpansive retraction of the parallel set Je = J + c: onto J.
Proof. We have observed above that Je is a nonempty ball intersection and so is itself a hyperconvex set. Let us construct the retract. First let D be any set such that JeD c Je and assume that ITD : D ~ J is a c:-constant nonexpansive retract. Let x E Je - D. Set D* = D U {x}. Define the target set
targ(x) = n B(ITD(w),d(w,x)) nJnB(x,c:). wED
We claim that targ(x) is not empty. Indeed, set J = nB(x",roJ Then targ(x) is an intersection of balls. Since H is hyperconvex, then to prove that targ(x) is not empty, it is enough to show that the balls intersect each other. Since x E Je, then B(x,c:) intersects any ball B(xa,ra) (coming from J). On the other hand, we have d(ITD(W),X):::; d(ITD(W),W) + d(w, x) :::; c:+d(x,w) because ITD is an c:-constant map. This clearly implies B(x,c:) nB(ITD(W),d(w,x)) # 0, for any wED. Since ITD maps into J, then B(xa,ra) nB(ITD(W),d(w,x)) # 0, for any a. Therefore targ(x) is not empty. Let z E targ(x) and define IT' : D* ~ J by
IT*(v) = {IID(V) ~f v E D z lfv = x.
It is easy to check that 11* is a c:-constant nonexpansive extension of lID. The rest of the proof is a simple induction. Start with D = J and II D to be the identity map on D. Zorn's lemma assures us of the existence of a maximal element (D, ITD). The argument described above will force the set D to be Je , which completes the proof. •
Remark 4.13 Now one may ask whether the previous result is also true for weakly externally hyperconvex or externally hyperconvex subsets. The answer is yes. In fact the same proof we used for admissible subsets still works for externally hyperconvex subsets without major modifications. But something more precise may be said about this problem since weak external hyperconvexity actually characterizes those sets for which the conclusion of the previous theorem holds. We will not prove this result. A proof may be found in [15], however the following theorem is a partial result in the same direction, in fact it provides a converse to Theorem 4.12.
Theorem 4.14 If E is a subset of a hyperconvex metric space H with the property that for each z E H \ E there exists a nonexpansive retraction R : E U { z} ~ E with d(z, R(z)) = dist(z, E), then E is weakly externally hyperconvex.
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Proof. First notice is that, from Theorem 4.4, such an E must be a hyperconvex subset of H, i.e. E is hyperconvex itself. We are going to assume that E is not weakly externally hyperconvex in order to find z E H such that
B (z, dist (z, E» n (nvEDB (v, d (v, z») n E = 0.
This will clearly contradict the hypothesis of the theorem. We state (claim 1) that if E is not weakly externally hyperconvex then there exist z E H, a family {v", : Q E A} in E, and a family {r",} in jR+ for which d (v""vf3) ~ r", + rf3, d (va, z) ~ r", + dist (z, E), and such that
B (z,dist(z,E» n (n",B (va,r",» nE = 0.
However, since M is hyperconvex it must be the case that
If B(z,dist(z,E» n (nVEEB(v,d(v,z») nE = 0 we are done. Otherwise we proceed as follows. Since DI n E = 0 there is no loss of generality (claim 2) in assuming that,
dist(B (z,dist(z,E» nE,D1 ) = d > O.
It is possible to choose WI E DI and wEB (z,dist (z,E» nE so that d (WI,W) = d + c for sufficiently small c > O. Now, by hyperconvexity of M,
( . d+c)n ( d+c)n ( d+c) B z, dlst (z, E) - -2- B WI, -2- B w, -2- =J 0.
Let Zl be any point in this intersection. Notice that if d (zJ, p) < 4¥ for any pEE, then
d(z,p) ~ d(z,zJ) +d(zJ,p)
d+c d+c < dist(z E) - --+--, 2 2
= dist (z,E)
which is a contradiction. Since dist (zJ, E) ~ d (Zl, w) ~ (d + c)/2, we conclude dist (zJ, E) = (d + c)/2.
By the assumptions of the theorem, the set
would have to be nonempty. However, since d (zJ, v"') ~ ra + (d + c)/2, for each a, we have
B (zJ, dist (Zl, E» n (naB (v""d (v"" Zl»)
~ B (z,dist(z,E» n (n",B (va,ra + d;c)),
and by Lemma 4.10,
B (z,dist(z,E»n (QB (va,ra + d;c) ) ~ N~ (D1).
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Clearly this neighborhood of D1 cannot intersect En B(z, dist(z, E)) for 10 > 0 sufficiently small, so we conclude
To complete the proof it remains only to prove the two claims.
Proof of Claim 1. Notice that what we get directly from the negation of the definition of weak externally hyperconvexity is that there exist z E M, a family {va : a E A} in E, and a family {ra} U {rz} in lR such that r z ::::: dist(z,E), d(va,v{3) <::: Ta + r{3, d (va, z) <::: ra + rz, and for which
B (z, dist (z, E)) n (naB (va, raj) n E = 0, (4.1)
something slightly different to what is required in our claim. In order to prove the claim, set T~ = Ta + Tz - dist(z, E), and r~ = dist(z, E) then we state that the intersection
B (z, dist (z, E)) n (naB (va, r~)) n E = 0.
Otherwise fix u in this intersection and consider naB (va, raj n B(u, rz - dist(z, E)). From the hyperconvexity of E, this intersection would also be nonempty but, since this set is contained in the set intersection of (4.1), this would be a contradiction.
Proof of Claim 2. To prove the second claim, assume d = 0 and let {IOn} be a decreasing null sequence of positive numbers. Select
U1 E B(z, dist(z, E)) n E n (naB(va, ra + 101)),
and consider B(z, dist(z, E))nEnB(ul, E:! +102) n (naB (va, Ta+C:2))' If this intersection is empty add B(U1,C:1) to the family {B(Va,Ta)} to obtain a new family for which d is positive. Otherwise, select U2 in this in this intersection and consider
and repeat the previous step. Either this process terminates after a finite number of steps, providing a new family for which d > 0, or we obtain a Cauchy sequence {un} whose limit lies in D1 n E, which is a contradiction. •
We finish this section with the announced characterization result which follows from Theorems 4.14 and 9.12.
Theorem 4.15 Let H be a hyperconvex metric space. Then E cHis weakly externally hyperconvex if and only if for any 10 > 0 there exists an c:-constant nonexpansive retract of the parallel set E + 10 onto E.
What else do we know about hyperconvex sets? Perhaps one of the most elegant results in hyperconvex spaces is Baillon's theorem. We will discuss this and other results in the next section.
5. More on Hyperconvex spaces
Hyperconvexity, as we mentioned before, is an intersection property. In other words, if a metric space is hyperconvex then the family of admissible sets has some kind of
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compactness behavior. Indeed, let M be a metric space and consider the family of admissible sets A(M). Following Penot's definition, we will say that A(M) is compact if for any family {A",} of elements in A(M), we have
provided finite intersections are not empty. Hyperconvexity obviously implies that A(M) is compact. But this kind of compactness is not equivalent to hyperconvexity in general. Indeed, hyperconvexity is about balls intersecting each others. For example, Lindenstrauss [45] showed that balls of It have a very nice and similar intersection property though 11 is not hyperconvex. Indeed, he showed that any collection of balls has a nonempty intersection provided the balls intersect three by three. In fact, in any reflexive Banach space or dual space X, the family A(X) is compact under Penot's formulation. Recall that loo(1) is basically the only hyperconvex Banach space. The reader will find more about the relation between hyperconvexity and Banach spaces in [19, 29, 42, 45, 46].
Early investigators of hyperconvexity wondered whether the compactness of the family of admissible sets holds also for the family of hyperconvex sets. Notice that this family is not stable under intersection. Indeed, the intersection of two hyperconvex sets may not be hyperconvex, even in the plane. So it was asked whether any descending chain of nonempty hyperconvex sets has a nonempty intersection. This question was answered by Baillon [3] using a highly technical proof. A simple proof of this result is not known to us.
Theorem 5.1 Let M be a bounded metric space and r a totally ordered index set. Let (H{3){3Er be a decreasing family of nonempty hyperconvex subsets of M, then n{3Er H{3 is not empty and is hyper-convex.
Proof. Consider the family
F = { A = n A{3 ; A{3 E A(H{3) and (A{3) is decreasing and nonempty }. (3Er
Since M is bounded, then each H{3 is bounded and therefore we must have H{3 E A(H{3). So F is not empty since I1{3H H{3 E F. Since H{3 is hyperconvex, A(H{3) is compact for every (3 E r. Therefore, F satisfies the assumptions of Zorn's lemma when ordered by set inclusion. Hence for every D E F there exists a minimal element A E F such that A c D. We claim that if A = I1{3H A{3 is minimal then there exists f30 E r such that diam(A{3) = 0 for every {3 :0:: {3o. Indeed, let (3 E r be fixed. For every D c M, set
cov{3(D) = n B(x, rx(D)).
Consider A' = I1"'Er A~ where
A' - { ",-
xEHf3
cov{3(A{3) n A", if a:::; {3,
An if a :0:: {3.
The family (A~~{3) is decreasing since A E F. Let a :::; I :::; {3. Then A~ c A~ since AI' C An and A{3 = cov (3 (A{3) n A{3. Hence the family (A~) is decreasing. On the other
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hand if Q :-s: (3, then cov,8(A,8) n A" E A(H,,) since H,8 C H". Hence A~ E A(H,,). Therefore, we have A' E F. Since A is minimal, A = A' which implies
A" = cov,8(A,8) n A", for every Q :-s: {3 .
Let x E H,8 and Q :-s: (3. Since A,8 C A", then rx(A,8) :-s: rx(A,,). Because cov,8(A,8) = nXEHIlB(x, rx(A,8)), then we have
cov,8(A,8) C B(x,rx(A,8)) ===? rx(cov,8(A,8)) :-s: rx(A,8)'
Additionally A" C cov,8(A,8) so rx(A,8) :-s: rx(Aa) :-s: rx(cov,8(A,8)) :-s: rx(A,8)' Therefore, we have rx(A,,) = rx(A,8) for every x E H,8. Using the definition of r, we get r(Aa) :-s: r(A,8). Let a E Aa and set s = ra(Aa). Then a E cov,8(A,8) since A" C cov,8(A,8). Hence a E nXEAIlB(x, s) n cov,8(A,8). So, from the hyperconvexity of H,8,
8,8 = H,8 n n B(x, s) n cov,8(A,8) of- 0. xEAIl
Let z E 8,8, then z E nXEAIlB(x,s) and, since A,8 = H,8 n cov,8(A,8), it follows that rz(A,8) :-s: s, which implies r(A,8) :-s: s = ra(A,,) for every a E A". Hence r(A,8) :-s: r(A",}. Therefore we have r(A,8) = rCA,,), for every Q, (3 E r. Assume that diam(A,8) > 0 for every (3 E r. Set A~ = C(A,8) for every {3 E r. The family (A~) is decreasing. Indeed, let Q :-s: (3 and x E A~. Then we have rx(A,8) = r(A,8). Since we proved that rz(A,8) = rz(A,,) for every z E H,8, then rx(A,,) = rx(A,8) = r(A,8) = rCA,,), which implies that x E A~. Therefore, we have A" = I1,8Er A~ E F. Since A" C A and A is minimal, we get A = A". Therefore, we have C(A,8) = A,8 for every {3 E r. This contradicts the fact that H,8 is hyperconvex for every {3 E r. Hence there exists (3o E r such that 8(A,8) = 0, for every {3 ::::: {3o. The proof of our claim is therefore complete since we have A,8 = {a} for every {3 ::::: {3o which clearly implies that a E n,8ErH,8 of- 0. In order to complete the proof, we need to show that 8 = n,8ErH,8 is hyperconvex. Let (Bi)iEI be a family of balls centered in 8 such that niEIBi of- 0. Set D,8 = niE1Bi n H,8 for (3 E r. Since H,8 is hyperconvex and the family (Bi) is centered in H,8, then D,8 is not empty and D,8 E A(H,8). Therefore, D,8 is hyperconvex. The above proof shows that n,8ErD,8 of- 0 which completes the proof of Theorem 5.1. •
Remark 5.2 This proof is different from Baillon's original one. It is little more complicated. But it has the advantage of being easy to adapt to I-local retracts. In other words, the conclusion of Baillon's result holds for sets that are I-local retracts.
While the intersection of two admissible subsets of a given hyperconvex space is again admissible, in general it is not the case that the intersection of two hyperconvex subspaces of a hyperconvex space is itself hyperconvex, even if one of them is admissible. However the following is true.
Lemma 5.3 Let H be a hyperconvex metric space. 8uppose E CHis externally hyperconvex relative to H and suppose A is an admissible subset of H. Then E n A is externally hyperconvex relative to H.
Proof. Suppose {x,,} and {r,,} satisfy d(x",x,8):-S: T" +T,8 and dist(x",EnA):-S: Ta· Since A is admissible, A = niEIB(Xi; ri) and since dist(x", E n A) of- 0 it follows
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that d(xa , Xi) ::; ra + ri for each i E I. Also, since A C B(Xi; ri), it follows that dist(xi, E n A) ::; ri and that d(Xi' Xj) ::; ri + rj for each i,j E I. Therefore by external hyperconvexity of E
• This leads to the following.
Theorem 5.4 Let {Hi} be a descending chain of nonempty externally hyperconvex subsets of a bounded hyperconvex space H. Then niHi is non empty and externally hyperconvex in H.
Proof. Theorem 5.1 assures that D = niHi # 0. To see that D is externally hyperconvex let {xa} CHand {raJ C IR satisfy d(xa ,xf3) ::; ra +rf3 and dist(xa , D) ::; ra. Since H is hyperconvex we know that A = n",B(xa; ra) # 0. Also, since dist(xa , D) ::; ra we have dist(xa , Hi) ::; r", for each i, so, by external hyperconvexity of Hi, we conclude A n Hi # 0 for each i. By Lemma 5.3 {A n H;} is a descending chain of nonempty hyperconvex subsets of H, so again by Theorem 5.1 ni(A n Hi) = AnD =10. •
Remark 5.5 Whether Lemma 5.3 holds for weakly externally hyperconvex subsets seems not to be known, however, it is not a very complicated exercise to prove that Theorem 5.4 is true for these subsets. An interesting open question is whether the intersection of two weakly externally hyperconvex subsets is still weakly externally hyperconvex. Such a property is an important one from a structural point of view and it should lead to fixed point results for weakly externally hyperconvex subsets that are already known for admissible subsets.
One of the implications of Theorem 5.1 is the existence of hyperconvex closures. Indeed, let M be a metric space and consider the family
1t(M) = {H;H is hyperconvex and M C H}.
In view of what we said previously, the family 1t(M) is not empty. Using Baillon's result, any descending chain of elements of 1t(M) has a nonempty intersection. Therefore one may use Zorn's lemma to ensure the existence of minimal elements. These minimal hyperconvex sets are called hyperconvex hulls. Isbell was among the first to investigate the properties of hyperconvex hulls. In fact he was the first to give a concrete construction of a hyperconvex hull [24]. We will discuss his ideas in Section 8.
It is clear that hyperconvex hulls are not unique. But they do enjoy some kind of uniqueness. Indeed, we have:
Proposition 5.6 Let M be a metric space. Assume that HI and H2 are two hyperconvex hulls of M. Then HI and H2 are isometric.
Proof. Since HI and H2 are hyperconvex, then they are injective. So there exists a nonexpansive map TI : HI -+ H2 such that the restriction of TI to M is the identity map. Keep in mind that HI as well as H2 contains M isometrically. For the same reason, there exists also another nonexpansive map T2 : H2 -+ HI such that the restriction of T2 to M is the identity map. We claim that TI 0 T2 is the identity map of H2.
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Indeed, the map Tl 0 T2 is defined on H2 into H2. Its restriction to M is the identity map of M. So we have M e Fix(TI 0 T2 ), where
In the next section, we will show that if T is nonexpansive, then Fix(T) is hyperconvex (see Theorem 6.1), so, Since Tl 0 T2 is nonexpansive, Fix(Tl 0 T2) is hyperconvex and contains M. The minimality of H2 implies
which completes the proof of our claim. A similar argument will show that T20Tl is the identity map of HI. So TI and T2 are inverse from each other and are nonexpansive. Therefore both are isometric maps. •
Remark 5.7 Though hyperconvex hulls are not unique, the previous proposition shows that up to an isometry they are indeed unique. This is quite an amazing result. From now on, we will denote a hyperconvex hull of M by h(M). Recall that if M is a subset of a hyperconvex set H, then there exists a hyperconvex hull h(M) such that Me h(M) e H.
Isbell, in his study of hyperconvex hulls, showed that if M is compact then h(M) is also compact. As a generalization of this result, we discuss next some ideas developed by the first author and L6pez in [14,16]. Since their work involves measure of noncom pact ness, let us first give some definitions. For a full treatment of these definitions the reader is referred to [2].
Definition 5.8 Let M be a metric space and let B(M) denote the collection of nonempty, bounded subsets of M. Then:
(i) The Kumtowski measure of noncom pact ness a: B(M) -+ [0,00) is defined by
alA) = inf { c > 0; A e ~Q Ai with Ai E B(M) and diam(A;) ::; c} . (ii) The Hausdorff (or bal0 measure of noncompactness X : B(M) -+ [0,00) is defined
by
X(A) = inf {r > 0; A e :Q B(xi,r) with Xi EM}.
These two measures are closely related to each other and to compactness. Indeed, the following classical properties are well known.
(1) For any A E B(M), we have 0 ::; alA) ::; diam(A).
(2) For any A E B(M), we have alA) = 0 if and only if A is precompact.
(3) For any A E B(M) and B E B(M), we have alA U B) = max{ alA), alB)}.
(4) For any A E B(M), we have X(A) ::; alA) ::; 2X(A).
(5) If (Ai)iEl is a decreasing chain of closed bounded sets such that infiEl alAi) = 0, then niElAi is nonempty and compact.
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In hyperconvex metric spaces, the two measures behave nicely. Indeed, we have:
Proposition 5.9 Let H be a hyperconvex metric space and A be a bounded subset of H, then a(A) = 2X(A).
Proof. From (4) it is enough to prove that 2X(A) ::; a(A). Let c > X(A). Then there exist subsets AI, ... , An of A such that A = Ul:oi:onAi with diam(A) ::; E, for i = 1, ... , n. From the hyperconvexity of H for any i E {I, ... , n}, there exists hi E H such that Ai C B(hi' E/2). Hence
which gives a(A) ::; E/2. Hence a(A) ::; X(A)/2 which completes the proof. •
The following technical result will be needed later on. Note that this result may be seen as an adaptation of the classical Arzela-Ascoli Theorem.
Lemma 5.10 Let M be a metric space. Consider the space '\[a,b] (M) of Lipschitzian real-valued functions defined on M with Lipschitz constant less than ,\ and taking values in the interval [a, bj. Then we have
Proof. Let EO > X(M). Without loss of generality, we may assume that there exist Xl, ... , Xn in M such that for any X E M, there exists i E {I, ... , n} such that d(x, Xi) ::; EO. Since [a, bj is compact, for any E > 0, there exist q, ... , em in [a, bj such that for any c E [a, bj there exists i E {I, ... , m} with Ic - cil ::; E. Let ¢ : {l, ... , n} -> {I, ... , m} be an application. Define
'\,p = {f E ,\[a,b](M); sup If(Xi) - C,p(i) I ::; E}. l:Si:Sn
Though these sets may be eventually empty, we still have that
u ,pE{I, ... ,m}{l . ... n}
Let f,g E '\,p. For any X E M, there exists i E {I, ... ,n} such that d(x,xi) ::; EO. Then we have
If(x) - g(x)1 ::; If(x) - f(Xi)1 + If(Xi) - g(x;ll + Ig(Xi) - g(x)l,
which implies If(x) - g(x)1 ::; '\EO + 2E + '\co. Hence sUPXEM If(x) - g(x)1 ::; 2'\co + 2c. Since the set {1, ... ,m}{I, ... ,n} is finite, we get a(,\[a,b](M)) ::; 2'\co + 2c. So, from the arbitrariness of c, we get a(,\[a,b](M)) ::; 2'\co , which clearly implies a(,\[a,b](M)) ::; 2,\ X(M). •
We will show in Section 8 how Isbell constructed for any bounded metric space M a hyperconvex hull h(M) included in '\[0,6j(M), where ,\ = 1 and 8 is the diameter of M. So from the above lemma, we deduce the following result.
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Corollary 5.11 Let M be any bounded metric space and h(M) a hyperconvex hull of M. Then we have X(h(M)) = X(M) and a(h(M)) = a(M).
Proof. Since h(M) may be chosen such that h(M) C A[O,6](M), where ,\ = 1 and 8 is the diameter of M, it follows a(h(M)) :::; 2X(M). Using the previous lemma, X(h(M)) :::; X(M). But, since h(M) contains M isometrically, X(M) :::; X(h(M)), which clearly implies the first part of the conclusion. The second part is a direct consequence of Proposition 5.9. •
6. Fixed point property and Hyperconvexity
Results of Sine [54] and Soardi [57] mark the beginning of the recent interest in hyperconvex metric spaces. Both Sine and Soardi showed that nonexpansive mappings defined on a bounded hyperconvex metric space have fixed points. Their results were stated in different contexts but the underlying spaces are simply hyperconvex spaces. Here we will give the proof based on Penot's [49] formulation of Kirk's fixed point theorem.
Theorem 6.1 Let H be a bounded hyperconvex metric space. Any non expansive map T : H --> H has a fixed point. Moreover, the fixed point set ofT, Fix(T), is hyperconvex.
Proof. Consider A(H) the family of admissible subsets of H. Set
F = {A E A(H); with A =J 0 and T(A) C A}.
Obviously, we have H E F. Since the intersection of any family of nonempty elements of A(H) is not empty and belongs to A(H) (because of the hyperconvexity of H), then F satisfies Zorn's assumptions. So it has minimal elements. Let Ao be one of them. Note that cov(T(Ao)) = Ao. This follows, since T(Ao) C Ao and cov(T(Ao)) is the smallest admissible set which contains T(Ao), we have cov(T(Ao)) C Ao. Using this, we get
T( COY (T(Ao))) C T(Ao) C COY (T(Ao)),
which clearly implies that cov(T(Ao)) E F. The minimality of Ao then implies
cov(T(Ao)) = Ao.
Now note that C(Ao) belongs to F. Indeed, we know that C(Ao) is not empty and it is in A(H) because
C(Ao) = n B(x,R(Ao)). xEAo
Let x E C(Ao). Then we have Ao C B(x,R(Ao)). Since T is nonexpansive, we get T(Ao) C B(T(x), R(Ao)), which implies
Ao = COy (T(Ao)) C B(T(x), R(Ao)).
Hence T(x) E C(Ao). In other words, C(Ao) is invariant under the action of T. So we have C(Ao) E F. Our claim is therefore proved. The minimality of Ao will then imply Ao = C(Ao). But we have seen that in hyperconvex metric spaces this is not possible for subsets with more than one point. This forces Ao to have one point which is a fixed point for T. In order to finish the proof of our theorem, we need to show that Fix(T) is hyperconvex. Let {Xi}iEI be a collection of points in Fix(T) such that
d(Xi, Xj) :::; ri + rj, for any i,j E I,
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for some positive numbers {rihE1' Set Ho = niElB(xi,ri). The hyperconvexity of H implies that Ho is not empty. Since the centers are in Fix(T) and T is nonexpansive, then we have T(Ho) C Ho. Moreover Ho is a bounded hyperconvex metric space, so the above proof implies that T has a fixed point in Ho, which in turn implies
Fix(T)n [nB(Xi,ri)] i= 0. iE1
This completes the proof of our theorem. • This result is quite amazing. Indeed, if we translate it into the hyperconvex Banach space 100 , we have, for example, that any nonexpansive mapping which leaves a ball invariant has a fixed point. Note that this space is quite bad from a geometrical point of view. It is also a universal space for separable Banach spaces. In other words, any separable Banach space sits inside 100 isometrically. So, for the classical fixed point property this space is very bad. In contrast with this theorem, we have a positive fixed point result.
Note that since Fix(T) is hyperconvex, then any commuting nonexpansive maps Ti, i = 1,2, ... , n, defined on a bounded hyperconvex set H, have a common fixed point. Moreover their common fixed point set Fix(TI) nFix(T2) n··· nFix(Tn) is hyperconvex.
Combining these results with Baillon's theorem, we get the following:
Theorem 6.2 Let H be a bounded hyperconvex metric space. Any commuting family of nonexpansive maps {Ti hEl, with Ti : H -t H, has a common fixed point. Moreover, the common fixed point set niE1 Fix(Ti) is hyperconvex.
Proof. Let r = 21 = {,8j ,8 C I}. It is obvious that r is downward directed (the order on r is set inclusion). Our previous theorem implies that for every ,8 E r, the set F(3 of common fixed points of the mappings Ti , i E ,8, is nonempty and hyperconvex. Clearly the family (F(3)(3Er is decreasing. Using Baillon's result, we deduce that
n Fix(T;) = n F(3 iEl (3Er
is nonempty and hyperconvex. The proof is therefore complete. • Remark 6.3 Baillon asked whether boundedness may be relaxed. More precisely he asked whether the conclusion holds if the nonexpansive map has a bounded orbit. In the classical Kirk's fixed point theorem, having a bounded orbit implies the existence of a fixed point. Prus answered this question in the negative. Indeed, consider the hyperconvex Banach space H = 100 and the map T : H -t H defined by
T((xn)) = (1 + I~Xn,Xl,X2' ... )
where U is a nontrivial ultrafilter on the set of positive integers. We could also use a Banach limit instead of a limit over an ultrafilter. The map T is an isometry and has no fixed point. On the other hand, we have
Tn(o) = (1,1, ... , 1, 0,0, ... )
where the first block of length n has all its entries equal to 1 and then 0 after that. So T has bounded orbits. This problem has been further studied in [35, 36, 40].
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Recently, we wondered whether this result holds for asymptotically nonexpansive mappings. Recall that a map T is said to be asymptotically nonexpansive if
and limn An = 1. The answer to this question is still unknown. But a partial positive answer is known for approximate fixed points. Before we state this result, recall that if T : H -+ H is a map, then x E H is an e-fixed point if d(x,T(x)) ::; e, where e:C:: O. The set of e-fixed points of T is denoted by FixE(T). Sine [55] obtained the following wonderful result:
Theorem 6.4 Let H be a bounded hyperconvex metric space and T : H -+ H a nonexpansive map. For any e > 0, FixE(T) is nonempty and hyperconvex.
Proof. FixE(T) is nonempty since T has fixed point. Let {x"JaEr be points in FixE(T) and {r"J"Er be positive numbers such that d(x", x(3) ::; r", + r{3 for any a, (3 E r. Set
J = n B(xa,r,,). "'Er
We know that J is nonempty as a subset of H. We wish to show that J n FixE(T) is not empty. Let x E J. Then T(x) E J + e because
d(Tx, x",) ::; d(Tx, Tx",) + d(Tx", x a ) ::; d(x, x",) + e ::; r", + e for all a.
Using Theorem 4.12, there exists a nonexpansive retraction II : J + e -+ J which is e-constant. The map R : J -+ J defined by R(x) = II 0 T(x) is nonexpansive. Since J is a bounded hyperconvex metric space, then R has a fixed point Xo E J. Since II is e-constant, we get
d(xo,T(xo)) =d(IIoT(xo),T(xo))::;e
which implies that J n FixE(T) i' 0. The proof is therefore complete.
Now we are ready to state the following previously unpublished result.
•
Theorem 6.5 Let H be a bounded hyperconvex metric space and T : H -+ H be asymptotically non expansive. For anye > 0, FixE(T) is nonempty, in other words we have
inf d(x,T(x)) = O. xEH
Proof. Using the convexity shown in Remark 4.9, we define
where a is a fixed point in H and An is the Lipschitz constant of Tn. The maps Tn are nonexpansive. Before we proceed with the proof we need to define the "ultrapower" of H. Consider the cartesian product H = I1n>l H, and let U be a nontrivial ultrafilter on the natural numbers. Define the equivaJence relation ~ on H by (xn ) ~ (Yn) if and only if limu d(xn, Yn) = O. The limit over U exists since H is bounded. Then we consider the quotient set ii. An element x E ii is a subset of H. If (xn} E x, then (Yn) E X if and only if limu d(xn, Yn) = O. On iidefine the metric d by d(x,fj) = limud(xn,Yn) where (xn ) (resp. (Yn)) is any element in x (resp. fj). It
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is easy to see that if endowed with the distance J has many nice properties similar to the linear ultrapower of a Banach space. Define the operators T and T by
T(x) = T((xn)) = (T-;;Z;::)) and f(x) = f((xn)) = (T(;:J).
Since T is asymptotically nonexpansive, the operator l' is nonexpansive. Moreover we
have f((xn)) = (~)). Since Tn is nonexpansive, it has a fixed point Xn. The point
x = (xn) is a fixed point of f. Hence the fixed point set Fix(T) is a nonempty subset of if. Since the two operators T and l' commute, then l' leaves invariant the set Fix(f). It is easy to show that l' restricted to Fix(T) is in fact an isometry (in particular it is nonexpansive). Fix c > O. Let Xi E Fix(T), i = 1, ... , N. Set
for n ~ 1, where Xi = (Xn(i)). Set
Hn = {x E Hj d(x, Tn(x)) :::; en}.
Hence xn(i) E Hn, for i = 1, ... , N and any n ~ 1. Since Tn is nonexpansive, Theorem 6.4 implies that Hn is hyperconvex. Therefore, there exist
for i = 1, ... , N. Consider, the point
z.; = (Zn(i)) , which we will denote eXI EB (1 - e)Xi.
Then we have, Zi E Fix(T), and d(z.;,zj):::; (l-c)d(Xi,Xj) for i, j = 2, ... ,N. Back to our maps T and 1', let X E Fix(T), and write X = Xl. Set
Then X2 E Fix(T). By induction, we will construct a sequence (xn) of points in Fix(T) defined by Xn+l = eXI EB (1- e)f(xn). For any n < m, we have
Since l' is nonexpansive when restricted to Fix(T), we get
This clearly implies that the sequence (xn) is a Cauchy sequence. Hence it converges to wE Fix(T). Moreover we have
If we set 8 = diam(H), we get dew, few)) :::; eO, so for anye > 0 there exists We E Fix(T) such that d(we, f(we)) :::; e. Classical arguments now imply that for any e > 0 there exists Xe E H such that d(xe, T(xe)) :::; e which completes the proof of Theorem 6.5. •
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7. Topological fixed point theorems and Hyperconvexity
Another important branch of fixed point theory comprises those results in which topological like conditions are considered. We may think of the well-known Schauder theorem as the starting point of this branch. This theorem states that any continuous mapping defined from a nonempty compact and convex subset of a Banach space into itself must have a fixed point. An easy improvement of this theorem is obtained when the compactness condition is imposed on the mapping instead of on its domain. New achievements came when the compactness condition on the mapping was treated in more general terms. Let M be a metric space and let B(M) be the collection of nonempty, and bounded subsets of M, then a mapping "( : B(M) --t [0, +00) is called a measure of noncompactness if it satisfies the following conditions:
(1) "((A) = 0 if and only if A is precompact.
(2) "((A) = "((if) for any A E B(M).
(3) "((A U B) = max{"((A) , "((Bn for any A, BE B(M).
Of course, as it was announced in Section 5, the mappings a and X given by Definition 5.8 are measures of noncompactness. A new kind of mapping arises naturally.
Definition 7.1 Let M be a metric space and D <:;; M. A mapping T : D --t M is said to be a ,,(-condensing (or condensing relative to "() mapping if T is continuous and if for each bounded A <:;; D, for which "((A) > 0, "((T(A)) < "((A).
A detailed studied of these mappings may be found in [2]. It is easy to see, however, that any compact mapping is condensing relative to any measure "(. The well-known Darbo-Sadovskii's theorem [52] states that if"( is a measure of noncompactness defined on a normed space such that "((B) = ,,((co(B)) for any nonempty and bounded subset of the normed space, and T is a ,,(-condensing mapping from a nonempty bounded closed and convex subset of the normed space into itself, then T has a fixed point. The condition "((B) = ,,((co(B)) is fundamental for this result and other related ones for which hyperconvex counterparts will seen below. The measures of Kuratowski and Hausdorff studied in Section 5 are among those that satisfy the above condition. Even more, Corollary 5.11 says that both measures satisfy an equivalent condition for hyperconvex spaces. We will make use of this later in the section. Surprisingly, as it was noted in [40], a hyperconvex version of the Darbo-Sadovskii theorem: does not require the use of this corollary.
Theorem 7.2 Let H be a bounded hyperconvex space and T : H --t H an a-condensing mapping. Then T has a fixed point.
Proof. From Section 3 we may assume that H is a bounded closed subset of a Banach space. We also know that there exists a nonexpansive retraction R : co(H) --t H (where co(H) denotes the closed convex hull of H). Then ToR: co(H) --t H, and if A <:;; co(H) satisfies a(A) > 0 then either a(R(A)) = 0 or
a(T 0 R(A)) < a(R(A)) :S alA).
In either case a(T 0 R(A)) < alA) so ToR is also a-condensing. Sadovskii theorem implies the existence of a fixed point x for ToR. that it must also be a fixed point for T.
Now the DarboIt is easy to see
•
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Remark 7.3 Notice that Q may be replaced by any measure of noncompactness for which the Darbo-Sadovskii theorem holds.
The classes of condensing operators defined relative to distinct measure of noncompact ness are not equal in general, but they nevertheless share a number of general properties. Bearing in mind these common properties Sadovskii [53] introduced the concept of limit operator without using the notion of measure of noncompactness. If X is a linear space and D is a subset of M, then a continuous operator T : D -> X is called a limit compact or ultimately compact operator if co(T(B n D)) = E, for E <:;; X, implies that E is compact. The concept of hyperconvex hull will help us to define limit compact operators in hyperconvex spaces.
Definition 7.4 Let D be a subset of a hyperconvex metric space H. Given an operator T: D ---+ H we will say that (Ta) is a transfinite sequence associated to T on D if
(1) To = h(T(D))
(2) Ta = h(T(D n Ta-ll) if Q - 1 exists,
(3) Ta = n{3<a T{3, if Q - 1 does not exist,
where Ta is a hyperconvex hull of T(D) or T(D n Ta-I) chosen so that the sequence (Ta) is nonincreasing.
Remark 7.5 It is easy to deduce from the properties of the hyperconvex hull that given D, Hand T as in the previous definition, there always exists a transfinite sequence associated to T on D.
The proof of the following lemma is rather easy, so we will omit it.
Lemma 7.6 If (Ta) is a transfinite sequence associated to T on D, then:
(1) Each Ta is closed.
(2) T(D n Ta) <:;; Ta+! for all Q.
(3) If f3 < Q, then Ta <:;; T{3.
(4) T(D n Ta) <:;; Ta for all Q.
(5) There exists an ordinal number TJ such that Ta = Try for all Q ~ TJ·
Hence we obtain the following corollary.
Corollary 7.7 Let (Ta) be a transfinite sequence associated to T on the bounded hyperconvex metric space H. Then the set Try given by Lemma 7.6 is nonempty.
Proof. It suffices to show that for any ordinal number Q, Ta # 0. We proceed by transfinite induction. The case Q = 0 is trivial since T(D) <:;; To, and therefore To is nonempty. Now we have to consider the following two cases: the ordinal Q has a predecessor, in which case, from the inductive hypothesis, T",-I is nonempty and hence, T", = h(T(M n T",-I)) is nonempty, or the ordinal Q has no predecessor, in which case
Hype1'CQnvex spaces 417
Tn = n/knT{3. By using the inductive hypothesis and Baillon's intersection result (Theorem 5.1), it follows that To is nonempty. •
This corollary allows us to define the concept of limit compact operator in hyperconvex spaces.
Definition 7.8 We say that TOO(D) is an ultimate range of the operator T on the set H if it is the limit set of a transfinite sequence associated to T on D. The operator T is said to be ultimately compact (or limit compact) if it is continuous and there exists an ultimate range TOO(D) of T such that T(D n TOO(D)) is relatively compact on H.
We offer the following lemma without proof, for details see [16].
Lemma 7.9 The following properties hold:
(1) TOO(D) = h(T(D n TOO(D))).
(2) If Dl ~ D, then T: Dl --> H has an ultimate range TOO(Dl) contained in TOO(D).
(3) 1fT is limit compact on D and Dl ~ D, T is limit compact on Dl.
(4) The operator T: D --> H is limit compact if and only if TOO (D) is compact.
(5) The operator T : D --+ H is limit compact if and only if for any B c H, the equality h(J(B n D)) = B implies that B is compact.
The following theorem states the relation of a-condensing and limit compact mappings.
Theorem 7.10 Let H be a bounded hyperconvex set, and suppose D ~ H is closed. If T: D --+ H is a-condensing, then T is limit compact on D.
Proof. Let TOO(D) be any ultimate range of T on D. By the previous lemma h(T(D n TOO (D))) = TOO(D). From this; h(T(D n TOO(D))) ;2 D n TOO(D). Since a is monotonous, we obtain a(h(T(DnTOO(D))) 2': a(DnTOO(D)). But, from Corollary 5.11, a(h(T(D n TOO(D))) = a(T(D n TOO(D))). And hence a(T(D n TOO(D))) 2': a(D n TOO(D)). Bearing in mind that T is condensing we conclude D n TOO(D) is relatively compact. Therefore T is ultimately compact on D. •
The following theorem states the existence of fixed points for limit compact mappings, so, in view of the previous theorem, it may be understood as an extension of Theorem 7.2.
Theorem 7.11 Let H be a bounded hyperconvex metric space and T : H --+ H a limit compact mapping on H. Then T has a fixed point in H.
Proof. The conclusion follows from Schauder's theorem since TOO(H) is nonempty, compact and hyperconvex, and T: TOO(M) --+ TOO(M). •
The following theorem was stated in [5] and is closely related to the previous one, although, as seen in [4], they are distinct.
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Theorem 7.12 Let H be a hyperconvex metric space, let Xo E H, and let T be a continuous mapping from H into itself. If every D S;; H such that D is isometric to h(T(D)) or D = T(D) U {xo} is relatively compact, then T has a fixed point in M.
Proof. The first part of the proof consists in proving that there exists Z S;; H such that T(Z) = Z. Let E = {U S;; H : U is T-invariant, and Xo E U}. E is nonempty since H belongs to it. It is easy to see that
E= n U UEE
is a nonempty, and minimal among all the elements of E, i.e. it does not contain any proper subset V such that T(V) U {xo} S;; V. From here it is possible to prove that B satisfies that T(B) U {xo} = B and hence, by hypothesis, that B is relatively compact and that this implies the existence of the set Z.The details may be found in [58].
Let n be the family of all sets A S;; H such that Z S;; A, A is hyperconvex and f(A) S;; A. n is nonempty since H belongs to it. Zorn's lemma leads to a minimal clement in n, let us say Ho. So T(Ho) S;; Ho, and, since Ho is hyperconvex, there exists a hyperconvex hull h(T(Ho)) of T(Ho) such that T(Ho) S;; h(T(Ho)) S;; Ho. Moreover we have T(h(T(Ho))) S;; T(Ho) S;; h(T(Ho)) and Z = T(Z) S;; T(Ho) S;; h(T(Ho)). Hence, since h(T(Ho)) is hyperconvex, we have h(T(Ho)) E n. Now, from the minimality of Ho, we obtain h(T(Ho)) = Ho and so, from the hypothesis, Ho is relatively compact, but additionally Ho is hyperconvex, and hence closed, which implies that Ho is compact. The conclusion follows since T : Ho ...... Ho is a continuous compact mapping and Ho is hyperconvex. •
Remark 7.13 It is not difficult to see that any a-condensing mapping satisfies the hypothesis of this theorem. Notice also that H is not required to be bounded in this theorem.
We finish this section with the following theorem stated in [40].
Theorem 7.14 Let H be a bounded hyperconvex metric space. Let T : H ...... H, be such that for some relatively prime pair i,j E N, r is nonexpansive and Ti is condensing. Then T has a fixed point.
Proof. Let A = {n EN: Tn is either nonexpansive or condensing}. Since A is additive, there exists an integer N such that if n 2:: N then n E A. It follows that there exist nand n + 1 such that rn is nonexpansive and Tn+! is condensing. Since Tn and Tn+! commute, and Fix(Tn) is hyperconvex (as was shown in Section 6)and we see that Tn+! must have a common fixed point with Tn. This point is also a fixed point for T. •
Remark 7.15 It is an interesting open question whether the previous theorem still holds if the nonexpansiveness condition is removed.
8. Isbell's Hyperconvex Hull
As we mentioned before, Isbell [24] showed that every metric space has an injectiv_e envelope. The injective envelope of a metric space M is an injective metric space M which contains an isometric copy of M and which is isometric with a subspace of any
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hyperconvex metric space which contains an isometric copy of M. In this section, we will discuss Isbell's ideas.
Let M be a metric space. For any x E M, define the positive real valued function fx: M -> [0,00) by fx(Y) = d(x,y). Let us discuss some of the properties satisfied by these iunctions.
1. Using the triangle inequality, we get
d(x, y) :::; fa(x) + la(Y),
and la(x):::; d(x,y) + fa(Y),
for any x,y,a E M.
2. Let I : M -+ [0,00) be such that d(x,y) :::; f(x) + fey), for any x,y E M, and for some a E M, assume that I(x) :::; la(x), for all x E M. In this case, we have f = fa. Indeed, first we have I(a) :::; la(a) = 0, which implies I(a) = O. Using the above inequality, we get
la(x) = d(x,a):::; I(x) + f(a) = f(x)
for any x E M. Combined with the assumptions on I(x), we get I(x) = la(x), for all x E M. This is a minimality property for the pointwise order.
Using these properties, Isbell introduced what he called the injective envelope of A, denoted e(A), for any subset A of M. The set e(A) is the set of all extremal functions defined on A. Here, a function f : A -+ [0,00) is extremal if
d(x,y):::; f(x) + fey) for all x,y in A
and is pointwise minimal, i.e. if 9 : A -+ [0,00) such that d(x,y) :::; g(x) + g(y) for all x, y in A and g(x) :::; I(x) for all x E A, then we must have I = g. In particular, we have fa E e(A), for any a E A. Consider the map e : A -+ e(A), defined by e(a) = la, for all a E A. The map e is an isometry. In other words, we have
d(e(a), e(b» = sup I/a(x) - fb(X) I = sup Id(a, x) - deb, x)1 = dCa, b). XEA :rEA
So, A and e(A) are isometric spaces, hence we may identify A with the subspace etA) of e(A), or a E A with e(a). Before we give some detailed properties of extremal functions, we will need the following easy to prove lemma.
Lemma 8.1 Let A be a nonempty subset 01 M. Let r : A -+ [0,00) be such that
d(x, y) :::; rex) + r(y)
lor all x,y E A. Then there exits R: M -> [0,00) which extends r such that
d(x,y) :::; R(x) + R(y)
lor all x, y EM. Moreover, there exists an extremal function I defined on M such that I(x) :::; R(x) lor all x E M.
Let us give some properties of extremal functions.
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Proposition 8.2 The following statements are true.
(1) If f E teA), then it satisfies f(x) :::; d(x,y) + fey) for all x,y in A. Moreover, we have
f(x) = sup If(Y) - fx(y)1 = d(f,e(x)). yEA
(2) For any f E teA), 6> 0, and x E A, there exists yEA such that
f(x) + fey) < d(x, y) + 6.
(3) If A is compact, then teA) is compact.
(4) If s is an extremal function on the metric space teA), then so e is extremal on A.
Proof. (1) Assume not. Then there exist xo, Yo E A, such that
d(xo, Yo) + f(yo) < f(xo).
Set
g(x) = { f(x) if x f= Xo d(xo, Yo) + f(yo) if x = Xo·
It is clear that we have g(x) :::; f(x), for all x E A. In particular, we have g(xo) < f(xo). Let us show that for any X,y E A, we have d(x,y) :::; g(x) + g(y). If both x and yare different from Xo, we use the properties of f. So we can assume x = Xo and y f= Xo. Then
d(x, y) = d(xo, y) :::; d(xo, Yo) + d(yo, y) ::; d(xo, Yo) + f(yo) + fey),
which obviously implies d(xo, y) ::; g(xo) + g(y). The minimality of f gives us f = 9 which is a contradiction.
Combining this inequality with the fundamental one (for extremal functions), we get If(y) - fx(y)1 ::; f(x) for any yEA. The equality holds for y = x which clearly implies
f(x) = sup If(Y) - fx(y)1 = d(f,e(x)). yEA
(2) Assume not. Then there exist x E A and 6 > 0 (we may take it less than f(x)) such that for any yEA, we have
d(x,y) + 6::; f(x) + fey).
Set
h(z) = { fez) if z f= x f(x) - 6 if z = x.
It is easy to check that dey, z) ::; hey) + h(z) for any y, z E A. Since h :::; f and hex) < f(x), we get a contradiction with the minimality of f. This completes the proof of Property 2.
(3) From property 1, we get
If(x) - f(y)1 :::; d(x,y)
for any x, YEA. This implies teA) c Lipl (A), where Lipl (A) is the space of all Lipschitzian real valued functions with Lipschitz constant equal 1. Hence, all extremal
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functions are equicontinuous. Also, it is quite easy to show that a pointwise-limit of extremal functions is an extremal function. Since A is compact, the Arzela-Ascoli theorem implies that e(A) is compact.
(4) Let s be an extremal function on the metric space e(A). Note that for any X,y E A, we have
d(x, y) = d(fx, fy) :::; s(fx) + s(fy) = so e(x) + s 0 e(y).
Assume that s 0 e is not an extremal function on A. Then there exists an extremal function hE e(A) such that h(x) :::; so e(x), for any x E A, and the inequality is strict at some point Xo. Define the function
t(f) = {s(f) if f =f. e(xo) h(xo) if f = e(xo).
Let us show that t satisfies the inequality d(f,g) :::; t(f) +t(g) for all f,g E e(A). Since t and s coincide almost everywhere and s is an extremal function, then we only need to prove the above inequality for 9 = e(xo) and f =f. e(xo), i.e. d(f, e(xo)) :::; t(f)+t(e(xo)).
For any 6 > 0, there exists yEA such that f(xo) + f(y) < d(xo, y) + 6. If y = xo, then we must have f(xo) :::; 1/26. Hence
1 d(f, e(xo)) = f(xo) :::; 2"6 + t(f) + t(e(xo)).
On the other hand, if y =f. Xo and f =f. e(xo), then
d(f, e(xo)) + f(y) - 6 = f(xo) + f(y) - 6 < d(xo, y)
and d(xo,y) :::; h(xo) + h(y) :::; h(xo) + s 0 e(y) = t(e(xo)) + t(e(y)).
Since s is an extremal function, then
t(e(y)) = s(e(y)) :::; s(f) + d(f,e(y)) = t(f) + f(y),
where we used the fact that f is an extremal function (to get d(f, e(y)) = f(y)). So we have the two inequalities
d(f, e(xo)) + f(y) - 6 < t(e(xo)) + t(e(y))
and t(e(y)) :::; t(f) + f(y)·
Adding the two inequalities, we get
d(f, e(xo)) + f(y) - 6 + t(e(y)) < t(e(xo)) + t(e(y)) + t(f) + f(y)
which leads to d(f, e(xo)) - 6 < t(e(xo)) + t(f). Since 6 is arbitrary, we get the desired inequality d(f,e(xo)) :::; t(e(xo)) + t(f). •
The link between Isbell's ideas and hyperconvexity is given in the following proposition.
Proposition 8.3 The following statements are true.
(1) e(A) is hyperconvex.
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(2) e(A) is an injective envelope of A, i.e. no proper subset of e(A) which contains A (metrically) is hyperconvex.
Proof. (1) In order to prove that e(A) is hyperconvex, let {f",} be a family of points in e(A) and {r",} be a family of positive numbers such that d(f"" f(3) :s: r", + 1'(3 for any a and (3. Define the map 1': {f",} -> [0,00) by 1'(f",) = 1'",. By Lemma 8.1, we extend l' to the entire set EtA) such that d(f,g) :s: r(f) + rig) for any f,g E e(A). Using the same lemma, there exists h an extremal function on EtA) such that h :s: 1'. Using Property 4 of extremal functions, we know that hoe is an extremal function on A, i.e. hoe E e(A). It is easy to see that
hoe E n B(f,r(f)) c nB(f""r",). JEf(A) '"
Indeed, we have
h 0 e(x) - f(x) = h 0 e(x) - d(f, e(x)) :s: h(f) :s: r(f)
and f(x) - h 0 e(x) = d(f, e(x)) - h 0 e(x) :s: h(f) :s: r(f)
for any x E A. Hence d(f,h 0 e) :s: r(f) for all f E EtA). The proof of our claim is therefore complete.
(2) Let H be a subset of e(A) such that e(A) C H. Assume that H is hyperconvex. Since e(A) is hyperconvex, there exists a nonexpansive retraction R : e(A) -> H. Let f E e(A). We have
d(R(f)'e(x)) = R(f)(x):S: d(f,e(x)) = f(x)
for any x E A. Since f is an extremal function, we must have R(f) = f. This clearly implies that H = e(A). So no proper subset of e(A) which contains e(A) is hyperconvex .
• Remark 8.4 The ideas of a linear injective envelope for a Banach space and metric injective envelopes were introduced by Cohen [9J and Isbell [24J respectively, both in 1964. Since then many interesting papers have studied properties of these elements as well as the question of whether the metric injective envelope of a normed spaces coincides with its linear one. This problem was finally solved in the affirmative by Rao [51J in 1992. For this and other interesting properties of injective envelopes the reader may consult [7, 9, 10, 21, 24, 25, 26, 51J.
9. Set-valued mappings in Hyperconvex spaces
Recall that A(M) denotes the family of all nonempty admissible subsets of a metric space M and throughout this section let £(M) denote the family of all nonempty bounded subsets of M which are externally hyperconvex (relative to M), in both instances endowed with the usual Hausdorff metric dH . Recall that the distance between two closed subsets A, B of a metric space in the Hausdorff sense is given by
dH(A,B) = inf{e > 0: A C No(B) and B C No(A)}
where No (A) denotes the closed c:-neighborhood of A.
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The main result of this section is the following selection theorem stated in [33].
Theorem 9.1 Let H be hyperconvex, and let T* : H -t £(H). Then there exists a mapping T : H -t H for which T(x) E T*(x) for each x E H and for which d(T(x),T(y»:S; dH(T*(x),T*(y» for each X,y E H.
Proof. Let ~ denote the collection of all pairs (D, T), where
T : D -t H, T(d) E T*(d) V d ED,
and d(T(x),T(y»:S; dH(T*(x),T*(y» for each X,y E D.
Notice that ~ i= 0 since ({xo},T) E ~ for any choice of Xo E Hand T(xo) E T*(xo). Define an order relation on ~ by setting
(Dl, T1 ) ~ (D2' T2) ¢} Dl C D2 and T21Dl = T1 .
Let {(D"" T",)} be an increasing chain in (~,~) . Then it follows that (U",D"" T) E ~ where T IDa = T",. By Zorn's Lemma, (~,~) has a maximal element, say (D, T) . Assume D i= H and select Xo E H\D. Set fJ = D U {xo} and consider the set
J = n B(T(x),dH(T*(x),T*(xo»)) nT*(xo). xED
Since T*(xo) E £(H) for each x E H, J i= 0 if and only if for each XED,
dist(T(x),T*(xo»:S; dH(T*(x),T*(xo».
Also, since T*(xo) is a proximinal subset of H, the above is true if and only if for each xED,
B (T(x), dH(T*(x), T*(xo»)) n T*(xo) i= 0.
By the definition of Hausdorff distance T*(x) C NdH(T*(x),T*(xO))+E(T*(xo)) for each c> o. However by assumption T(x) E T*(x) so it must be the case that for each c > 0,
B(T(x),dH(T*(x),T*(xO») +c) nT*(xo) i= 0.
Since T*(xo) is proximinal in H, this in turn implies
Thus we conclude J i= 0. Choose Yo E J and define
T(x) = { YO if x = XOi
T(x) ifxED.
Since d(T(xo),T(x» = d(Yo,T(x»:S; dH(T*(x),T*(xo» we conclude that
(D U {xo},T) E~,
contradicting the maximality of (D, T). Therefore D = H. •
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Looking in detail at this proof it becomes obvious that the hyperconvexity of H has not been used, only the extremal hyperconvexity of the images was needed. Hence the following corollary holds.
Corollary 9.2 The above theorem remains true if T* : M --> £(H) with M and H any metric spaces and £(H) the class of bounded externally hyperconvex spaces of H (assuming this class is nonempty).
The following corollary is a direct consequence of Theorem 9.1 combined with Theorem 6.1.
Corollary 9.3 Let H be bounded and hyperconvex, and suppose T* : H --> £(H) is non expansive. Then T* has a fixed point, that is, there exists x E H such that x E
T*(x).
The method of the proof of Theorem 9.1 also gives the following. In this theorem Fix(T*) = {x E H: x E T*(x)}.
Theorem 9.4 Let H be hyperconvex, let T* : H --> £(H) be nonexpansive and suppose Fix(T*) of. 0. Then there exists a nonexpansive mapping T : H --> H with T(x) E T*(x) for each x E H and for which Fix(T) = Fix(T*). In parlicular, Fix(T*) is hyperconvex.
Proof. Let ~ denote the collection of all pairs (D, T), where D J Fix(T*), T : D --> H, T(d) E T*(d) for all d E D, T(x) = x for all x E Fix(T*), and d(T(x),T(y)) ::; d(x,y) for all x,y E D. By assumption (Fix(T*),Id) E ~, so ~ of. 0. The argument is now a simple modification of the proof of Theorem 9.1. Define an order relation on ~ by setting
(DI' T I ) ::5 (D2, T2) ¢} DI C D2 and T2 IDl = T I ·
Let {(Da, Ta)} be an increasing chain in (~,::5) . Then it follows that (UaDa, T) E ~ where T IDo = Ta. By Zorn's Lemma, (~,::5) has a maximal element, say (D, T) . Assume D of. H and select Xo E H\D. Set jj = D U {xo} and consider the set
J = n B(T(x),d(x,xo)) nT*(xo). xED
Since T*(xo) E £(H) for each x E H, J of. 0 if and only if for each xED,
dist (T(x),T*(xo)) ::; d(x,xo).
Also, since T*(xo) is a proximinal subset of H, the above is true if and only if for each xED, B(T(x), d(x, xo)) n T*(xo) of. 0. Using the definition of Hausdorff distance and the fact that T* is nonexpansive, for each c > 0
T*(x) C NdH(T*(x),T'(xo))+,(T*(xo)) C Nd(x,xo)+' (T*(xo)).
However by assumption T(x) E T*(x) so it must be the case that for each c > 0,
B(T(x), d(x, xo) + c) n T*(xo) of. 0.
Since T*(xo) is proximinal in H, this in turn implies B(T(x); d(x, xo)) n T*(xo) of. 0. Thus we conclude J of. 0. Choose Yo E J and define
T(x) = { Yo if x = Xo; T(x) if xED.
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Since d(T(xo),T(x)) = d(yo,T(x)) ::; d(x,xo) we conclude that (D U {xo},T) E J, contradicting the maximality of (D, T). Therefore D = H. To conclude the proof, we need to show that Fix(T*) is hyperconvex. This is a direct consequence of Theorem 6.1 applied to T. •
Remark 9.5 Due to its importance in different branches of mathematics, selection problems have been widely studied over the last fifty years. The problem is usually as follows: given a certain multivalued mapping to be able to find an univalued selection of it with certain properties such as, for instance, continuity or measurability. Theorem 9.1 is very surprising since it is not common at all to be able to guarantee that a nonexpansive multivalued mapping admits a nonexpansive selection, in fact this seems to be quite characteristic of hyperconvex geometry. One of the most challenging open problems in hyperconvex metric spaces is whether this theorem may be improved or not in the sense that £(H) be replaced by a wider class of subsets of M. More specifically the most natural question at this moment is whether W(H), the class of nonempty bounded weakly externally hyperconvex subsets, could replace £(H) in Theorem 9.1. Counterexamples are not known for the case 1t(H). The reader will find a large collection of results and references on multi valued selection problems in the recent book [50].
We finish this section with some applications of Theorem 9.1. First we show that the family of all bounded A-lipschitzian functions of a hyperconvex space M into itself is itself hyperconvex and second we will study best approximation problems in hyperconvex spaces. This leads naturally to Ky Fan type theorems for hyperconvex spaces.
Let f and 9 be two bounded A-lipschitzian functions of a hyperconvex space Minto itself, we define the distance between them in the usual way, that is, if f, 9 : M --> M, set
d(f,g) = sup d(J(x),g(x)). xEM
Theorem 9.6 Let M be hyperconvex and for A > 0 let J>. denote the family of all bounded A-lipschitzian functions of Minto M. Then J>. is itself a hyperconvex space.
Proof. Suppose {f,,} C J>. and {r,,} C lR satisfy dU", fj3) ::; r" + rj3. Then for each x E M d(fa(x), f,,(x)) ::; ra + rj3, so in view of the hyperconvexity of M
J(x) = nB(J,,(x),r,,) =I 0
We show that dH(J(X), J(y))::; Ad(x,y) for each X,y E M. To see this it clearly suffices to show that J(x) C N>'d(x,y)(J(y)). for each x,y E M. However if z E J(x) then for each a
d(z, f,,(y)) ::; d(z, f,,(x)) + d(fa(x), fa(Y)) ::; d(z,f,,(x)) + Ad(x,y)
::; ra + Ad(x,y).
Using Sine's Lemma (Lemma 4.10) we now have
z E nB(Ja(y), r" + Ad(x,y)) = N>'d(x,y) (J(y))
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In view of Theorem 9.1 it is possible to select f(x) E J(x) for each x E M so that f E J>.. Since f E naB(fa, raj, J>. is hyperconvex. •
This leads to the following.
Corollary 9.7 Let M be a bounded hyperconvex metric space and let f E Fl. Then the family
R = {r E Fl : r(M) c Fix(fl}
is a nonexpansive retract of Fl.
Proof. The mapping Tf : Fl ---+ Fl defined via the formula Tf(9) = fog is nonexpansive and has a nonempty fixed point set Fix(Tf ) which is hyperconvex. Theorem 4.4 will then imply that Fix(Tf ) is a nonexpansive retract of Fl. But r E Fix(Tf) if and only if r E R. •
One of the most important concepts in approximation theory is that of metric projection. We now turn to a study of the problem of finding nonexpansive selections for the metric projection. Let us introduce some definitions. Recall that the concept of proximinality was introduced in Definition 3.7.
Definition 9.8 Let M be a metric space and A a proximinal subset of M, then the mapping R : M ---+ 2A defined as
R(x) = B(x, dist(x, A)) nA
for every x E M is called the metric projection onto A (relative to M).
Notice that the proximinality of A guarantees that R(x) # 0 for all x E M. We will also deal with the following concept.
Definition 9.9 A subset A of a metric space M is said to be a proximinal non expansive retract of M if there exists a nonexpansive selection of the metric projection on A, i.e. if there exists a nonexpansive retraction r : M ---+ A such that r(x) E B(x, dist(x, A)) n A for each x E M.
The first one to take up the problem of characterizing proximinal nonexpansive retracts in hyperconvex metric spaces was Sine [56J. The following theorem was stated in [56J for admissible subsets, we adapt Sine's proof to externally hyperconvex subsets.
Theorem 9.10 Let E be a nonempty externally hyperconvex subset of a metric space H. Then E is a proximinal nonexpansive retract of H.
Proof. For each x E H we define A(x) = B(x,dist(x,E)) nE and
C(x)=n{A(y)+d(x,y): YEH}.
Since A(x) is one of the sets that define C(x), the inclusion C(x) ~ A(x) is clear. We will show that C : H ---+ 2E is a nonexpansive multivalued mapping whose values are
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nonempty externally hyperconvex subsets of H. Due to the externally hyperconvexity of E, to establish that C(x) is nonempty, it will be enough to prove that
B(YI, dist(YI,E) +d(X,YI)) nB(Y2,dist(Y2,E) +d(X,Y2)) i= 0
for each YI and Y2 in H. But x belongs to both of these sets, so C(x) i= 0. Additionally, the externally hyperconvexity of C(x) follows as an easy consequence of Lemma 5.3. Let us see now that Cis nonexpansive. For this, pick u and v in H. We have to show that C( u) <;;; C( v) + d( u, v), or equivalently
n (A(z) + d(u, z)) <;;; n (A(z)+d(z,v)+d(u,v)), zEH zEH
which is clear if one recalls that d( u, z) :S d( u, v) + d( v, z). Now the theorem follows as an application of Theorem 9.1. •
This problem was taken up again in [15J where some improvements of the above theorem were given. We state these results without proofs. The proofs may be found in [15J.
Theorem 9.11 A compact subset E of a hyperconvex metric space H is a proximinal nonexpansive retract of H if and only if E is weakly externally hyperconvex (relative to H).
The situation for the noncompact case has not been solved yet although the following theorem states a partial result that gets very close to completely remove the compactness condition from Theorem 9.11.
Theorem 9.12 Let E be a weakly externally hyperconvex subset of a hyperconvex metric space H. Then given any 10 > 0 there is a nonexpansive retraction Rc of H onto E with the property that given any n E H\E there exists x E H such that d (u, x) :S 10 and d (x, Rc (x)) = dist (x, E) . Moreover if int (E) i= 0 then Rc may be chosen so that Rc (H\E) c BE.
Sine made use of Theorem 9.10 to obtain certain Ky Fan [17J type theorems for hyperconvex spaces. Even though Theorem 9.12 does not quite solve the "nonexpansive proximinal retract" problem it still enjoys enough good properties to lead to new improved versions of the Ky Fan type results given by Sine. We conclude this section with these fixed point results.
The first result we will see is a topological one, like those seen in Section 7, with a boundary condition. In fact it extends the Darbo-Sadovskii Theorem.
Theorem 9.13 Suppose E is a bounded weakly externally hyperconvex subset of a hyperconvex space H with non empty interior, and let T : E -> H be a uniformly continuous condensing mapping for which T (BE) c E. Then T has a fixed point.
Proof. Let 10 > 0 and choose 10' :S 10 so that d (n, v) :S 10' => d (T (u), T (v)) :S e. Now let Rc' be the nonexpansive retraction assured by Theorem 9.12. It is easy to see that the mapping Rc' 0 T : E -> E is condensing, and since E is hyperconvex R E, 0 T has a fixed point, say Xc E D. If T (xc) E E, then Rc' 0 T (xc) = T (xc) = xc. If T (xc) ric E, then there exists Y E BE such that d (xc, y) :S 10'. In this case (since T (y) E E) we have
d(y,T(y)) :S d(y,x,J + d(xc,T(y))
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::; c: + d(R" 0 T(x,), R" 0 T(y))
::; c:+d(T(x,),T(y))
::; 2c:.
This proves that inf {d (y, T (y)) : y E E} = O. Since T is condensing it easily follows that T has a fixed point in D. •
The following is an easy consequence of Theorem 9.12.
Theorem 9.14 Suppose E is a bounded weakly externally hyperconvex subset of a hyperconvex metric space H, and suppose T : E -t H is a non expansive mapping for which T (8E) C E. Then T has a fixed point.
Proof. Let R be a nonexpansive retraction of H onto E for which R(H \ E) ~ 8E. Then RoT: E -t E is nonexpansive and has the same fixed point set as T. •
Another consequence of Theorem 9.12 is the following.
Theorem 9.15 Suppose E is a bounded weakly externally hyperconvex subset of a hyperconvex metric space H, and let T : E -t H be a condensing mapping for which T (8E) C E. Then T has a fixed point.
Finally, since compact hyperconvex spaces have the fixed point property for continuous mappings (e.g., [31, 47]), Theorem 9.11 yields a Ky Fan's approximation principle for compact weakly externally hyperconvex sets.
Theorem 9.16 Let E and H satisfy the assumptions of Theorem 9.11 and suppose T : E -t H is a continuous mapping. Then there exists x E E such that
d(x,T(x)) =inf{d(y,T(x)) :YED}.
Proof. Let R be the retraction given by Theorem 9.11, then RoT: E -t E is a continuous mapping so it has a fixed point in E which confirms the statement of the theorem. •
Ky Fan's fixed point theorem for hyperconvex metric spaces may be stated in the following way.
Corollary 9.17 Let E and H satisfy the assumptions of Theorem 9.11 and suppose T : E -t H is a continuous mapping such that T( 8E) ~ E (if the interior of E is empty, assume T(E) ~ E). Then T has a fixed point in E.
Proof. It is easy to see that the fixed point of RoT in the theorem must be a fixed point of T under the additional boundary condition in the statement of the corollary .•
10. The KKM theory in Hyperconvex spaces
Among the results equivalent to the Brouwer's fixed point theorem, the Theorem of Knaster-Kuratowski-Mazurkiewicz (in short KKM) occupies a special place. Historically Brouwer's fixed point theorem failed to establish itself in the metric setting in
Hyperconvex spaces 429
comparison with Kirk's fixed point theorem. In our opinion, this is due to the fact that the first theorem depends heavily on the convex structure of the set while the second one depends on set theoretical convexity. Since hyperconvex metric spaces exhibit some kind of convexity, it was natural to investigate Brouwer's fixed point theorem in this setting. This problem was first studied by the second author in [31J and later by others in [41, 47J. This section is devoted to some of the results appearing in these three works. The literature on the KKM principle is quite large, the reader will find a more precise and exhaustive treatment of it in [22, 59J where the KKM principle is mainly developed in topological and nonlinear settings (see also the chapter by).
Let H be a metric space. A subset A cHis called finitely closed if for every X1,X2, ... ,Xn E H, the set COV({Xi}) nA is closed. If A is closed then obviously it is also finitely closed. Recall that a family {A"'},,Er in 2H is said to have the finite intersection property if the intersection of each finite subfamily is not empty.
Definition 10.1 Let H be a metric space and X C H. A multivalued mapping G : X -t 2H is called a Knaster-Kuratowski-Mazurkiewicz map (in short a KKM-map) if
for all X1, ... ,Xn E X.
COV({X1, ... ,Xn}) C U G(Xi) l:S;i:S;n
We have the following result:
Theorem 10.2 (KKM-mapping principle) Let H be a hyperconvex metric space, and X be a nonempty subset of H. Let G : X -t 2H be a KKM-map such that each G(x) is finitely closed. Then the family {G(x); x E X} has the finite intersection property.
Proof. Assume not, i.e. there exist Xl, ... , Xn E X such that
i=n n G(Xi) = 0. i=l
Set L = cov( {Xi}) in H. Consider the hyperconvex Banach space loo(H) and set Hoo = cov(H) in loo(H). Let C = CO(Xi) in Hoo. Here we consider the linear convex hull. By Theorem 4.4, there exists a nonexpansive retraction r : Hoc -t H. Note that r(C) C L. Our assumptions imply that L n G(Xi) is closed for every i = 1,2, ... , n. Since niG(Xi) n L = 0 then, for every c E C, there exists io such that r(c) does not belong to L n G(Xio). Hence dist(r(c),L n G(Xio)) > 0 because L n G(Xio) is closed. Therefore, the function
a(c) = fdist (r(c),LnG(xi)) i=l
is not zero for any c E C. Define the map F : C -t C by
1 i=n F(c) = a(c) L dist (r(c), L n G(Xi) )Xi.
-z=1
Clearly, F is a continuous map. Since C is compact, then Brouwer's theorem implies the existence of a fixed point Co of F, i.e. F(co) = co. Set 1= {i; dist(r(co), LnG(Xi)) =I O} .
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Clearly we have
Co = 0'(1 ) Ldist (r(co),LnG(xi»)xi. CO iEI
Therefore, r(co) fj. UiEIG(Xi) and r(co) E cov( {Xi; i E I}), contradicting the assumption cov( {Xi; i E I} C UiEIG(Xi) . The proof of Theorem 10.2 is therefore complete. •
As an immediate consequence, we obtain the following theorem.
Theorem 10.3 Let H be a hyperconvex metric space and X C H be a nonempty subset. Let G : X -t 2H be a KKM-map such that G(x) is closed for any X E X and G(xo) is compact for some Xo EX. Then we have
n G(x) # 0. xEX
Notice that the compactness assumption on G(xo) is over strong. We can reach the same conclusion if one involves an auxiliary multivalued map and a suitable topology on H (such as the ball topology for example).
Theorem 10.4 Let H be a hyperconvex metric space and X C H be a nonempty subset. Let G : X --+ 2H be a KKM-map. Assume there exists a multivalued map K : X -t 2H such that G(x) C K(x) for every X E X and
n K(x) = n G(x). xEX xEX
If there exists a topology T on H such that each K (x) is compact for T, then
n G(x) # 0. ",EX
The proof is obvious.
The concept of a KKM mapping was generalized in [41J in the following way.
Definition 10.5 Let H be a metric space and X ~ H. A multivalued mapping G : X --+ 2H \ {0} is called a generalized metric KKM mapping (GMKKM) if for each finite set {Xl, ... , x n} ~ X, there exists a set {yl, ... , Yn} of points of H, not necessarily all different, such that for each subset {Yiu ... ,Yik} of {Yl, "',Yn} we have
COV{Yi;: j = 1, ... k} ~ U~=l G(Xi;).
It is easy to check that KKM mappings are generalized KKM mappings while the converse is not true, the interested reader may consult [41, 59J for more about this topic. The following theorem is an extension of Theorem 10.2 where generalized metric KKM mappings substitute for KKM mappings. The proof, although more complicated, follows similar ideas to those in the proof of Theorem 10.2 and we will omit it.
Theorem 10.6 (Generalized metric KKM principle) Let X be a nonempty subset of a hyperconvex metric space H. Suppose G : X --+ 2H \ {0} has finitely closed values. Then the family {G(x) : X E X} has the finite intersection property it and only if the mapping G is a generalized metric KKM mapping.
Hyperconvex spaces 431
Kirk at al. exhibit a large number of consequences of this theorem in [41] (see also [40, 47, 59]). These consequences have to do with Minimax inequalities, fixed point theorems for multivalued mappings, saddle points, and Nash equilibria. The following theorem is ,among these consequences. Note that this theorem is a multivalued version of Ky Fan's approximation principle already seen in the previous section.
Theorem 10.7 Let H be a hyperconvex space and A a nonempty admissible compact subset of H. Suppose T : A -+ A(H) is a multivalued continuous mapping. Then there exists Xo E A such that
dist (xo,T(xo)) = inf dist (x,T(xo)). xEA
Proof. Define the mapping G : A -+ 2H \ {0} by
G(x) = {y E A: dist (y,T(y)) :s; dist(x,T(y))}
for each x E A. As T is continuous, G(x) is closed and nonempty for each x E A. We want to prove that G is a KKM mapping. Suppose it is not, then there exists a nonempty and finite subset {Xl, ... ,Xn } and y E COV({Xi: i = 1, ... ,n}) such that dist(xi,T(y)) < dist(y,T(y)) for i = 1, ... ,n. Let e > 0 be such that dist(xi,T(y)):S; dist(y,T(y)) - e for i = 1, ... ,n. Let r = dist(y,T(y)) - e. Then Xi E T(y) + r for i = 1, ... , n. From Lemma 4.10 T(y) +r E A(H), thus COV{Xl' ... ,xn } <;;; T(y) + r. This in turn implies y E T(y) + r and hence dist(y, T(y)) :s; r = dist(y, T(y)) - e, which is not possible by assumption. Therefore G must be a KKM mapping.
Note that nxEAG(X) 1 0 since X is compact. Take Xo E nXEAG(X). Then it is clear that dist(xo,T(xo)) :s; dist(x,T(xo)) for all x E A, which completes the proof of the theorem. •
11. Lambda-Hyperconvexity
Since the beginning it was known that the Hilbert space [2 fails to be hyperconvcx. By studying this case closely, the second author, Knaust, Nguyen and O'Neill [34] introduced a property very similar to hyperconvexity, called )"-hyperconvexity. The idea is to expand the radius of the given balls by a uniform factor. For example, every pairwise intersecting collection of balls in [2 has non-empty intersection if the radius of the balls are increased by the factor v'2. In light of this, the following definition becomes natural.
Definition 11.1 Let M be a metric space and let ).. ~ 1. We say that the metric space !vI is )..-hyperconvex if for every non-empty admissible set A E A(M), for any family of closed balls {B(xo<l ranaEA, centered at Xa E A for a E A, the condition
d(x", , x(3) :s; r", + r(3 for every O!, f3 E A,
implies
An (n B(xa,)..ra)) 10. aEA
Let )"(M) be the infimum of all constants).. such that M is )..-hyperconvex, and say that )"(M) is exact if M is )..(M)-hyperconvex.
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Griinbaum [20J and other authors have studied a similar property not involving an underlying admissible set A. But its introduction becomes essential when we try to connect this concept to the fixed point property via the normal structure property.
Let us recall Griinbaum's definition: For a metric space M, let the expansion constant E(M) be the infimum of all constants f.l such that the following holds: Whenever a collection {B(x""r",): Q E A} intersects pairwise, then
n B(x""f.l· r",) f 0. ",EA
We say E(M) is exact, if the condition is satisfied for f.l = E(M).
Trivially, E(M) ::::: A(M) holds in metrically convex spaces (see Definition 2.3). On the other hand, if M is a two element metric space, then E(M) = 1, while A(M) = 2, so the two concepts do not coincide in general.
Let us first summarize some basic properties of A-hyperconvex metric spaces, some of which are trivial, while others can be easily derived from corresponding results for expansion constants:
Theorem 11.2 Let M be a metric space.
(1) M is hyperconvex if and only if it is I-hyperconvex.
(2) Every A-hyperconvex metric space is complete.
(3) Reflexive Banach spaces and dual Banach spaces are 2-hyperconvex.
(4) There is a subspace X of II which fails to be 2-hyperconvex.
(5) Hilbert space is v'2-hyperconvex.
We will finish this section, and hence the chapter, by studying the connection between A-hyperconvexity and the fixed point property. The theorem we will finish with is based on a metric generalization of Kirk's fixed point theorem established in [30J. In order to state this generalization we need to know what uniform normal structure in a metric space means.
Let M be a metric space and :F a family of subsets of M. Then we say that :F defines a convexity structure on M if it contains the closed balls and is stable under intersection. For instance A(M), the class of the admissible subsets of M, defines a convexity structure on any metric space M. We say that :F is a uniform normal structure on M if there exists c < 1 such that R(A) ::::: c· diam(A) for every A E :F with diam(A) > 0, where R(A) and diam(A) are, respectively, the Chebyshev radius and diameter of A as defined in Section 3.
Now we may state the generalization of Kirk's fixed point theorem.
Theorem 11.3 Let M be a bounded complete metric space. If M admits a uniform normal structure then it has the fixed point property for non expansive mappings.
The connection between A-hyperconvexity and the fixed point property is giving by the following theorem.
REFERENCES 433
Theorem 11.4 Let M be a bounded >..-hyperconvex space. If>.. < 2, then M admits a uniform normal structure and hence any nonexpansive mapping T : M -+ M has a fixed point.
Proof. Let M be a bounded A-hyperconvex space with>" < 2. Theorem 11.2 assures that M is complete, so from the previous theorem it suffices to prove that M has a uniform normal structure. The family A( M) defines a convexity structure on M, we will show that A(M) is actually a uniform normal structure on M. Let A E A(M) with diam(A) > O. For each x E A let B(x, rx) denote the ball centered at x with constant radius rx = ~ diam(A). Then d(x, y) :S rx + ry for every x, yEA. Since M is >..-hyperconvex we can find
Xo E An ( n B(X,Arx»). xEA
Thus we have d(x,xo) :S A/2diam(A) for every x E A. It follows that R(A) :S A/2 diam(A). Finally since >../2 < 1 we obtain that A(M) is a uniform normal structure on M, and hence the theorem is proved. •
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[20] B. Griinbaum, On some covering and intersection properties in Minkowski spaces, Pacific J. Math. 27 (1959), 487-494.
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[22J C. Horvath, Contractibility and generalized convexity, J. Math. Anal. Appl. 156 (1991), 341-357.
[23J O. Hustad, Intersection properties of balls in complex Banach spaces whose duals are Ll spaces, Acta Math. 132 (1974), 283-213.
[24J J. R. Isbell, Six theorems about injective metric spaces, Comment. Math. Helvetic 39 (1964), 65-76.
[25] J. R. Isbell, Injective envelopes of Banach spaces are rigidly attached, Bull. Amer. Math. Soc. 70 (1964), 727-729.
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[27J E. Jawhari, D. Misane, and M. Pouzet, Retracts: graphs and ordered sets from the metric point of view, Contemp. Math. 57 (1986),175-226.
[28] J. Jaworowski, W. A. Kirk, and S. Park, Antipodal Points and Fixed Points, Notes of the Series of lectures held at the Seoul National Univesity, 1995.
[29J J. L. Kelley, Banach spaces with the extension property, Trans. Amer. Math. Soc. 72 (1952), 323-326.
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[32J M. A. Khamsi, One-local retract and common fixed point for commuting mappings in metric spaces, Nonlinear Anal. 27 (1996), 1307-1313.
[33J M. A. Khamsi, W. A. Kirk, and C. Martinez Yanez, Fixed point and selection theorems in hyperconvex spaces, Proc. Amer. Math. Soc. 128 (2000), 3275-3283.
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Chapter 14
FIXED POINTS OF HOLOMORPHIC MAPPINGS: A METRIC APPROACH
Tadeusz Kuczumow
Maria Curie-Skfodowska University
20-031 Lublin, Poland
tadeklOgolem.umcs.lublin.pl
Simeon Reich
Department of Mathematics
The Technion-Ismel Institute of Technology
32000 Haifa, Ismel
sreich(Qtx.technion.ac.il
David Shoikhet
Department of Applied Mathematics
ORT Bmude College
21982 Karmiel, Ismel
davs<atx.technion.ac.il
1. Introduction
Let Xl and X2 be two complex normed linear spaces and let DI be a domain (that is, a nonempty open connected subset) in Xl. A mapping f : Dl --> X2 is said to be holomorphic in DI if it is Fnkhet differentiable at each point of DI. If DI and D2 are domains in Xl and X 2, respectively, then H(Dl,D2) will denote the family of all holomorphic mappings from DI into D2.
A system of pseudodistances which assigns a pseudodistance to each domain in every complex normed linear space is called a Schwarz-Pick system [68] if the following two conditions hold:
(i) The pseudodistance assigned to the open unit disc l!. of the complex plane <C is the Poincare distance, that is,
pa (z, w) = argtanh ---= , I z-w I l-zw
z,WEl!..
437
WA. Kirk and B. Sims (etls.), Handbook o/Metric Fixed Point Theory, 437-515. Ii;) 2001 Kluwer Academic Publishers.
(Ll)
438
(ii) If PD, and PD2 are two pseudodistances assigned to the domains Dl and D2, respectively, then
PD2 (J(x),f(y))::; PD, (x,y) (1.2)
In particular, taking D = Dl = D2 and f E H(D, D) = H(D) in (1.2) we obtain the inequality
PD (J(x),f(y)) ::; PD (x,y), x,yED. (1.3)
In other words, each holomorphic self-mapping f of D is nonexpansive with respect to any pseudodistance PD assigned to D by a Schwarz-Pick system. This fact is the basis of our metric approach to holomorphic fixed point theory.
The present exposition is by no means a complete survey of all aspects of this theory. Rather, our primary focus will only be on the following three issues: existence and uniqueness of fixed points, the behavior of iteration and approximation processes, and the structure of fixed point sets.
2. Preliminaries
We begin this section with a few notions which are of a general metric character.
Let (M, d) be a metric space.
Definition 2.1 [106].An admissible subset of M is any intersection of closed balls.
Definition 2.2 A mapping f : M -t M is nonexpansive if
d(J(x),f(y))::; d(x,y)
for all X,y E M.
Remark 2.3 The above definition can, of course, be extended to the case of two metric spaces and a mapping from one of them to the other.
The identity operator on M will always be denoted by I. An isometry from M onto M will be denoted by Y. The fixed point set of a mapping f : M -t M will be denoted by Fix(J). Now we recall the following theorem.
Theorem 2.4 [106]. Let (M, d) be a metric space and let f : M -t M be nonexpansive. Then any nonempty admissible subset A of M which is f -invariant contains a nonempty admissible subset Ao of M which is also f-invariant and in addition satisfies
diam (Ao) ::; ~ (diam (A) + r (A)) ,
where
r(A) = inf {r(x,A) = sup {d(x,y) : YEA}: x E A}.
We will also need a result due to A. Calka [32] concerning the behavior of the iterates of a nonexpansive mapping on a finitely compact metric space.
Fixed points of holomorphic mappings 439
Theorem 2.5 [32]. Let f be a non expansive mapping of a finitely compact metric space (M, d) into itself. If for some Xo E M the sequence of iterates {jk(xO)} contains a bounded subsequence {jk= (xo)}, then for every x E M the whole sequence {jk (x)} is bounded.
Remark 2.6 In his 1971 paper [104] about local isometries, W. A. Kirk asked when an isometry I of a metric space with some bounded subsequence of iterates {(I)k~(xo)} had to have bounded orbits. Calka's theorem solves Kirk's problem. In view of the example due to M. Edelstein ([47]); see also Section 12), the assumption that M is finitely compact seems to be quite natural.
Now we turn our attention to Banach spaces. Let (X, 11·11) be a Banach space and let o =f- D c X. We say that a mapping f : D -t D is O'II.II-condensing with respect to Kuratowski's measure of noncompactness 0'11.11 [137] if
0'11·11 (f (C)) < 0'11.11 (C)
for each bounded C c D with 0'11.11 (C) > O.
Theorem 2.7 [176J (see also [13], [17], [19J, [38J and [148]). Let D be a nonempty bounded closed convex subset of a Banach space (X, II . II) and let 0'11.11 denote the Kuratowski measure of noncompactness in X. If f : D -t D is a continuous and O'lir condensing mapping, then f has a fixed point.
Remark 2.8 [176]. Let D be a nonempty closed convex subset of a Banach space (X, 11·11) and let 0'11.11 denote the Kuratowski measure of noncom pact ness in X. Suppose f : D -t D is a continuous and O'II'II-condensing mapping. If there exists a point xED
such that the sequence of iterates {fk(x)} is bounded, then 0'11.11 ({fk (x)}) = O.
In the sequel, all Banach spaces will be complex, unless stated otherwise.
In order to formulate a characterization of holomorphic mappings, we need the concept of a norming set.
Definition 2.9 [45]Let (X, II . II) be a Banach space and let N be a nonempty subset of its dual X*. If there exist positive constants c and C such that
sup {II (x)1 : lEN, 11111 ::; C} ::: c Ilxll for each x E X, then we say that N is a norming set for X.
It is obvious that a norming set generates a Hausdorff linear topology a(X,N) on X which is weaker than the weak topology a(X,X*).
Theorem 2.10 [15J, [16], [20], [34], [37], [45], [53], [75]. Let (Xl, 11·lId and (X2' II· 112) be Banach spaces, D C Xl a domain in Xl, and let N be a norming set for (X2,1I·1I2). ForaED and x EX\{O}, letD(a,x) denote the set
D(a,x) = {z E C: a+zx ED}.
Then the mapping f : D ---t X2 is holomorphic in D if and only if f is locally bounded on D and for each a E D, x E X\{O} and lEN, the function
10 fID(a,x) : D (a, x) -t C
440
is holomorphic in D (a, x) in the classical one-variable sense.
Corollary 2.11 Let Dl and D2 be bounded convex domains in the Banach spaces (XI, II . 111) and (X2, II . 112), respectively. Let N be a nonning set for (X2, II . 112). If {f>.heJ is a net of holomorphic mappings f>. : Dl -> D2 which is pointwise convergent in a(X2,N) to a mapping f : Dl -> D2, then f is also holomorphic.
Finally, we recall the Maximum Modulus Theorem for unit balls. First we need two definitions.
Definition 2.12 [192J A point x in a convex subset C of a Banach space (X, II ·11) is said to be a complex extreme point of C if there is no non-zero vector y E X such that
{x + oy EX: 0 E C, 101 = I} C C.
Definition 2.13 [39J, [41J A Banach space (X, 11·11) is said to be strictly convex if the following implication holds for all x, y EX:
Directly from these definitions we get the following fact.
Corollary 2.14 [192J. In a strictly convex Banach space (X, II . II) each point on the boundary of its closed unit ball 13 is a complex extreme point.
Theorem 2.15 [192J. Let (XI, II . 111) and (X2 , 11·112) be Banach spaces, D a domain in Xl and f : D -> X2 a holomorphic mapping. If IIfOll2 has a maximum at a point in D, then IIfOll2 is a constant function on D.
Corollary 2.16 [192J. Let f satisfy the hypotheses of Theorem 2.15. If every point on the unit sphere of (X2, 11·112) is a complex extreme point of the closed unit ball 13, then f is a constant mapping on D.
Corollary 2.17 [192J. Let f satisfy the hypotheses of Theorem 2.15 and let (X2, 11·112) be a strictly convex Banach space. Then f is a constant mapping on D.
3. The Kobayashi distance on bounded convex domains
Let !:J. be the open unit disc in the complex plane C. Recall that the Poincare distance on !:J. is given by (see (Ll))
kll (z,w) = Pil (z,w) = argtanh ---= = argtanh(l- a(z,w))2, I z -w I 1
1-zw
where
z,wE!:J.
Fixed points of holomorphic mappings 441
(see [60) and [61)).
Note that following the usual practice in complex analysis we call the metric kll a distance (although it is, in fact, a metric in the topological sense).
Next, for any a E Ll, consider the Mobius trans/ormation ma : Ll -+ Ll defined by
z+a ma(z) = 1 +za' z E Ll.
This mapping is not only a biholomorphic automorphism of Ll, but also a kll-isometry. Using these Mobius transformations and the Schwarz Lemma ([30), [42)), we obtain the following simple but fundamental fact.
Theorem 3.1 Each holomorphic function / : Ll -+ Ll is non expansive in the metric space (Ll, kll ), i.e.,
(3.1)
for all z, wEll.
From now on we let D be a bounded convex domain in a complex Banach space (X, 11·11). We will use the following two equivalent definitions ([43) , [138) , [172)) of the Kobayashi distance kD on D ([109) , [110)).
The first definition is, in fact, that of the Lempert function ti ([138); see also [82)).
kD (x,y) = tiD (x,y)
= inf {kll (0, {) : there exists / E H(Ll, D) s.t. / (0) = x and / ({) = y} .
To arrive at the second definition we first define the function aD : D x X -+ 1R+ by
aD (x, v) = inf {'1/ > 0: 3 g E H(Ll, D) with g (0) = x, g' (0) '1/ = v}
for all xED and vEX. The function aD is called the infinitesimal Kobayashi pseudometric for D. Directly from its definition we get the equality
aD (x, tv) = taD (x, v),
and the following lemma.
t > 0, (3.2)
Lemma 3.2 1/ / : Dl -+ D2 is a holomorphic mapping between the bounded convex domains Dl C Xl and D2 C X 2, then
aD. (J(x) ,/' (x) v) ~ aDl (x,v)
for all x E Dl and v E Xl.
Given x,y E D, we now consider the family of all curves;Y: [0,1]-+ D that join x and y and have piecewise continuous derivatives. We call such curves admissible and define
Then we have
kD (x,y) = inf{L(;Y) :;Y is admissible with ;yeO) = x and ;Y(1) = y}.
442
This second definition means that the Kobayashi distance kD for a bounded convex domain D is the integrated form of 0D [172J.
Our next lemma follows directly from the definition of the Lempert function.
Lemma 3.3 [85], [135J. Let D be a bounded convex domain in a Banach space (X, /1./1).
(i) Ifx,y,w,z E D and S E [0, IJ, then
kD (sx + (1- s)y,sw+ (1- s)z) ::; max [kD(x,w) ,kD (y,z)J;
(ii) ifx,y E D and s,t E [0, 1J, then
kD (sx + (1- s)y,tx+ (1- t)y)::; kD (x,y).
Another immediate consequence of the first definition is the following important extension of (3.1): If DI and D2 are bounded domains in the Banach spaces (Xl, /I . III) and (X2, II . 112), respectively, and kDI and kIJ, are the Kobayashi distances on DI and D2 ,
respectively, then each holomorphic f : DI ....... D2 is nonexpansive, i.e.,
kDI (I(x), fey)) ::; kD2 (x, y) (3.3)
for all x, y E DI.
As a matter of fact, Kobayashi's original definition [109J applies to all domains, not just to the convex ones, and leads to a Schwarz-Pick system which, of all Schwarz-Pick systems, assigns the largest possible pseudodistance to each domain [68].
We also recall at this point that all the pseudodistances assigned to a convex domain D by Schwarz-Pick systems coincide ([43], [138], [144J; see also [82]). In the special case of a bounded convex domain D they are all equal, of course, to the Kobayashi distance kD. This unique distance is sometimes called the hyperbolic metric on D.
The following theorem shows that the Kobayashi distance kD is locally equivalent to the norm /I. /I. We denote by dist II. II (x, aD) the distance in (X, II· /I) between the point x and the boundary aD of the domain D and by diamll.1I D the diameter of D in (X, 11·11).
Theorem 3.4 [68J. If D is a bounded convex domain in a Banach space (X, /I. /I), then
for all X,y E D and
argtanh ( ~Ix - y/l ) ::; kD(X,y) dlamll·IID
( IIx - yll ) kD(x,y) ::; argtanh d· (aD)
1stll.11 x,
whenever IIx - yll < distll.lI(x, aD).
A subset C of D is said to lie strictly inside D if distll.II(C, aD) > O. Thus any closed subset C lying strictly inside a bounded convex domain D is complete with respect to
k D •
Theorem 3.5 [68J. Let D be a bounded convex domain in a Banach space (X, II . II)· A subset C of Dis kD-bounded if and only i/C lies strictly inside D.
Fixed points of holomorphic mappings 443
We continue with the following simple lemma on sequences of holomorphic mappings. We denote the norm closure of a subset C by C.
Lemma 3.6 [83], [123]. Let Dl and D2 be bounded convex domains in the Banach spaces (Xl, 11·111) and (X2, 11·112), respectively, and let N be a norming set for (X2, II . 112). If {f>.hEJ is a net of holomorphic mappings f>. : Dl --> D2 which is pointwise convergent in the topology u(X2,N) to a mapping f : Dl --> D2 and there exists a point Zo E Dl such that Wo = f(zo) E D2, then f maps Dl holomorphically into D2.
Proof. By Corollary 2.11, f maps Dl holomorphically into D2. Next, we observe that the mapping
gk = ~wo + (1 -~) f (k E N) transforms Dl into Ak = iwo + (1 - i)D2, and that the set Ak lies strictly inside D2. Finally, let z be any point in Dl. Since
kD2 (wo, gk (z)) = kn, (gk (zo) ,gk (z)) ::; kDl (zo, z),
the sequence {gk (z)} lies strictly inside D2 by Theorem 3.5. Therefore
f (z) = limgk (z) k
is an element of D2 and this completes the proof of Lemma 3.6. • We end this section by studying properties of certain sequences in (D, k D). First we observe that in analogy with the norm, the Kobayashi distance is lower semicontinuous with respect to a suitably chosen topology.
Theorem 3.7 [83], [117] , [123]. Let X be a Banach space, N a norming set for X and let D c X be a bounded convex domain such that its norm closure 15 is compact in u(X,N). If {X/3}/3EJ and {Y/3}/3EJ are nets in D which are convergent in u(X,N) to x and y, respectively, and x, y E D, then
kD (x, y) ::; liminf kD (x/3, Y/3) .
If in place of the nets {X{3}{3EJ and {y{3}{3EJ we have sequences {xkhEN and {YkhEN, then the compactness of15 in u(X,N) can be replaced by its sequential compactness in u(X,N).
Proof. Without loss of generality we may assume that
sup kD (x/3, Y(3) < 00. {3
Let {/hhEJ' be an ultranet in J ([14], [50), [87]) such that
limkD (x{3)..,Y{3)..) = liminf kD (x/3,Y/3)· ~ /3
Let € > O. Then there exist holomorphic functions f(3).. : ~ --> D and points 'Y(3).. in the open unit disc ~ such that
444
for each >. E J'. Hence
s~p k~ (0, 'YPJ ~ [s~p kD (xp~, YP~)] + E < 00.
Since {,B~hEJ' is an ultranet in J, 15 is compact in the topology a (x,N) , and
x = a (X,N) -limxp~ = a (X,N)-limfp~ (0), ~ ~
Lemma 3.6 implies the existence of a holomorphic mapping f : D. --> D and a point 'Y E D. such that
a (X,N)-limfp~ (z) = f (z) ~
for each zED., and 'Yp" --> 'Y. These observations and Theorem 3.4 yield ~
f (0) = x,
k~ (0, 'Y) = lim k~ (0, 'YPJ < lim kD (xp" , ypJ + E = lim inf kD (xp, yp) + E < 00, ~ - ~ p
lim k~ (-y, 'YPJ = 0, ~
limsup [argtanh (1Ifp~ (~) - fp~ ('YpJII)] ~ d1amll'lI D
~ limsupkD (Jp~ (-y) ,fp~ (-ypJ) ~ limk~ (-Y,'YPJ = 0, ~ ~
and finally,
f(-y) -y = a(X,N)-lim[fp~ (-y) -yp~l = a(X,N)-lim[fp~ (-y) - fp~ ('YpJl = 0. ~ ~
This means that
kD (x, y) = kD (J (0) ,J (-y)) ~ k~ (0, 'Y)
and
kD (x, y) ~ liminf kD (xp,yp) + E
for each f > 0. Therefore kD(x, y) ~ lim infp kD(Xp, yp), which completes the first part of the proof. If in place ofthe nets {xp}PEJ and {yp}pEJ we are given sequences {xkhE]\[ and {YkhE]\[, and in place of the compactness of 15 in a(X,N) we have its sequential compactness in a(X,N), then we can repeat our earlier arguments taking into account the following facts: (D., k~) is a separable metric space, the distance k~ is locally equivalent to the Euclidean metric and holomorphic mappings are nonexpansive with respect to Kobayashi distances. We also replace the ultranet technique by the usual diagonal procedure. •
Remark 3.8 Theorem 3.7 can be used to show that Lemma 3.6 has a nonexpansive analog.
We now replace the general bounded convex domain D with the open unit ball in X. We begin with the definition of the Kadec-Klee property with respect to a(X,N).
Fixed points of holomorphic mappings 445
Definition 3.9 [17], [58].Let N be a norming set for a Banach space (X, 11·11). The space (X, II . II) is said to have the Kadec-Klee property with respect to a(X,N) if for every sequence {Xk} in X the following implication holds:
IIXkl1 ::; 1 } sep{xd = inf {lIxk - Xjll : j 'I k} > 0 => Ilxll < l. a (X,N)-limk->ooxk = x
Lemma 3.10 [83], [123]. Let N be a norming set for a strictly convex Banach space (X, 11·11) and let B denote the open unit ball in (X, 11·11). Assume that Ii is sequentially compact in the topology a(X,N) and (X, 11·11) has the Kadec-Klee property with respect to a(X,N). IJ{xk} and {Yk} are sequences in B, {Xk} is strongly convergent to ~ E DB, and
sup{kB(Xk,Yk): k = 1,2, ... } = c < 00,
then {Yk} is also strongly convergent to ~.
Proof. We can assume that the sequence {Yk} is a(X,N)-convergent to 7]. If ~ = 7],
then {Yk} is strongly convergent to ~ by the Kadec-Klee property of X. Assume ~ 'I 7].
In this case by the strict convexity of B each point of the open segment
(~, 7]) = {z EX: z = s~ + (1 - s) 7] for some 0 < s < I}
lies in B. Next, for 0 < s, t < 1 the sequences {SXk + (1- S)Yk} and {txk + (1- t)Yk} tend in a(X,N) to s~ + (1- s)ry and t~ + (1- t)7], respectively. Now, applying Lemma 3.3 and Theorem 3.7, we get
kB (s~ + (1 - s) 7], t~ + (1- t) 7]) ::; liminf kB (SXk + (1 - s) Yk, tXk + (1 - t) Yk) k->oo
::; lim sup kB (Xk, Yk) ::; c < 00 k->oo
for all s, t E (0,1). By Theorem 3.5 this means that (~, 7]) lies strictly inside B, which is impossible since ~ E DB. •
In a similar way we can prove the following lemma.
Lemma 3.11 [85]. Let D be a strictly convex bounded domain in a Banach space (X, 11·11). Let {xd and {Yd be two sequences in D which converge in norm to ~ E DD and to 1] E 15, respectively. If
then ~ = 1].
To prove the last result of this section we will need still another lemma.
Lemma 3.12 [84]. Let X be a Banach space with the open unit ball B. Let {wd be an arbitrary sequence in B with limk Wk = ~ E DB and Wk 'I 0 for k = 1,2, ... . Let 0 < a < 1, and let Projk denote, for each k, the projection of X onto the one-dimensional space line Wk) generated by Wk given by
446
where lk is a linear functional with Illkll = 1 and lk(wk) = IIWkli. If
Zk = Projk (a~) ,
then
Zk E B, lim zk=a~, lim kB(zk,a~)=O and lim (Zk,wk)=a, k--+oo k--+oo k--+oo
(3.4)
where for wE X, W # 0, 81. S2 E C and VI = (sdllwll)w, V2 = (s2/1Iwll)w, the scalar product (Vl,V2) in lin(w) is defined by (Vl,V2) = 8182.
Proof. It is obvious that Zk E B. We also have
Hence we get limk->oo kB(Zk, a~) = O. Next, taking ak = alk(~) we obtain
Therefore
• Theorem 3.13 [84). For every ~ E 8B and 0 < a < 1 we have
lim [kB (a~,w) - kB (O,w») = -kB (O,a~). w->!
(3.5)
Proof. Let us observe that
kB (0, w) = argtanh IIwll = .! log 1 + IIII willi = -21 log (1 + Ilwllt (3.6) 2 1 - w 1 - Ilwll
for wEB, and that
1 1- a 2 kB (O,a~) = argtanh lIa~1I = -2 log ---2.
(1 - a) (3.7)
Now let us take an arbitrary sequence {Wk} in B with Wk # 0 for all k and limkwk =~. For each k, let Projk denote the projection of X onto the one-dimensional space lin(wk) given in Lemma 3.12. If Zk = Projk(a~), then by (3.4) we get Zk E B, limk Zk = a~, and limk kB(Zk, a~) = O. Therefore, in place of (3.5), we need to prove that
lim [kB (Zk, Wk) - kB (0, Wk») = -kB (0, a~) . k->oo
(3.8)
Since for each k, the points Zk and Wk belong to lin(wk), we have
Fixed points of holomorphic mappings 447
By (3.6) and (3.9) we get
kB(Zk, Wk) - kB(O, Wk)
1 {II - (Zk, wk)1 + [11 - (Zk' wk)12 - (1 - IIZkIl2) (1 - IIWkIl2) 1 ~} 2
= "2 log (1-lIzkIl2)(1 + IIwkl1)2 '
and by (3.4) and (3.7) this inequality implies that
. 1 (1- Qi hm [kB (Zk, Wk) - kB (0, Wk) 1 = -2 log 2 = -kB (0, a~) .
k-+oo 1- a
Thus the formula (3.8) is indeed valid. • 4. The Kobayashi distance on the Hilbert ball
Let BH denote the open unit ball of a complex Hilbert space (H, (., .)). For each a E BH consider the Mobius transformation defined by
where x E BH, Pa is the orthogonal projection of H onto lin (a) = {Aa : A E IC} and Qa = 1- Pa. Each Ma is a kBH-isometry and has a norm continuous injective extension from BH onto itself. We also note that Ma (0) = a, M;;l = M_a, Mo = I, and for every a, b E BH the mapping Mb a M_a takes a to b. Applying these properties of Mobius transformations we get the following explicit formula for kBH' namely,
1
kBH (x, y) = argtanh (1 - u (x, y))2 ,
where x, y E BH and
(see [60J , [61]).
Theorem 4.1 The metric space (BH' kBH) has the following properties:
(i) [61]. For each pair of distinct points x,y E BH, there exist a unique geodesic line passing through them and a unique geodesic segment [x, yJ joining them. For each O:S t:S 1 there is auniquepointz = (l-t)xE9ty satisfyingkBH(x,Z) = tkBH(X,y) and kBH(Z,y) = (1- t)kBH(X,y).
(ii) [60J. If x, y, wand Z are in BH and ° :S t :S 1, then
k BH «1 - t) x E9 tw, (1 - t) x E9 tz) :S tkBH(W, z)
448
and
Strict inequalities occur if 0 < t < 1 and the relevant points do not lie on the same geodesic.
(iii) [60]. For any a E BH and R > 0, the open ball
B(a,R) = {x E BH: kBH (a,x) < R}
is the "ellipsoid"
where
(iv) [121], [122]. If the sequences {Xk} and {Yk} of elements of BH are weakly convergent to x E BH, then for any Y E BH we have
CIM U(Yk, y) = U (x, y) CIM 1 - IIYkll 2 ,
U(Xk, x) 1 - IIXkll2 where CIM denotes any of the following limits (the same on both sides of the above equality),
1· . f U(Yk, y) imill -(--)'
U Xk,X
whenever it makes sense.
(v) [61], [122]. All the holomorphic automorphisms of (BH, kBH) are weakly continuous.
(vi) [61], [122]. The metric space (BH,kBH) has the Opial property [145] with respect to the weak topology in H, i.e., if a bounded sequence {Xk} in (BH,kBH ) tends weakly to x, then
limsupkBH (Xk,X) < lim sup kBH (Xk,Y)
for each Y E BH \ {x}.
(vii) [60], [61]. The metric space (BH,kBH ) is kBH-uniformly convex, i.e.,
kBH (a,x) ::; R } 1 1 kBH (a,y)::; R ~ kBH (a, 2'xEB 2'Y) ::; (1- 6(R,E))R,
kBH (x,y) 2 ER
where 6( E, R) > 0 for all R > 0 and 0 < E < 2, or equivalently,
kBH(a,X)::;R} 1 1 kBH (a,y) ::; R ~ kBH (a, 2'x EB 2'Y) ::; argtanh kBH (x,y) 2 ER
tanh2 (R) - tanh2(~)
1 - tanh2(,:)
FiJ;ed points of holomorphic mappings 449
(viii) [122]. The metric space (BH, kBH) is linearly uniformly convex, i.e.,
kBH (a, x) ~ R } (1 1) kBH(a,y)~R ,*kBH a'"2x+"2Y ~(l-'h(a,R,€))R, kBH (x,y) ~ ER
where ol(a,R,€) > 0 for all R > 0 and 0 < € < 2.
(ix) [150]. Let {Xk} and {Zk} be two sequences in BH and let R ~ O. Suppose that for some yin BH, limsuPkkBH(xk,Y) ~ R, limsuPkkBH(Zk,y) ~ R, and liminfk kBH«Xk + zk)/2,y) ~ R. Then limk IIXk - zkll = limk kBH(Xk, Zk) = o.
We say that a subset C of BH is a kBH-Chebyshev set if to each point x in BH there corresponds a unique point Z in C such that
In this case we define the nearest point projection rc : BH -> C by assigning Z to x. The following result is an application of Theorem 4.1.
Theorem 4.2 (i) [122]. Every kBH -closed and convex subset of BH is a kBH -Chebyshev set.
(ii) [61]. Every kBH-closed and kBH-convex subset C of BH is a kBH-Chebyshev set. Moreover, the nearest point projection rc : BH -> C is a kBH-nonexpansive mapping and for each x E BH \ C, Y E [x,rc(x)] \ {x} and Z E C we have rc(x) = rc(y) and kBH(Z,y) < kBH(Z,X),
The next lemma will be essential when we look for common fixed points of holomorphic mappings in Bli (Section 7).
Lemma 4.3 [181]. Let {XO}oEJ be a kBH-unbounded net in BH satisfying
sup {kBH (xo ,x/3) - kBH (x,xf3)} = R < 00 a5f3
for some x E B. Then there is a point ~ E 8B such that ~ = limo Xa.
Proof. [28]. It is obvious that the point x can be replaced by the origin. We first note that limo kBH (0, xo) = 00. Otherwise, there would exist RI > 0 and, for each index i E J, an Qi ~ i such that kBH(O,xoJ ~ RI. But then, for all i, we would have,
kBH (0, Xi) ~ kBH (0, XaJ + kBH (Xo;, Xi) ~ 2kBH (0, XaJ + R ~ 2RI + R,
contradicting the kBH-unboundedness of {Xa}oEJ. Now, without loss of generality we can assume that Xa f 0 for all Q E J. Then there exists 0 < r < 1 such that for each
r Za = -lixall Xa, Q E J,
we have kBH (za, x"') = kBH (0, xa) + R. This implies
kBH (M- zf3 x o , M-zf3xf3) = kBH (xa, xf3) ~ kBH (0, xf3) + R
= kBH (zf3,xf3) = kBH (O,M-zf3 x f3)
450
for a :S (3, or equivalently,
11 - (M-zpxa,M-zpXfJW:s l-IIM_zpXaI1 2
for a :S (3. Next, we observe that for Ilx",1I ~ r we have
and therefore
min kBH (M_zx"" 0) = kBH (x"" 0) - R II zll=r
lim (min IIM-zx",lI) = l. '" IIzll=r
Now, let {X"';,hlEJl and {XfJj,}i2 Eh be two subnets of {Xa}",EJ which are weakly convergent to 6 and 6, respectively. Then we obtain
1 = lim lilll (M- zp . x",. , M-z{3. xfJ. ) )1 32 J2 3I J2 32
and finally we have
(6,6) +r r(6,6) + l'
0=1- (6,6)+r =(I-r) 1-(6,6) r(6,6)+1 r(6,6)+1
which implies 6 = 6 E aBo This concludes the proof of Lemma 4.3.
5. Fixed points in Banach spaces
•
The basic fixed point theorem for holomorphic mappings in Banach spaces is due to C. J. Earle and R. S. Hamilton [46]. The one-dimensional and finite-dimensional cases were proved earlier by J. F. llitt [171] and H. J. Reiffen [170], respectively.
Theorem 5.1 [46]. Let D be a bounded convex domain in a Banach space (X, II . II). If a holomorphic f : D --> D maps D strictly inside itself, then there exists 0 :S s < 1 such that
kD (I(x), f(y)) :S SkD (x, y)
for all x, y E D, and therefore f has a unique fixed point. Moreover, for any x in D the sequence of iterates {/k(x)} converges to this fixed point.
Proof. Suppose that IIxll < R for all xED. Let f> 0 be such that
distll'lI (I(D), aD) > f
and set t = f/(2R). Choose Xo E D and let 9 : D --> X be defined by
9 (y) = (1 + t) f(y) - tf (xo) = f(y) + t(f(y) - f (xo))
for each y E D. Since for any y E D,
t IIf(y) - f (xo) II :S t(llf(y)1I + IIf (xo)lI) :S 2tR = f,
Fixed points of holomorphic mappings 451
we see that 9 (D) c D. Hence by Lemma 3.2,
aD (g (x) ,g' (x) v) :::; aD (x,v)
for all v EX. Since
9 (xo) = f (xo) and g' (xo) = (1 + t) f' (xo) ,
we also have by (3.2),
aD (J (xo) ,f' (xo) v) = aD (g (xo) , 1 ~ / (xo) v )
= _l_aD(g(xo),g'(xo)v) < -11 aD (xo,v). l+t - +t
Since Xo E D was chosen in an arbitrary way, the second definition of Kobayashi's distance shows that for s=I/(I+t), we have
kD (f(x), f(y)) :::; SkD (x, y)
for all x,y E D. Therefore f has a unique fixed point by Banach's fixed point theorem. Moreover, for any x in D, the sequence of iterates U"'(x)} converges to this fixed point .
• Remark 5.2 As a matter of fact, the proof of Theorem 5.1 shows that it holds for all bounded domains [46).
Corollary 5.3 [71). Let D be a bounded convex domain in a Banach space (X, II ,11) with 0 ED. Suppose f : D ...... D is holomorphic, and for a E t!.. let
z (a) = af (z (a))
be the unique fixed point of af. Then the mapping z : t!.. ...... D is holomorphic.
Proof. It is sufficient to observe that
and that each z'" is holomorphic. • Before stating the next fixed point theorem we recall briefly the definitions and results of R. Ryan [175) which are needed in its proof. For 1 :::; p < 00 let CP(T) denote the space of measurable complex-valued functions defined on the unit circle T = at!.. for which
.cOO(T) denotes the space of measurable and essentially bounded complex-valued functions on T. If f E .cOO(T), then Noo(f) = esssup If(eiO)I. If (X, II . II) is a separable Banach space, then .c~(T) denotes the space of functions defined a.e. on T with values in X which are measurable in the norm topology of X and such that Ilf011 E .cP(T). Next, for f E .c~(T) denote
452
For x E X and I E X', the dual of X, denote by (x,l) the value I(x) of the norm continuous linear functional I at the point x. Since X is separable, a function f from T to X is measurable if and only if ([(·),1) is measurable for every IE X*. Let £~. denote the space of functions which are defined a.e. on T with values in X*, and such that (x, f(·)) is measurable for each x E X and Ilf(-) II E £P(T). Once more the separability of X implies that IIf(-)11 is measurable if (x, f(·)) is measurable for all x EX. In general, £~. C £~., but we have equality if X is reflexive because then
X' is separable too. If f E £~:, then (x,f(')) E £P(T) for each x E X and the weak integral or Pettis integral of f i~ the bounded linear functional on X given by
The following two theorems are due to R. Ryan [175].
Theorem 5.4 [175]. Let (X, II . II) be a separable Banach space and f E £~. (T) for
1 ::; p ::; 00. If (p).fo27r einO f(eio)dO = ° for n = 1,2, ... , then Np(iT) ::; Np(f) for 0< r < 1, where
j(z) = ~(P) r f(t) dt 2m J1tl=1 t - z
and iT (eiO ) = i (reiO ). Furthermore, NP(jT - f) -+ ° as r -+ 1- and ir(eiO ) -+ f(e iO ) a.e. in the norm of X'. If f is norm continuous, then ir(eiO ) -+ f(e iO ) uniformly for o E [0,271"].
Theorem 5.5 [175]. Suppose that X is a separable Banach space with dual X*. Let i be holomorphic in ~ with values in X' and assume Np(jr) ::; 1 for 0< r < 1, where 1 < p ::; 00 and lr(eiO ) = j(reiO). Then there exists a function f E £~: (T) such that
i(z) = ~(P) r f(t) dt 27rl J1tl=1 t - z
and furthermore, Np(f) ::; 1, (P) J~7r einO f(eio)dO = ° for n = 1,2, ... , and f is uniquely determined modulo a set of measure zero.
Now we can state and prove the following remarkable theorem.
Theorem 5.6 [71]. Let B be the open unit ball of a separable and reflexive Banach space X and let f : B -+ B be norm continuous on Band hoi om orphic in the open unit ball B. Then for almost all 0 in [0,271"] the mapping eiO f has a fixed point in B.
Proof. For lal < 1 let
z (a) = af (z (a))
be the unique fixed point of af. By Corollary 5.3 the function {z(a)}"ELl. is holomorphic. Since
Fixed points of holomorphic mappings 453
we have Np (zr) ::; 1 for 0 ::; r < 1 and p > 1. Since X is reflexive and since the function z has values in X, the dual of X·, by Theorem 5.5 there exists a function
Y E .c~:. (T) = .c~ .. (T) = .c~ (T)
such that
z(w) = -21 . (P) r yet) dt, Np(Y)::; 1 and 7r~ J1tl=1 t - w
for n = 1,2, ... Now we can apply Theorem 5.4 to the function y E .c~ (T) and conclude that
a.e. in the norm of X. We complete the proof by noting that the fixed points of reiO f are precisely z (reiO ) and that these have a limit as r -> 1-, so that y (eiO ) is a fixed point of eiO f. •
The following simple example shows that Theorem 5.6 fails in the nonreflexive separable space co.
Example 5.7 [71]. Let X = Co with the supremum norm. Let f : X -> X be defined by
f (Zl, Z2, Z3, ... ) = (~, Zl, Z2, Z3, .. .) .
It is obvious that f is holomorphic, feB) C B and that for no () in [0,27r] does the function eiO f have a fixed point.
Remark 5.8 It seems that in the setting of Theorem 5.6 no example is known of a fixed-point-free mapping f.
As we have already seen in Corollary 5.3, the Earle-Hamilton theorem (Theorem 5.1) is sometimes useful even if f does not map D strictly inside itself. Indeed, if f is any holomorphic self-mapping of a bounded convex domain D, then Theorem 5.1 shows that for each t E [0,1) one can define a holomorphic mapping hf{t,·) : D -> D by
hf(t,x) = (l-t)x+tf(hf(t,x», xED. (5.1)
In other words, hf(t,·) coincides with the so-called nonlinear resolvent
Jr = (I +r(I - f)-l, (5.2)
where r = l':t. These resolvents are used, for instance, in the proofs of the following results.
Theorem 5.9 [91], [161]. Let D be a bounded convex domain in a Banach space X and let f : D -> D be holomorphic. Suppose that f admits a uniformly norm continuous extension to 75 which satisfies the following boundary condition:
IIx - f(x)1I ~ f > 0
454
for all x E 3D. Then f has a unique fixed point.
See [91] for an application of this theorem to operator theory on spaces with an indefinite metric.
As a matter of fact, the resolvent method is also applicable to a wider class of holomorphic mappings which are not necessarily self-mappings of D. This wider class consists of those mappings of the form h = 1- g, where 9 : D --> X is a holomorphic mapping for which the resolvent Jr = (I + rg)-l is a well-defined holomorphic self-mapping of D for each r 2: O.
It is shown in [161] (see also [162] and [164]) that for a bounded convex domain D, a holomorphic mapping g, which is bounded on each subset strictly inside D, has this property if and only if it is the infinitesimal generator of a semigroup of holomorphic self-mappings of D (see Section 15). When D is the open unit ball B in X, this class can be described in terms of the numerical range of holomorphic mappings defined by L. A. Harris [67]. More precisely, let D be a convex domain in a Banach space X and suppose D contains the origin. Let h : D --> X be a holomorphic mapping. For x E 3D, let J(x) be the set of all norm continuous linear functionals which are tangent (in the real sense) to D at x, i.e.,
J (x) = {I E X* : l(x) = 1, Re( l(y)) ::; 1 for all y E D}.
If hE H(D,X) has a norm continuous extension to D, then the numerical range of h is the set
V(h) = {l(h(x)) : I E J(x), x E 3D}.
It is shown in [69] that for a holomorphic mapping h : B --> X, the nonlinear resolvent Jr = (I + r (I - h))-l, r 2: 0, is a well-defined holomorphic self-mapping of B for each r 2: 0 if and only if
L(h) = lim sup Re( V (hs )) ::; 1, 8---+1-
where hs(x) = h(sx) for s E [0,1) and x E B.
Theorem 5.10 [69]. If h: B --> X is holomorphic and L(h) < 1, then h has a unique fixed point.
Finally, we mention a related result in Hilbert space.
Theorem 5.11 [12], [161]. Let BH be the open unit ball of a Hilbert space H and let h : BH --> H be a holomorphic mapping in BH which has a norm continuous extension to B II. If L (h) ::; 1, then h has at least one fixed point in B H . Moreover, if the set of interior fixed points of h is nonempty, then it is an affine subset of BH.
The proof of this result is based on Theorem 6.4 in the next section which is devoted to a detailed study of fixed points in B H .
6. Fixed points in the Hilbert ball
inxxFixed point, in the Hilbert ball Let BH be the open unit ball of a Hilbert space H. Applying the asymptotic center method due to M. Edelstein [48] and the properties of kBH we get the following fixed point theorem.
Fixed points of holomorphic mappings 455
Theorem 6.1 [61], [122] , [130]. Let f : BH -t BH be a holomorphic mapping or more generally, a kBH -nonexpansive mapping. Then the following statements are equivalent:
(i) f has a fixed point;
(ii) there exists x E BH such that {!k(x)} lies strictly inside BH (this means that {!k(x)} is kBH-bounded);
(iii) there exists a ball B(x,r) in (BH,kBH ) which is f-invariant;
(iv) there exists a nonempty, kBH-bounded, kBH-closed and kBH-convex subset of BlI which is f -invariant;
(v) there exists a nonempty, kBH-bounded, kBH-closed and convex subset of BH which is f -invariant.
If f : BH -t BH has a fixed point, then the approximating curve (cf. Corollary 5.3 and (5.1))
[0,1] :3 t ---> Zo (t) = tf (zo (t))
behaves very well:
Lemma 6.2 [61] If a kBH-nonexpansive f : BH -.,> BH has a fixed point, then the approximating curve zo(t) = tf(zo(t)), 0 ::; t < 1, tends strongly, as t --> 1-, to the unique fixed point of f with the smallest norm.
We also know the structure of the fixed point set of a kBH-nonexpansive (holomorphic) mapping.
Theorem 6.3 [61], [173], [174].
(i) The fixed point set of a kBH-nonexpansive mapping f : BH -.,> BH is kBH-closed and kBH-convex and therefore it is a kBH-nonexpansive retract of B H .
(ii) The fixed point set of a holomorphic mapping f : BH -t BH is affine and therefore it is a holomorphic retract of B H .
Proof. (i) It is sufficient to apply Theorems 4.1 and 4.2.
(ii) Let C be a maximal affine subset of Fix(f). Suppose that z E Fix(f) does not belong to C and let w E C. Then the metric segment [w, z] is contained in Fix(f) by (i). Since f is holomorphic, it follows that the entire one-dimensional disc spanned by z and w in BH is contained in Fix(f). Thus the affine set spanned by z and C is contained in Fix(f). This contradicts the maximality of C. Hence C = Fix(f) and the proof is complete. •
When we study the possible existence of a fixed point of a norm continuous mapping f : BH --> BI{ which is holomorphic inside BH, then the following theorem will turn out to be very useful.
456
Theorem 6.4 [56], [60] , [173] , [174] , [214]. If a kBH -nonexpansive mapping f : BH ---t
BH is fixed-point-free, then there exists a unique point ~f of norm one such that all the "ellipsoids"
E(C R)={XEB :ll-(X'~f)12<R} R 0 "f, H 1 _ IIxl12 ' >,
are invariant under f, E( ~f' R) n BB H = {~f} and for every x E B H there exists R > 0 such that x E E(~f,R). Moreover, all the approximating curves defined by
Wa (t) = (1 - t) a Ell tf(wa (t)) and Za (t) = (1 - t) a + tf (za (t)),
t E [0,1), a E B H , converge strongly to ~f as t ---t 1-.
Remark 6.5 [56],[174]. If
<I> (x) = 11- (x,OI2 € 1 -llxl1 2
for x E BH and ~ E aBH, then for R > 0 the inequality <I>e (x) :s; R is equivalent to
where, R = 0/(1- 0).
Lemma 6.6 [150]. Let the point ~ belong to the boundary of B H , and let {xd and {zd be two sequences in B H . Suppose that for R 2:: 0, we have limsuPk<I>€(xk) :s; R, lim sUPk <I>€(Zk) :s; R, and lim infk <I>€((Xk + zk)/2) 2:: R. Then limk IIXk - zkll = o.
Proof. By Remark 6.5, we have
Using the parallelogram law, we see that
Therefore limk Ilxk - zkll = 0, as claimed. • Remark 6.7 [56], [601. If a kBH-nonexpansive mapping f: BH ---t BH is fixed-pointfree, then <I>e,(j(x)):s; <I>€,(x) for each x E BH.
Theorem 6.8 [61]. If a norm continuous mapping f : BH ---t BH maps BH into BH and is kBH -nonexpansive on BH, then it has a fixed point in BH.
Proof. If the kBH-nonexpansive mapping flBH : BH ---t BH is fixed-point-free, then by Theorem 6.4 there exists a unique point ~f of norm one such that all the ellipsoids
E(C R)={XEB : 11-(x'~f)12 <R} R>O "f, H 1 _ IIxII2 ' ,
Fixed points of holomorphic mappings 457
are invariant under fIBH'
n E(~f,R) = {~f}· R>O
This means that ~f must be a fixed point of f. • Corollary 6.9 [60]. If a norm continuous f : BH ...... BH is kBH-nonexpansive on BH and Fix(f) n BH i= 0, then the set Fix(f) n BH is weakly compact.
Observe that in the setting of Corollary 6.9 the set of fixed points in BH may be larger than the norm closure of the nonempty fixed point set of f in B H.
Example 6.10 [60]. Consider H = «::2 and the mapping f : BH ...... BH given by f(z, w) = (z, w 2 ). In this case Fix(f) = {(z,O) : Izl :s: I} U {(O, I)}.
Now we investigate the possible existence of common fixed points of commuting holomorphic mappings.
Theorem 6.11 [117], [118], [121]. If {/k}k=l is a finitefamily of commuting kBH -non expansive self-mappings of BH and if Fix(fk) is nonempty for each k = 1,2, ... m, then nk=l Fix(/k) is a nonempty kBH -nonexpansive retract of BH.
Proof. By Theorem 6.3, each Fix(/k) is kBH-closed, kBH-convex, and fk-invariant for 1 :s: k :s: m. To see that nk=l Fix(/k) is nonempty, it is now sufficient to apply Theorem 6.1 and induction with respect to m. Applying Theorem 4.2 we also see that this nonempty intersection is a kBH-nonexpansive retract of BH •
The next example shows that Theorem 6.11 is no longer valid if the family of commuting holomorphic mappings is infinite.
Example 6.12 [126]. Let BH be the open unit ball in a separable Hilbert space H with the orthonormal basis {ekhEN. With each affine subset Y = (x + Y) nBH of BH, where Y is a closed subspace of H and x E BH is the unique point of least norm in Y, we associate the nearest point holomorphic retraction of BH onto Y given by the formula
where Projy denotes the orthogonal projection onto Y. We construct the family {fd kEN in the following way: We choose a sequence of positive real numbers {OOk hEN such that this sequence is strictly decreasing, 0 < OOk < ~7f and limk-->oo OOk = O. We set
Xo = H, Co = BH, ao = 0, and Ro = IIe1 - aoll· For k = 1, the mapping r1 is the nearest point holomorphic retraction of BH onto the affine set C1 in B H , where
458
and
1 hI = IlgIlIgI,
al = ao + Ro [ G + ~ cos ell) e~ + G sin ell) hI] ,
RI = Ilq -alII,
XI={XEXo:x=al+Y and (al,Y)=O}
If we already have Ro,RI, ... ,Rk E JR, aO,al,a2, ... ,ak E H, e~,e~, ... ,e~ E H, gl,g2, ... ,gkEH, hl,h2, ... ,hkEH, XO,XI, ... ,Xk, Co,CI,···,Ckand rl,r2,· .. ,rk,whereeach Xi is an affine set in H, el,ai E Ci C Xi, em E Xi for m :2: i + 2 (i = O, ... ,k), Xi+1 C Xi for i = 0, ... , k - 1 and each ri is the nearest point holomorphic retraction of BH onto Ci for i = 1, ... , k, then the next holomorphic retraction rk+! is the nearest point holomorphic retraction onto
where
1 hk+1 = Ilgk+!lI gk+!,
ak+! = ak + Rk [ (~ + ~ cos elk+! ) e~+! + (~sin elk+!) hk+!] ,
Rk+! = Ilel - ak+!11 ,
and
Now it is sufficient to take
for k = 1,2, ... to get
Ik 0 1m = 1m 0 ik
for every k, mEN,
for each kEN,
lim diamll.11 Ck = 0, k~oo
and
n Fix (ik) = 0. k~1
Fixed points of holomorphic mappings 459
On the other hand, we have the following positive result when at least one of the holomorphic mappings is condensing.
Theorem 6.13 (126]. Let BH be the open unit ball in a Hilbert space H and let {joj",EJ be a family of commuting holomorphic self-mappings of BH. If Fix(f",) is nonempty for each a E J and at least one fa is condensing, then n.:xEJ Fix(fa) is nonempty.
Proof. It is sufficient to observe that Fix(f,,) is affine for each a E J (see Theorem 6.3), n k=l Fix(f"k) is nonempty and affine for an arbitrary choice of a finite number of indices and Fix(fa) is a finite-dimensional affine subset of BH because fa is condensing .
• The next example shows that in Theorem 6.13 we cannot replace the holomorphic mappings by kBH-nonexpansive mappings.
Example 6.14 [126]. If BH = b. is the open unit disc in H = <C, let Pc be the projection of BH onto the real segment C = (-1,1) defined by Pc(x + iy) = x, where x,y E lR and x + iy E B H. Next let Qk : (-1,1) -t (-1,1) be given by Qk(X) = max{x,l- n for x E (-1,1) and kEN. Then we set fk = QkPC for kEN, and we obtain ik 0 fm = fm 0 ik for every k, mEN, Fix(ik) = (1 - hi) for each kEN, and n~l Fix(fk) = 0.
We end this section by establishing a theorem about the existence of common fixed points of a commuting family of mappings on BH.
Theorem 6.15 [117], [121]. Suppose {j,,}aEJ is a commuting family of norm continuous mappings of BH into BH such that each fo: : BH -t BH is either a constant mapping or maps BH into BH and is kBH -nonexpansivc on BH. Then naEJ Fix(fo:) # 0.
Proof. We consider three cases.
Case 1. Fix(f"'k) n BH # 0 for each a E J. In this case the result is clear because the set
m n "'F""ix"""f "'-k-:::n~B"'Hk=l
is nonempty and weakly compact (see Corollary 6.9) for every choice of finite number of indices.
Case 2. One f!3 is a constant mapping: f!3 == c, where c E RH. Then the point c is the common fixed point of the whole family.
Case 3. Neither case 1 nor case 2 is valid. Then one mapping, say f!3' maps BH into BH and Fix(f!3) n BH = 0. By Theorem 6.4 there exists ~f(3 E 8BH such that
for each R > O. Let
460
for an arbitrary a E J. Then
and since Fix(f,B) = 0, the set Ca is unbounded in (BH, kBH)' This implies that
~ff3 E C" C BH
and therefore there exists a sequence {Xk} of points in E(~ff3' R) such that la(Xk) --> ~ff3' But then we also have Xk --> ~ff3 and hence la(~f{3) = ~f{3' •
7. Fixed points in finite powers of the Hilbert ball
It is not difficult to observe that the Kobayashi metric in the Cartesian product B'H of n open unit balls BH is given by
In this section N(B'H) will denote the class of all kB'H -nonexpansive self-mapping on B'k The class of those mappings in N(B'H) which have a continuous (in norm) extension to B'H will be denoted by C N (B'H ). It will also be convenient to consider the slightly more general class of mappings N (B'H) which consists of all norm continuous mappings
I: B'H --> B'H such that t/IBj, E N(B'H) for all 0 < t < 1 [118].
Note that when I E N(B'H) it may happen that I (x) E 8(E'H) for x E B'H. But then, if f (x) = v with
then
for all Y E B'H.
The next lemma is an extension of Lemma 4.3. It will be used in the proof of the main theorem of this section.
Lemma 7.1 [181]. Let {zo,JaEJ be a kBj,-unbounded net in E'H such that
sup [kBn (za, z,B) - kBn (0, z,B)] < 00. a~,B H H
Then there are indices 1 :S ]1 < i2 < ... < ]r :S n (1 :S r :S n) and points {el}t=l on aBH such that lor any I E CN(E'H) lor which there is an index ao with {kBj,(za,/(za))}azao bounded by 0 < R < 00, the face
K = {y E Bli: Yjl = el, ···,Yjr = er }
is f -invariant.
Proof. By passing to a subnet and reordering indices if necessary, we may assume that for some r with 1 :S r :S n we have
(7.1)
Fixed points of holomoryhic mappings 461
for 1 :::: j :::: 1', and
(7.2)
for l' + 1 :::: j :::: n. Then for 1 :::: j :::: l' and j3 2: a we get
for some M. Hence by Lemma 4.3 there are el, ... , er in 8BH such that lima(za)j = ej for j = 1, .. , .1'. If l' = n, then
is a fixed point of I by Lemma 3.10. So we may assume that l' < n. We will show that
K = {y E B'H : Yl = el, ... , Yr = er}
is I-invariant. To this end, fix (xr+1' ... , xn) in B7£-r and for each a let
We have
and hence
lim Va == (el' ... , er , Xr+I, ... , xn) == v, a
liml (va) = I (v) = W a
exists. We claim that Wj = ej for j = 1, ... , r. Indeed, for a 2: aD and 1 :::: j :::: l' we have
kBH G/(Va)j + ~(Za)j, (Za)j) :::: kBH((.t(Va)j, (za)j) :::: kB'h(f(va),Za) (7.3)
:::: kB'H(f(va),!(za)) +R
:::: max kBH(Xk, (Za)k) + R. r+ 1::'0 k::'On
Hence, if Wj =I ej and Wj E BH for some 1 :::: j :::: 1', then there would exist al such that
sup kBH(O,/(v",)j) < 00 62:a l
and therefore, by (7.1), (7.2) and (7.3), for a 2: al we would get the following contradiction:
00 f--- [kBH(O, Za) - max kBH(Xk, (za)k)] a:2:a, r+l::'Ok::'On
:::: kBH(O, za) - kBH((f(Va)j, (za)j) + R
:::: sup {[kBH (0, z"') - kBH (0, (z",)j)] + kBH (f(v"')j, 0) + R} < 00. &2:0!1
Next, if Wj =I ej and Wj E 8BH , then by (7.1), (7.2) and (7.3) this would lead to the following contradiction:
462
1 1 ::; kBH(O, z"') - kBH (2 I (v",)j + 2(z",)j, (z",)j) + R
1 1 ::; s~p {[kBH (0, za) - kBH (0, (za)j)]) + kBH (2 f (va )j + 2(Za)j, 0) + R
1 1 ---;;' s~p {[kBH (0, za) - kBH (0, (za)j)]) + kBH (2Wj + 2ej, 0) + R < =.
Thus Wj = ej for all 1 ::; j ::; r. • Remark 7.2 Note that the face K obtained in Lemma 7.1 depends only on the net {z"'}"'EJ and that its I-invariance results from the boundedness of{kBjf (za,J(za))}a2ao.
We begin to study the existence of fixed points in Bli with the following simple generalization of Theorem 6.1. Its proof uses induction with respect to n and is based on the asymptotic center method.
Theorem 7.3 [55], [117], [121]. Let I: Bli -> Bli be a holomorphic mapping or more generally, a kBjf -nonexpansive mapping. Then the following statements are equivalent:
(i) I has a fixed point;
(ii) there exists x E Bli such that {Jk(x)} lies strictly inside Bli (this means that {Jk(x)} is kBjf-bounded);
(iii) there exists a ball B(x,r) in (Bli,kBjf) which is f-invariant;
(iv) there exists a nonempty, kBjf -bounded, kBjf -closed and convex subset of Bli which is f -invariant.
Remark 7.4 An analogous theorem holds in a more general situation, where I: C ->
C is a kBjf -nonexpansive self-mapping of a nonempty convex and kBjf -closed subset of H'k
Here we quote a result from [118] (see also [117] and [119]).
Theorem 7.5 [118]. Let h, ... ,Jm be m commuting mappings in N(Bli) (H(Bli)) such that Fix(fj) =f. 0 for 1 ::; j ::; m. Then n~l Fix(fj) is a nonempty k Bjfnon expansive (holomorphic) retract of H'k
We omit the proof of this theorem because the existence of such a kBjf -nonexpansive retraction of Bli follows directly from a modification of Bruck's method [27] and we will use similar modifications in the proofs of Theorems 11.3 and 14.5. A different proof of the existence of a common fixed point can be found in [117].
Theorem 7.6 [181], [187]. Let C be a nonempty convex and kBjf-closed subset of Bli. Let {Ja}",EJ be a commuting family of kBjf-nonexpansive mappings I",: C -> C with a kBjf -bounded common invariant subset C1 . Then naEJ Fix(j",) =f. 0.
Proof. We will use the asymptotic center method and induction on n. First we make the following observation. Let {gj LEJ' denote the algebraic semigroup generated by {f",}aEJ via composition. Each j E .I' may be identified with a function from .I
Fixed points of holomorphic mappings 463
to the nonnegative integers which is zero except for a finite number of entries. This identification induces a natural order on J'. Fix x E C l and consider the function r(·, {gj(x)}jEl') : C -> [0,00) defined by
r (Y' {gj (X)}jEJ') = lim sup kBn (y,gdx )) JEY j''2j H
for y E C. It is easy to see that
r (ia(Y), {gj (x)} jEJ') :s: r (Y' {gj (x)} jEJ')
for each a E J and Y E C. In addition (see Lemma 3.3) we have
kB'jf GYI + ~Y2,gj' (x)) :s: max [kB'jf (Yl, gj' (x)),kB'jf(Y2,gj' (x))]
for all Yl, Y2 E C. Therefore, if
ro=inf{r(y,{gj(x)}jEY) :YEC}
is the asymptotic radius of {gj(x)}jEJ', then the asymptotic center of {gj(X)}jEJ',
{y E C: r(y,{gj (x)}) = ro},
is a nonempty, closed and convex invariant subset for {!a}aEJ, which by Theorem 4.1 is either a singleton if n = lor, after a possible reordering of indices, of the form {Xl} x 0, where 0 is a nonempty kBn-I-bounded closed and convex subset of B'lf- l
H if n > 1. In the first case we have found the desired common fixed point and in the second one we can use the induction hypothesis. •
Now we state and prove the main theorem of this section.
Theorem 7.7 [181]. A commuting family of mappings {fa}aEJ in N(B'H) has a common fixed point in B'H.
Proof. We use induction on n. For n = 1 see Theorem 6.15. For n > 1 we consider three cases.
Case 1. There exists ao E J such that
fao (B'H) c 8(B'H).
This means that there exist el, ... , er E 8BH and 1 :s: h < j2 < ... < jr :s: n such that
fao (BYj) C K = {y E 8(B'H) : YiI = el, · .. ,Yjr = er}.
We have
fa (fao (K)) = fao (fa (K)) c K
for each a E J. If r = n, then fao is a constant mapping and (el, ... , en) is a common fixed point of the family {fa}aEJ' If r < n, then the face K can be shown to be invariant under each fa and the result follows by induction.
Case 2. Each fa belongs to CN(B'H) and there exists fao for which
464
We construct a sequence {f~;,"(0)} = {zm} in the following way:
k1 = 1
and
The sequence {zm} is kBjf-unbounded and for j 2: m we have
We also have for each Q,
SUpkBjf (zm,/a(zm)) :s kBjf (0, fa (0)) < 00. m
Therefore by Lemma 7.1 we obtain a face of B'H which is invariant under {fa}aEJ (see Remark 7.2) and we can use induction to get a common fixed point of {!a}aEJ.
Case 3. Each fa belongs to CN(B'H) and
Fix (fa) n B n =J 0
for all Q. Let S be the set of all finite subsets of J. The set S is directed by inclusion. Then by Theorem 7.5 for each s = (Q1, ... ,Qk) E S there exists a kBjf-nonexpansive retraction
k
Ps : B'H ...... n (Fix (f aJ n B'H) . ;=1
Let us consider the net {PS(O)}SES. If {Ps(O)}SES is kBjf-bounded, then the asymptotic center
{Y E B'H : T (y, {Ps (O)}sES) = TO},
where
and
TO = inf {T (Y, {Ps (O)}SES) : Y E c},
is a nonempty, kBjf -bounded, closed and convex subset of Bir. Next, it is easy to observe that if Q E J and Q EsE S, then
fa (Ps (0)) = Ps (0)
and therefore the inequality
T (fa (x), {Ps (a)}) = lim sup kBj} (f" (x), Ps (0)) S
Fixed points of holomor-phic mappings 465
:::; limsupkBjf (x,Ps (0)) s
:::; r (x, {Ps (a)})
is valid for each a E J and each x E B'Jr, This implies that the asymptotic center is fa-invariant for each a. Hence by Theorem 7.6 there is a common fixed point in B'H.
Finally, let us assume that {Ps(O)}sEJ is kBjf-unbounded. For s',s" E S, s':::; s", we have
Next, for any singleton So = {ao} and for any s 2: So it is obvious that
faD (Ps (0)) = Ps (0)
and we may apply Lemma 7.1 to get an invariant face for {foJaEJ' The result now follows from the induction hypothesis. This completes the proof of Theorem 7.7. •
Corollary 7.8 [118], [131]. If f E N(B'H) , then Fix(j) f 0.
Remark 7.9 Earlier and different proofs of the existence of a fixed point for a single f E N(B'H) can be found in [118] and [131].
Remark 7.10 When we look at Example 5.7 and Corollary 7.8 and compare them with Theorem 5.6, we see that these are the two extreme cases which may occur in infinite-dimensional Banach spaces.
Remark 7.11 At this point it is worthwhile recalling that there exist two continuous and commuting mappings f,g : [0,1] -+ [0,1] such that Fix(j) n Fix(g) = 0 ([24], [76]).
8. Isometries on the Hilbert ball and its finite powers
By an isometry on (B'H, kBjf) we always mean a kBjf -isometric automorphism of B'H onto B'H. If 0 fee B'H, then a mapping f : C -+ B'H satisfying
for all x, y E B'H is called an isometric mapping. We begin this section with a complete characterization of kBH-isometric automorphisms.
Theorem 8.1 [68], [70], [79], [124], [146]
(i) Every isometry from BH onto BH is of the form I = U 0 M a, where Ma is a Mobius transformation with a E BH and U is either a unitary linear operator or an antiunitary linear operator from H onto H.
(ii) Every isometry from BH onto BH is of the form I = Ma 0 U, where Ma is a Mobius transformation with a E BH and U is either a unitary linear operator or an antiunitary linear operator from H onto H.
In (B'H, kBjf) , n 2: 2, we also have a nice situation.
466
Theorem 8.2 [64], [124]. If n 2': 2 and I : Eli -t Eli is a kB'H -isometry, then there exist a permutation 7r of (1, ... , n) and n kBH-isometries II, ... ,In such that
More information on isometric mappings between finite powers of finite-dimensional open unit balls can be found in [82].
Corollary 8.3 [124]. Every kBH-isometry I : EH -t EH transforms affine sets onto affine sets.
Corollary 8.4 [124]. Every kBf, -isometry I : Eli -t Eli has a unique norm continuous extension to Eli. We will call this unique norm continuous extension to Eli a kB'H-isometry I: Eli -t Eli.
Remark 8.5 [124]. As we already know, every kB'H-isometry I: Eli -t Eli is of the form
where 7r is a permutation of (1, ... , n) and II, ... , In are kBH-isometries. If 7r consists of one cycle (this kind of an isometry is called a cyclic isometry), then, after reindexing if necessary, we can write
Now we define the isometry
It is easy to see that a point (Xl, ... , x n) E Eli is a fixed point of I if and only if
Xl E Fix(r) and (Xl, ... ,Xn) = (xl,(I2o ... oIn)(xIl, ... ,In(xl)).
We now turn our attention to the structure of the fixed point sets of isometries.
Theorem 8.6 [70], [124].
(i) If I: BH -t EH is an isometry and Fix(liBH) = 0, then I has either one or two fixed points on aBH.
(ii) If n 2': 2 and I: Eli -t Eli is a cyclic isometry with Fix(IIB'H) = 0, then I has either one or two fixed points on (aEH)n.
Proof. By Remark 8.5 it is sufficient to prove (i). Suppose that I has at least two fixed points on aBo Without loss of generality we may assume that they are ~T and -~T' As we know, I is of the form I = MT(o) 0 U, where U is either a unitary operator or an antiunitaryone. Since I transforms affine sets onto affine sets, the set BH n lin (~T) is invariant under I and therefore I (0) = a~T with 0 < lal < 1. If
v = f3~I + wE Fix(I),
Fixed points of holomorphic mappings 467
where 1(11 :::; 1 and (~T, w) = 0, is another fixed point of Ion 8BH, then (U (w), ~T) = 0 and
(1~T + w = v = I (v) = (MT(o) 0 U)((1~T + w)
= MT(O)(U((1~T) + U(w)) ,--------
\h -III(0)11 2
= MT(O)(U((1~T)) + 1 + (U((1~T),I (0)) U (w)
V1 -III (0)11 2
= I((1~T) + 1 + (U((1~T),I (0)) U (w) .
This means that (1~T E Fix (I). Next, for 'Y E K we have
if U is unitary,
if U is anti unitary,
where U('Y~T) = ei°'Y~T if U is unitary and Ub~T) = ei°'Y~T if U is antiunitary. Since I(~T) = ~T and I( -~T) = -~T we get eiO = 1 and a = a > 0 (the last inequality is a consequence of the definition of ~T). This implies that IIBHnlin(~I) has exactly two
fixed points ~T and -~T. Thus (1 E {-I, I} and therefore B H 3 v = (1~T + w = (1~T E {-~T, ~T}. In other words, I has exactly two fixed points. •
If we have a kB'H -isometry I : BN --+ BII , then, as we know, it consists of cyclic isometries of
B'H (1:::; k :::; n)
and the fixed point set of I is equal to the Cartesian product of the fixed point sets of these cyclic isometries. As we saw above (Remark 8.5), each fixed point of a cyclic isometry I is of the form (a, (I2 0 ... 0 In)(a), ... ,In(a)), where a E Fix(I) and I = Il 0 ... 0 In. Thus we only need to characterize the fixed point sets of kBH-isometries of BH.
Theorem 8.7 [124]. If I: BH --+ BH is a kBH-isometry and Fix(IIBH) =F 0 is not a singleton, then for each a E Fix(IIBH) there exist an orthonormal basis {ej hEJ of H and 0 =F J' c J such that
{ Ma (BH n lin (ej)jEJ1)
Fix(I) =
Ma (BH n linjR (ej )jEJI)
if M_a 0 I 0 Ma is unitary
if M_ a 0 I 0 Ma is antiunitary ,
where linjR(ej)jEJI consists of all x in lin(ej)jEl' with real coefficients.
As an immediate corollary of Theorems 8.1, 8.2, and 8.7 we get the following fact.
Theorem 8.8 [124]. If I : BN --+ BN is an isometry and Fix(IIB'H) =F 0, then
Fix(IIBi) = Fix(I)
468
Remark 8.9 [124J. Theorem 8.8 does not hold for all mappings in eN(Bli). Indeed, consider z ---+ J(z) = z2 in E Then Fix(fILl) = {O}, but Fix(f) = {O, I}. See also Example 6.10.
Now let HJR be a real Hilbert space and let BH .... be the open unit ball in HJR. Then HJR and BH .... can be identified in a natural way with a subset of a complex Hilbert space H and with a subset of the open unit ball Bll, respectively. Thus the Kobayashi distance kBH in Bll may be restricted to BllR. We denote this restriction by k EH ..... There are at least three reasons to investigate (BllR , k BH.):
(a) there i~ an obvious connection between (BH .... ,kBH .... ) and the Klein model of hyperbolic geometry;
(b) the distance kBH .... in Bll .... is a projective invariant [111J;
(c) the distance kBHR in BllR has some metric properties which are different from the properties of the distance kBH in Bll.
It is easy to observe that the ball BHR is a kBH-closed and kBH-convex subset of (Bll,kBH ) and that geodesics in (BuR,kBH ) are simply linear segments with their endpoints on the boundary of B llR' R
As a direct consequence of Theorem 8.2 we get that each mapping U 0 Ma, where U is a unitary operator from H'II, onto H'II, and Ma is a Mobius transformation with a E B llR , is an isometry in (Bll .... , kBH .... ). In fact we know a bit more.
Theorem 8.10 [133].
(i) Every isometry from Bll .... onto BllR is of the form I = U 0 Ma, where Ma is a Mobius transformation with a E BllR and U is a unitary linear operator from H'II, onto H'II,.
(ii) Every isometry from BUR onto BllR is of the form I = Ma 0 U, where Ma is a Mobius transformation with a E BHR and U is a unitary linear operator from H'II, onto HR.
Corollary 8.11 [133J. If I is an isometry from BllR onto B ll .... , then it has a natural norm continuous extension to B ll .....
Corollary 8.12 [133]. If I is an isometry from B llR onto B llR and has no fixed point in B llR , then its norm continuous extension to BllR has either one or two fixed points on the boundary of BllR.
Corollary 8.13 [133]. If I is an isometry from BHR onto BllR with no fixed point in BllR and if its norm continuous extension to BHR has two fixed points on the boundary of B llR , then there exists a fixed point ~I E 8BHR of this extension such that for each x E BHR , the sequence of iterates {(I)k(x)} converges to ~I. This convergence is uniform, on each subset of BHR which lies strictly inside BUR'
The form of an isometry in BliR is analogous to the one in Bli.
Fixed points of holomorphic mappings 469
Theorem 8.14 [124]. If n ~ 2 and I: B'Ha -+ B'HR is a kBj}a -isometry, then there
exist a permutation 7r of (1, ... , n) and n kBH'lI. -isometries (Ih, ... , (I)n such that
Theorem 8.15 [124]. If I: BHR -+ BHa is a kBHR -isometry and Fix(IIBHR) =F 0 is not a singleton, then for each a E Fix(IjBHR) there exist an orthonormal basis {ej hEJ of Hand 0 =F J' c J such that
Fix(I) = Ma (BH n lin (ej)jEJ') .
9. The extension problem
We saw in the previous sections that some metric properties of (BH, kBH) are close to the corresponding proper,ties of the Hilbert space H. Therefore it is natural to ask whether the Kirszbraun-Valentine extension theorem ([52], [108], [177], [193]) is also valid in (BH,kBH). In the present section we give the answer to this question.
Theorem 9.1 [132]. If dimH = 1, 0 =F C c ~ and f : C -+ ~ is ka-nonexpansive, then there exists a mapping h : ~ -+ ~ which is a ka-nonexpansive extension of f.
Theorem 9.2 [124]. If dimH = 1, 0 =F C c ~ and f : C -+ ~ is aka-isometric mapping, then there exists a ka-isometry I : ~ -+ ~ which is an extension of f.
The following examples show that both Theorems 9.1 and 9.2 fail in higher dimensions.
Example 9.3 [124],[132]. In H = C2 consider the points al = (a,O), a2 = (ia,O), a3 = (-ia, 0), bl = (a',O) , b2 = (0, a) and b3 = (0, -a), where a, a' E (0,1) and
For r = arg tanh a we have
3
kBH(aj,ak)=kBH(bj,bk), j,k=I,2, OE nB(ak,r) k=1
3
and n B (bk, r) = 0. k=1
To obtain a counterexample it is sufficient to take C = {al,a2,a3} and f(aj) = bj.
We can construct similar counterexamples in B'iI = ~ x ~ C C x C = H2.
Example 9.4 [124], [132]. If
470
and r = ! arg tanh a, where 0 < a < 1, then we obtain
3 3
OE nB(ak,r), and n B(bk,r) = 0. k=l k=l
Example 9.5 [124]. Take the points
Then
for j, k = 1,2,3, but by Theorem 8.2 there is no kB~ -isometry I such that IIG = f.
We now turn our attention to kBHR . Theorem 9.8 below will be used in the construction of the counterexample we will present in the next section.
Our considerations are based on the following key result.
Theorem 9.6 [133]. If aI, ... ,am,bl , ... ,bm,p are points of BHR such that
for j, k = 1, ... , m, then there exists a point p' in BHR such that
for j = 1, ... , m.
Proof. For every sufficiently large fl 2': 0 the set
is nonempty, kBHR -bounded, kBRR-ciosed and convex, hence weakly compact. Moreover, o ::; fl ::; >. implies PI' CPA. Let flo be the smallest nonnegative number for which the set PI'O is nonempty. If flo::; 1, then the proof is finished.
Suppose that flo > 1 and let qo be an element of Pl'o. Without loss of generality we may assume that p = qo = 0,
and
Fixed points of holomorphic mappings 471
Hence we have Ilbjll > Ilajll for j = 1, ... ,1. Now we may apply the method due to Schoenberg [177J. The element ° has to lie in the kBHR -convex hull (equal to the linear convex hull) of the set {bl, ... , bk }. Therefore we get
I
L{3jbj = 0, j=1
where {31, ... , {3k ~ ° and L}=I {3j = 1. Next, we have the inequalities
for j, k = 1, ... , I, which imply that
<1
for j, k = 1, ... , I, and finally,
for j, k = 1, ... , I. Therefore we obtain
2 I I I I
OS; L{3jaj j=1
= L (3j{3k(aj,ak) < L (3j{3k(bj ,bk) = L{3jbj j,k=1 j,k=1 j=1
This contradiction completes the proof of Theorem 9.6.
= 0.
• As a simple consequence of this theorem we obtain the following two equivalent theorems.
Theorem 9.7 [133J. Let {B(a",r")},,EJ and {B(b",ra)}aEJ be two families of closed balls in (BHl<' kBH,,). If
for all a, {3 E J and the intersection naEJB (aa, ra) is nonempty, then so is the intersection naEJB (ba , r",).
Theorem 9.8 [133J. If 0 =F C C BHR and f : C ....... BHl< is a kBR/iI. -nonexpansive mapping, then there exists a mapping it : BHTI/. ---+ BHfI. which is a kBRR -nonexpansive extension of f.
Now let us consider the case of (BHfI.' kBHfI.) with n ~ 2. Here we have two distinct situations. For n ~ 2 and dimHlll. ~ 2 Theorem 9.7 fails as the following example shows.
Example 9.9 [133J. It is sufficient to consider the case n = 2 and dimHlll. = 2. If 0< 0' < 1,
al = (0,0,0,0), a2 = (0',0,0,0), a3 = (0,0,0',0),
472
and 1
r = 2" argtanha,
then kB2 (bj , bk) = kB2 (aj, ak) for 1 5, j, k 5, 3, HJR HR
3 3 n B(ak,r) =J 0 and n B(bk,r) = 0. k=l k=l
The case HIT!. = ~ and BNR = (-1, 1 t is different.
Theorem 9.10 [133]. If HIT!. = ~, BNR = (-l,lt and 0 =J C c B N", then every kBn -nonexpansive mapping f : C --t B Hn has a kBn -nonexpansive extension II to all
HJR 'R HR
ofBN[/.·
The proof of this result is based on Helly's theorem [73].
Finally, we consider kBJi -isometric mappings and the problem of extending them to isometries. In this case we have a more complicated situation.
Theorem 9.11 [124]. If 0 =J C C BHR and f : C --t BHR C HIT!. is a kBH" -isometric mapping, then there exists a real Hilbert space H1,IT!. with the open unit ball BH,,[/. such that HIT!. C H1,IT!.I BHR C BH"RI kBH[/. = kBHl ,., on BHJR. and there exists a kBHl,r<
isometry I : BH"r< --t BH"R with IIBHR = f.
Remark 9.12 The following example shows that to obtain extensions of kBH"isometric
mappings to kBHl,JR -isometries the larger real Hilbert spaces BH"r< are sometimes really needed,
Example 9.13 Consider the real Hilbert space 12 with the standard basis {ed. If C = Hek} U {O} and the isometric mapping is the shift operator
f (0) = 0, f Gek) = ~ek+l' for k = 1,2,3, ... , then we can extend f to a kB,2-isometric mapping (defined on all of B/2), but not to a kB,2-isometry from B/2 onto B/2.
10. Approximating sequences in the Hilbert ball
First we recall a theorem due to K. Goebel and W. A. Kirk [57] which generalizes and unifies earlier results of M. A. Krasnoselskii [113], M. Edelstein and R. C. O'Brien [49]' S. Ishikawa [77] and of W, A. Kirk [107]. To state this theorem we need several definitions.
Let (M, d) be a metric space and let ~ denote the real line, We say that a mapping c : ~ --t X is a metric embedding of ~ into M if
d (c(s), c(t)) = Is - tl
Fixed points of holomorphic mappings 473
for all real sand t. The image of lR under a metric embedding will be called a metric line. The image of a real interval [a, b] under such a mapping will be called a metric segment.
Assume that (M, d) contains a family C of metric lines such that for each pair of distinct points x and yin M there is a unique metric line in C which passes through x and y. This metric line determines a unique metric segment joining x and y. We denote this segment by [x,y]. For each 0::; t::; 1 there is a unique point z in [x,y] such that
d(x,z)=td(x,y) and d(z,y)=(I-t)d(x,y).
This point will be denoted (1 - t) x EI1 tV.
We shall say that M, or more precisely (M, d, C), is a hyperbolic space if
( 1 1 1 1) 1 d "2x EI1 "2Y' "2x EI1"2z ::;"2d (y, z) (10.1)
for all x, y and z in M.
An equivalent requirement is that
( 1 1 1 1) 1 1 d "2x EI1 "2Y'"2w EI1 "2z . ::; "2d (x, w) + "2d (y, z) (10.2)
for all x, y, z and win M.
It is clear that all normed linear spaces are hyperbolic. A discussion of metric convexity and of more examples of hyperbolic spaces and, in particular, of the Hilbert ball can be found, for instance, in [57] , [60] , [105] , [107] , [160] and in references therein.
Proposition 10.1 [107]. Suppose (M, d) is a hyperbolic space, let 0 < >.. < 1, and suppose the sequences {xd and {Yk} in M satisfy
(i) Xk+1 = (1 - >..) Xk EI1 >"Yk, and
(ii) d(Yk+1' Yk) ::; d(Xk+b Xk),
for all kEN. Then
1 d(Yk+m, Xk) 2: (1 _ >..)m [d(Yk+m, Xk+m) - d(Yk, Xk)] + (1 + m>..) d(Yk, Xk)
for all k,m E N.
Theorem 10.2 [57]. Let (M, d) be a hyperbolic space, Xo E M, 0 < b < 1, and let {>"k} C [0, b] satisfy 2:%:0 >"k = +00. Let f : M -> M be non expansive, and for each k = 0,1,2, ... , suppose that Xk+1 = (1 - >"k)Xk EI1 >"kf(xk). If {Xk} is bounded, then limk d(Xk, f(xk» = O.
Given a kBH-nonexpansive mapping f : BH -> BH, we associate with it two kinds of kBH-nonexpansive averaged mappings as follows:
h,>. = (1 - >.) I EI1 >..f
(an averaged mapping of the first kind) and
12.>. = (1 - >..) I + >..f
474
(an averaged mapping of the second kind), where 0 < A < 1. If Fix (I) =f 0, then these averaged mappings have the same fixed point set as the mapping f. They also have several remarkable properties which are not shared by all kBH-nonexpansive mappings.
Corollary 10.3 [150J. If a kBH-nonexpansive mapping f : BH -t BH has a fixed point, then for each 0 < A < 1 and each x E BH we have
liJ?kBH (it>. (x) ,Jut>. (x))) = liJ?kBH (ft>. (x) ,f~tl (x)) = O.
In other words, 11,>. is asymptotically regular [25]' [58J.
It turns out that an averaged mapping of the second kind enjoys the same property.
Theorem 10.4 [150J. If a kBH-nonexpansive mapping f: BH -t BH has a fixed point and
12,>. = (1 - A) I + Af,
where 0 < A < 1, is an averaged mapping of the second kind associated with f, then 12,>. is asymptotically regular.
Proof. Choose x E BH and denote f~>.(x) by Xk and f(Xk) by Wk. Let y be a fixed
point of f (and h,>.). Assume without l~ss of generality that 0 < A :S ! and let
Then
and
Zk = (1 - 2A) Xk + 2AWk.
1 1 Xk+l = "2Xk + "2Zk,
lim kBH (Xb y) = L exists, k
limsup kBH (Wk,Y) :S L, k
Since any kBH-ball is convex, we also have
lim sup kBH (ZbY) :S L. k
By the linear uniform convexity of each kBH-ball (see Theorem 4.1 (viii) and (ix)) we conclude that
Hence
as well. But this implies (see Lemma 3.3) that
liJ? kBH (Xk, 12,>. (Xk)) = 0,
Fixed points of holomorphic mappings 475
or equivalently,
• Theorem 10.5 [150]. Let h>.. (j = 1,2,0 < A < 1) be an averaged mapping associated with a kBH-nonexpansive mapping f : BH -t BH. If f has a fixed point, then for each x E BH the sequence of iterates {fl.>,,(X)}~l converges weakly to a fixed point of f·
Proof. We know (see Theorem 4.1) that (BH' kBH ) has the Opial property with respect to the weak topology in H. By Corollary 10.3 and Theorem lOA we also have
lir kBH (fl.>.. (x), fUl.>.. (x))) = o.
It follows that the weak limit point z of every weakly convergent subsequence
{fl,:\(x)} :=1 of the sequence {fl.>.. (x )}k=l is a fixed point of f (and h>..). Hence
l~kBH (!l,:\(x),z) =lirkBH (fl.:..(x),z)
and by the Opial property of (BH, kBH ) with respect to the weak topology the whole sequence {fl.>..(X)}~l converges weakly to z, as claimed. •
A modification of the Genel-Lindenstrauss example [54] shows that the convergence established in Theorem 10.5 is not strong in general.
Example 10.6 [133]. Let HlF,. be the real space Ii with the standard orthonormal basis {ern}~=l' First we define inductively sequences {Xk} and {f(Xk)} which satisfy
1 1 Xk+1 = 2Xk + 2f (Xk)
for k = 1,2, .... We start the construction of the sequence {xd by picking
Let ml and <PI satisfy the conditions
fiT OJ ml > 10, 7r
<PI = "-3 -:-( m-l---l--C-) and
Next, the points Xb k = 2, ... , ml and f (Xk), k = 1, ... , ml - 1 will be chosen in the plane PI = lin (el' e2) = lin (Xl, e2) according to the following rules:
for k = 1, ... , ml - 1. It is clear that for every 1 :::; j, k :::; ml - 1 we have
476
and
At this point we modify the Genel-Lindenstrauss example [54J. We choose the point !(xm1 ) in the following way. Let Yl be the next point after xm1 (in the plane PI) chosen according to the above rules. This means that
lIyIiI = IIxmlll , and (xmu Yl) = IIXmll12 cos (2'Pl) .
We set
Hence we get
for k = 1, ... , ml - 1 because
cos((ml - k)'Pl) < cos 'PI cos((mi - k - 1)'Pl)
and
for k = 1, ... ,ml -1. Now we put
1 1 Xm1+l = 2Xm1 + 2! (xm1 ) .
The point xm1+1 belongs to P2 = lin(xmue3) (and so will all the points {xklZ'~ml+2 which we will construct next) and
Since the angle between the half-planes
and
is acute, the orthogonal projections of !(xm1 ) and xm1 on P2 show that there exists an angle 1/J2 > 0 such that for each u E 122 with
Ilull = Ilxmlll and (u,xm1 ) > Ilxmlll2cos1/J2
we have
Fixed points of holomorphic mappings 477
Analogously, applying the orthogonal projection of
onto lin(xm1+l,J(Xm1 ) - xm1 ) we get
kBH'Il. (u',J(xm1 )) < kBH'Il. (u',xm, ).
Next, taking
and observing that the angle between Wl and J(Xk) is strictly less than the angle between Wl and Xk (k = 1, ... , ml - 1), we obtain
kBHIR (>,Wl, J(Xk)) < kBHIR ('\Wl, Xk)
for ° < II'\wlII < 1, ,\ > ° and k = 1, ... , ml - 1. Hence for each
uE {'\wl+/Le3 :'\>0, /L2::0}
with Ilull < 1 we have
kBH,,(U,J(Xk)) < kBH'Il.(U,Xk)
for k = 1, ... , ml - 1. Therefore the number
E2 = min { kBH'Il.(U',Xk) - kBH,,(U',J(Xk)): u' E R2, k = 1, ... ,ml -I},
where
is positive. It is clear that we can find m2 and 'P2 which satisfy
and
N '3 m2 - ml > 10,
'If 'l/J2 'P2 = <-,
3(m2-ml-1) 2
3 ( )m2_m1 1 + ~ 5 8 COS'P2 > -2- = 16
argtanh (1 - W2 2) < E2·
(l-lcos(2'P2))
Now we can repeat the procedure we have just used to construct {Xk};;''';l by starting with X m, +1 and rotating always in the plane P2 by the fixed angle 2'P2. We have, of course, to check whether J is kBH'Il. -nonexpansive on its domain of definition up to now, i.e., whether the inequalities
1:::: j, k :::: m2 - 1,
478
are valid. For ml + 1:0:::: j, k :0:::: m2 -1 we have them by the same reason as in the first case. Next, applying the orthogonal projection of f(xm ,) and xm , onto P2 we obtain
kBHR (f(xm,), I(Xk)) < kBHR (XmuXk)
for ml + 2:0:::: k :0:::: m2 - 1. By the choice of '/f!2 and 'P2 we also have
kBHJ& (f(xm,), I(xm,+l)) < kBHR (xmu Xm,+l)'
For 1 :0:::: j :0:::: ml - 1 and ml + 1 :0:::: k :0:::: m2 - 1 we get
kBHR (f(Xj).J(Xk)) :0:::: kBHR (f(Xj), Xk) + kBHR (Xk.!(Xk))
< kBHR (f(Xj), Xk) + 102
:0:::: kBHR (Xj, Xk) - 102 + 102 = kBHR (Xj,Xk).
All other cases were considered earlier. Now it is clear how to continue the inductive definition of {xd and {/(Xk)}. The sequence {Xk} is kBHR -bounded from above by
arg tanh( ~) and from below by arg tanh ( !). It does not converge strongly to 0, but it does tend weakly to the origin. Next we use the extension property of (BHR,kBHJ&) to obtain a kBHR -nonexpansive mapping I : BHR -+ BHR' It is easy to see that
{/t~(Xl)}:l and {1:~(Xl)}:l tend only weakly to O. Since there is a kBH-nonexpansive retraction of BH onto BHR' an analogous example of a kBH-nonexpansive I : BH -+ BH can be constructed in (BH, kBH)' Note, however, that in this example I is not a holomorphic mapping.
The next theorem is of particular interest when I is fixed-point-free. It can be considered an extension of Corollary 10.3. See [23] and [160] for more general results (which extend, inter alia, Theorem 10.2) and [154] for a continuous analog.
Theorem 10.7 [150]. Let I: BH -+ BH be a kBH-nonexpansive mapping, 0 < A < 1, and let II,)' be an averaged mapping 01 the first kind associated with I. Then lor each x in BH and all m 2': 1,
. ( k+l k ) . kBH (ittm(x).!f,)'(x)) . kBH (if,),(x),x) hmkBH II). (x),/I).(x) =hm =hm k .
k ' , k m k
Proof. In order to prove the first equality, we fix x E BH and m 2': 1. Since II,)' is kBH-nonexpansive, the limits
exist. To see that R :0:::: mL, we note that
m
kBH (ft,),(x),lttm(x)) :0:::: ~kBH (i;,tj-\x).!ttj(x)) :0:::: mkBH (it),(x).Jf,tl(x)). j=l
To show that R 2': mL, we first recall that
h,)' = (1 - >.) I EEl Af·
Fixed points of holomorphic mappings
Applying Proposition 10.1, we see that
kBH (iuttm(x)),ft>.(x))
~ (1_\)m [kBH (l(Jttm(x)),/~.tm(x)) - kBH (iut>.(x)),ft>.(x))]
+ (1 + m,x) kBH (i(Jt>.(x)), It>. (x) ) .
Now we observe that
Hence
Since
kBH (fUttm(x)),ft>.(x))
::; kBH (i(Jttm(x)),fut>.(x))) +kBH (l(Jt>.(x)),ft>.(x))
::; kBH (/~.tm(x),ft>.(x)) + kBH (iut>.(x)),ft>.(x)).
kBH (Ittm(x), ff,>.(x))
~ (1_\)m [kBH (IUttm(x)),fttm(x)) - kBH (iut>.(x)),lt>.(x))]
+ (m,x)kBH (IUt>.(x)),ft>.(x)).
we see that
kBH (ittm(x),ft>.(x))
~ ,x(1 ~ ,x)m [kBH (Ittm+1(x),lttm(x)) - kBH (itt1(x),lt>.(x))]
+mkBH (Itt 1 (x), It>. (x)) .
Letting k --+ 00, we obtain R ~ mL, as claimed.
To establish the second equality we first note that
kBH (ittm(x),/t>.(x)) ::; kBH Ui,>.(x),x).
Therefore the first equality shows that
( k+l k ) kBH (fi,>.(x),x) liJ?kBH 11,>. (X),/1,>.(X) ::; m
for all m ~ 1. Hence
( k+1 k) kBH (Ii,>. (x),x) limkBH f1>. (x),/1>. (x) ::; liminf ----'----.!-
k ' , m m
kBH (/1">. (x),x) ::; lim sup ----'--' --"-
m m
479
480
Thus
exists and equals
This completes the proof of Theorem 10.7. • To end this section we mention two strong convergence results. The first one verifies a conjecture formulated in [60] and is used in the proof of the second one. It is also used [93] in the study of the asymptotic behavior of the nonlinear resolvent Jr defined by (5.2).
Theorem 10.8 [182]. Let f : BH -+ BH be a kBH-nonexpansive mapping with a nonempty fixed point set Fix(f). For each 0 :-s: A < 1 define the mapping h>. : BH -+ BH by
h>. (x) = (1 - A) x + Af(h>. (x)), x E BH .
Then the strong lim>'-+I- h>.(x) = q(x) exists for each x in BH and belongs to Fix(f).
The mapping q : BH -+ Fix(f) is the unique retraction of BH onto Fix(f) which is firmly kBH-nonexpansive of the second kind (Definition 12.13). See [60] for an analogous result involving the approximating curve {gt}09<1 defined in Theorem 12.14.
Theorem 10.9 [157]. Let f : BH -+ BH be a kBH-nonexpansive mapping with a fixed point, Xo a point in BH, and {Ak} a real sequence in [0,1) satisfying the following three conditions (where ILk = 1 - Ak):
limAk = 1, k
00
LILk = 00 and k=1
Then the sequence {Xk} defined by
converges strongly to a fixed point of f.
1· ILk-1 - ILk 0 1m 2 =. k ILk
Remark 10.10 The conditions required of the sequence {Ak} in Theorem 10.9 are satisfied, for example, if Ak = 1 - k-f3 with 0 < f3 < 1. Analogous results for nonexpansive mappings in Banach and Hilbert spaces can be found in [149] and references therein.
Fixed points of holomorphic mappings
11. Fixed points in infinite powers of the Hilbert ball
Let J be a set of indices,
and Efl the open unit ball in lOO(H) with the supremum norm.
The Kobayashi distance in Efl is given by
kB= (x,y) = SUpkBH (Xj,Yj) H jEJ
481
[129]. The proof of this equality is based on the fact that for each a = {aj}ju, the mapping
Ma = {Maj} jU : Efl ----4 Efl,
where each Maj is a Mobius transformation, is biholomorphic.
Theorem 11.1 [128]. (Generalized Hartogs' Theorem). Let X be a Banach space and D a nonempty open subset of X. Let {Yj} be a family of Banach spaces. If f : D ----4 lOO(Yj) is locally bounded, then the following statements are equivalent:
(i) I = {Ii} is holomorphic;
(ii) each fj : D ----4 Yj is holomorphic.
Theorem 11.2 [128]. Let f : Efl -> Efl be a holomorphic mapping. Then the following statements are equivalent:
(i) I has a fixed point;
(ii) there exists a ball E (x, r) in (Efl, kBfj) which is f-invariant;
(iii) there exists an I-invariant, kBfj-bounded product ITjUCj of closed convex subsets of EH.
Proof. The implications (i)=}(ii)=Hiii) are obvious. The implication (iii)=}(i) is proved by a simple modification of the classical proof of Kirk's theorem [103]: we can find a set which is minimal with respect to inclusion, admissible in (ITjEJ Cj , kBfj) and f-invariant, and after applying Theorems 2.4 and 4.1 we see that this minimal set is a singleton. •
Remark 11.3 [128]. One can show that the Kobayashi distance in Efl nco(H), where co(H) = {x = {Xj hEN E IT}:1 H : limj Xj = O}, is given by
kB=nco(H) (x,y) = sUPP(Xj,Yj) H jEN
[129]' but it is obvious that conditions (i) and (ii) are not equivalent in this case. To see this, it is sufficient to consider the mapping
482
Remark 11.4 [128]. We observe that in contrast with the case of the open unit ball BH, there exists in B'{j a holomorphic fixed-point-free self-mapping f with a kB'll'
bounded iteration {fk(x)} for each x.
Example 11.5 [128]. Let C be Cayley's transform:
C : b. --t n+ = {z E C : 1m (z) > O}
[30]. For H = C and J = N, f : B'{j ---t B'{j is defined in the following way:
where LIM is a fixed Banach limit ([39], [41]). Then
(a) f is a holomorphic isometric mapping (with respect to kB'H) from B'{j into B'{j;
(b) f is fixed-point-free;
(c) {Jkx} is kB'll-bounded for every x E B'{j.
Now we are prepared to prove an analog of Theorems 6.3 and 7.5. Note that in the case of reflexive Banach spaces P. Mazet and J.-P. Vigue [142] obtained a holomorphic retraction onto the fixed point set of a holomorphic self-mapping by using complexanalytic arguments, the function h),(u) = (1- >.)u + >.f(h),(u)), and its convergence to fixed points when >. ---t 1-. They also showed that their method failed in the case of the open unit ball in loo. Therefore our approach shows the strength of Bruck's metric method ([26], [27]).
Theorem 11.6 [129]. If f : B'{j ---t B'{j is holomorphic (kB'll·nonexpansive), then Fix(f) is either empty or a holomorphic (kB'll-nonexpansive) retract holomorphic re· tract of B'{j.
Proof. We prove this result only in the holomorphic case. Our metric approach works equally well in the kB'H·nonexpansive case. Set
Noo = {g : 9 is a holomorphic self-mapping of B'{j, Fix (f) c Fix (g)}.
Fix Xo = {XOj} E Fix (f) . Note that
xEB'H jEJ xEB'll jEJ
If each exj is equipped with the weak topology, then each exj is weakly compact and by Tychonoff's Theorem [50] I1xEB'll I1 jE J exj is compact in the product topology. The set Noo is closed in this topology, i.e., in the topology of coordinate pointwise weak convergence (see Theorems 2.10 and 3.7). This allows us to apply Bruck's method [27].
Preorder N 00 by setting 9 ::; h if and only if
kB'H (g (x), w) ::; kB'H (h (x), w)
for all w E Fix(f) and x EB'{j. Let us choose a descending chain
Fixed points of holomorphic mappings 483
in (Noo , SJ . By the compactness of I1xEBif I1jEJ Oxj, {G>.} has a subnet {g,v hIEA' for which A' is an ultranet and
w-limg,vJo (x) = gj (x), x E Ell and j E J. ,v
The mapping 9 = {gj}jEJ is holomorphic (see Theorem 11.1). Since by Theorem 3.7, kBH is weakly lower semicontinuous, the following inequalities are valid for each wE Fix(f) and x E Ell:
kBoo (g (x), w) < lim kBoo (g,v (x) , w) < kBoo (g;>.. (x) , w), ,\ E A, H -,v H - H
and this means that 9 is a lower bound of the chain. So Zorn's Lemma implies that Noo contains a minimal element r.
Now we need to show that r maps Ell onto Fix(f). Suppose there exists y E Ell such that r(y) ¢:. Fix(f). Since r 0 r S rand r is minimal,
kBif (r (Yo) , w) = kBif (Yo, w) > 0
for Yo = r(y) and all w E Fix(f). Since for each j E J, after interchanging j-coordinate functions between two arbitrarily chosen mappings from Noo we also get a mapping from Noo , and since g, hE Noo and 0 S f3 S 1 imply that (3g + (1- (3)h E Noo too, the set Noo is equal to I1jEJDj, where each D j is convex and weakly compact. Let
0= {(g 0 r)(yo) : 9 E Noo }.
Using the same arguments as above we see that 0 is kBif-bounded and 0 = I1 j EJ OJ, where each OJ is convex and weakly compact. The definition of Noo implies that 0 is f-invariant and therefore by Theorem 11.2, On Fix(f) =I 0. Let us choose an arbitrary point (g 0 r)(yo) EOn Fix(f). Since go r S rand r is minimal, we get the following contradiction:
0= kBif (g 0 r) (Yo), (g 0 r) (Yo)) = kBif (r (Yo), (g 0 r) (Yo)) > O.
This completes the proof of Theorem 11.6.
12. The Denjoy-Wolff theorem in the Hilbert ball and its powers
Denjoy-Wolff theorem We begin by recalling the original Denjoy-Wolff theorem.
•
Theorem 12.1 [40], [212], [213]. Let bo. be the open unit disc in the complex plane <C. If f E H(bo.) is not the identity and is not an automorphism of bo. with exactly one fixed point in bo., then there is a unique point O! in the closed unit disc is: such that the iterates {fk} of f converge to O!, uniformly on compact subsets of bo..
In the next two sections we will only consider the case where f E H(bo.) does not have fixed points in bo.. This means that we will actually deal with the following simpler version of the above theorem.
Theorem 12.2 [40], [212], [213]. Let bo. be the open unit disc in the complex plane <C. If f E H (bo.) with no fixed point in bo., then there is a unique point £.1 E abo. such that the iterates {fie} of f converge to £'1, uniformly on compact subsets of bo..
484
The usual proof of Theorem 12.2 is divided into the following four steps [31].
Step 1. There are a point f"f E Bfl and J-invariant horocycles at f"f with radii R> 0 (d. Theorem 6.4),
{ 11 - Zf"l} E (f"f' R) = z E fl : 2 < R , l-lzl
in fl. Here f"j is obtained as the limit point of any convergent subsequence of the approximating sequence
(the existence of such a sequence is guaranteed by Rouche's theorem [30] applied to the mapping 1- (1- y-l)J). The above horocycles are obtained by applying the SchwarzPick Lemma. Observe that E(f"f, R) c fl is a disc with the point f"f on its boundary which is internally tangent (in R2) to Bfl at f"f.
Step 2. By Montel's theorem and the Open Map Theorem [30] each convergent subsequence Uk~} of the sequence of iterates Uk} has a constant function g with Igl = 1 as its limit.
Step 3. For each z E fl there is R > 0 such that z E E( f"f, R).
Step 4. According to steps 1 and 3 for every z E fl there is R > 0 such that each point Jk(z) lies in E(f"f, R) and therefore Jk(z) ---t f".
Somewhat more complicated arguments yield the following generalization of the DenjoyWolff theorem in en.
Theorem 12.3 [74], [116] ,[139]. Let B be the open unit ball in the complex space en. If J : B ---t B is holomorphic with no fixed point in B, then there is a unique point f"j E oB such that the iterates Uk} of f converge to f"j, uniformly on compact subsets of B.
If we wish to extend the above convergence results to other domains in en or more generally, in a complex Banach space X, then we should keep in mind the following three negative examples.
Example 12.4 [36]. If e2 is equipped with the maximum norm, then its open unit ball B is equal to the Cartesian product fl x fl. Let us choose any fixed-poi nt-free h : fl ---t fl. If we define J : fl x fl ---t fl x fl by
J(z,w) = (iz,h(w))
for z, w E fl, then J is also fixed-point-free and it is clear that its sequence of iterates Uk} does not converge to any boundary point on o(fl x fl). As we will see, the main reason for such a behavior of the sequence of iterates is the lack of strict convexity of the closed unit ball B = fl x fl.
To construct the next example we need the following two facts.
1) If Bl2 is the open unit ball in the complex infinite-dimensional Hilbert space 12 with the standard inner product and if e is a point in [2 of norm 1, then B12 is
Fi:xed points of holomorphic mappings 485
biholomorphically equivalent to the domain
O={(A,w)ECX[2; ImA>lIwll}
under the Cayley transform C ; B -+ 0 given by
( .1 + (z,e) i ) B I23z-+C(Z)=(A,w)= z ( )' ( )(z-(z,e)e) EO.
1- z,e 1- z,e
The domain 0 is the analog of the upper half-plane [60].
2) In [47] M. Edelstein shows that there exists a fixed-point-free isometry 9 of the complex infinite-dimensional Hilbert space [2 such that the sequence of iterates {gk(O)} is unbounded, but a suitably chosen subsequence {gk~(O)} is convergent to O. We recall this isometry;
g(w) = g({Wj}) = {e 2;' (Wj -1) + I}.
One can compute that for km = m! (m = 1,2, ... ) the subsequence {gkm(o)} tends to 0 and for
(m = 1,2, ... ) we have "gk~(O)II-+ 00. We also have that
g(w) =g(w)+a,
where a = g(O) and 9 : [2 -+ [2 is a linear isometry.
Now we are ready to present the second example in which we use Edelstein's isometry 9 = g+a.
Example 12.5 [188]. In the complex infinite-dimensional Hilbert space [2 convergence fails even for biholomorphic self-maps. To see this, consider the above-mentioned domain O. For the mapping cp : 0 -+ 0 defined by
cp(A,W) = (A+illaIl 2 +2i(g(w),a),g(w))
we have
lim cpk~(i,O)=O for km=m! (m=I,2, ... ) and limsupllcpk(i,O)II=oo. m~oo k~oo
Setting f = C-l 0 cp 0 C : Bl2 -+ Bl2 we see that f is a biholomorphic automorphism of B12, Fix(f) = 0, and
In the third example we show that there is a complex Banach space X and a fixedpoint-free holomorphic self-mapping f of the open unit ball B in X such that
486
for each x E B.
Example 12.6 [128]. Consider the complex Banach space of all bounded sequences 100 = lOO(q and let
be the Cayley transform
C:A-t{ZEIC: Imz>O}
C(z) = i(1 +z) 1- z
for z E A [30]. If B't is the open unit ball in 100 , LIM is a fixed Banach limit on [00,
and f : B't -t B't is given by the formula
f(x) = f({xj}) = {C-1 (I+LIM({C(xj)})),Xl,X2, ... },
then f is holomorphic, Fix(f) = 0, and for each x E B't the sequence of its iterates {jk(x)} lies strictly inside B't, i.e.,
The above examples show that in order to obtain a generalization of the Denjoy-Wolff theorem, not only do we need additional properties of the boundary of the open unit ball B, but we also have to impose some restrictions on the holomorphic self-mapping f: B -t B.
We begin our discussion with a special class of kBH-nonexpansive self-mappings of the Hilbert ball B H . As we already know (see Example 12.5), the Denjoy-Wolff theorem is false in general in BH. However, one can prove that this theorem is valid for averaged mappings. To accomplish this we need two preliminary lemmas.
Lemma 12.7 If a kBH -nonexpansive f : BH -t BlI is fixed-point-free and fJ,>, (j = 1,2 and 0 < A < 1) is an averaged mapping associated with f (see Section 10), then the mapping fj,), is also fixed-paint-free and ~f = ~h' .
Proof. It is obvious that fj,), is fixed-point-free. Next it is sufficient to observe that the ellipsoid E(~f,R) is convex, kBH-convex and f-invariant. •
Lemma 12.8 [150]. If a kBH-nonexpansive mapping f : BH -t BH is fixed-point-free and limk II fk (x) II = 1 for some x in B H, then the strong limk fk (y) = ~f for all y in B H ·
Proof. For this x there exists R > 0 such that x E E(~f' R). The ellipsoid E(~f' R) is f-invariant and therefore limkfk(x) = ~f. Next we apply Lemma 3.10 to get limk fk(y) = ~f for all y in BH . •
Theorem 12.9 [150]. Let f : BH -t BH be a kBH-nonexpansive mapping. If f is fixed-point-free, then for each x in BH the sequence of iterates {fj,)' (x)} of the averaged
mapping fj,), (j = 1,2 and 0 < A < 1) converges strongly to ~f' a point on the boundary ofB.
Fixed points of holomorphic mappings 487
Proof. The case of iI,>.' Denote ff,>.(x) by Xk and assume that {Xk} has a kBHbounded subsequence {Xk=}' Then
by Theorem 10.7 and hence limkkBH (h,>.(Xk),Xk) = O. It follows that the asymptotic center x of {Xk=} is a fixed point of h,>. and therefore x is a fixed point of f, a contradiction. Hence limk Ilxkll = 1 and the result follows from Lemmas 12.7 and 12.8.
The case of h,>.. Denote f~,>.(x) by Xk and f(Xk) by Wk. Assume without loss of
generality that 0 < ). :S ~, and let
Since
for all y in BH, the sequence {q,~f(Xk)} decreases to a limit Rand
The convexity of the ellipsoids E(t;j,R) now implies that
too. Since
lim sup q,~f (Zk) :S R k
lir q,~f (~Xk + ~Zk) = lir q,~f(Xk+1) = R,
we can apply Lemma 6.6 and conclude that limk Ilxk - zkll = O. Hence
aswell. Since f does not have a fixed point, this implies that {xd cannot have a kBH-bounded subsequence. Thus
and once again the result follows from Lemmas 12.7 and 12.8. • Now we prove the Denjoy-Wolff theorem for a certain class of kB'H isometries.
Theorem 12.10 [124].
(i) Assume that I: BH -+ BH is a kBH -isometry. If
Fix (IIBH) = 0
and I has two fixed points {t;,t;r} on the boundary 8BH, then the sequence of iterates {(I)k(x)} tends to t;r for each x E BH \ {t;,}o
488
(ii) Let I : B'H -> B'H be a cyclic kBj, -isometry (n ~ 2) without fixed points inside B'H and with exactly two fixed points on the boundary a( B'H ). Then there exists ~I E Fix(I) such that {(I)k(x)} tends to ~I for each x E B'H.
Proof. (i) Without loss of generality we may assume that Fix(I) = {~I, -~I}. Then we have
lim (I/ (0) = ~I k
(see the proof of Theorem 8.6) and by Lemma 3.10 (I is a kBH-isometry) we get
lim (I/ (x) = ~I k
for each x E BH. Next, it is easy to see that for
we also have
lim (I)k (x) = ~I. k
Finally, for k = 1,2, ... and for every x = Q~I + u E aBH with (~I,U) = 0, we get
//(I/(x)//=l and (I)k(x)=(I)k(Q~I)+Uk' with (~I,Uk)=O.
Therefore {(I)k(x)} converges to ~I for each x E BH \ {-~I}.
(ii) Let
and
i = (Ih 0 (Ih 0 •.. 0 (I)n .
Let x E B'H, k = jn + m, j = 1,2, ... , and 0 ~ m < n, then we have
(I)k(Xb ... ,Xn ) = ((i)i(X1)' ((I)2o ... 0 (I)n 0 (i)i-l 0 (I)I)(X2), ...
... , ((I)n 0 (i)i-l 0 (I) 1 0 .. ·0 (I)n_l) (Xn)),
ifm = 0, and
Ik(Xb ... , xn) = ( ((i)i 0 (I) 1 o· ··0 (I)m) (Xm+1),
((I)2 0 ... ° (I}n 0 (i);-1 0 (I) 1 0 .. ·0 (I)m+l) (Xm+2), ...
... , ((I)n_m 0 ... 0 (I}n 0 (i);-1 0 (I) 1 o· ··0 (I)n_l) (Xn),
((I)n+1-m 0 ••. 0 (I}n 0 (i);) (Xl),
((I)n+2_m 0···0 (I)n 0 (i); 0 (I) 1) (X2), ...
Fixed points of holomorphic mappings 489
if 0 < m < n. By (i), {(I)k(Xl' ... , xn)} tends to
~I = (~i' ((I)2 0 ..• 0 (I)n)(~i)"" , (I)n (~i)) E Fix(I).
• Next we consider firmly kBH-nonexpansive mappings of the first and second kinds.
Definition 12.11 [60].A mapping f : BlI -> BlI is said to be firmly kBH-nonexpansive of the first kind iffor each x and y in B 1I, the function ip : [0,1) -> [0, (Xl) defined by
ip (s) = kBH((l- s) x E9 sf (x), (1 - s) y E9 sf(y))
is decreasing.
Remark 12.12 [60). The nearest point projection rc of Theorem 4.2 (ii) is firmly kBH-nonexpansive of the first kind.
Definition 12.13 [60].A mapping f : BlI ->BlI is said to be a firmly kBJinonexpansive mapping of the second kind if for each x and y in B 1I, the function 1jJ : [0, 1) -> [0, (Xl) defined by
1jJ(s) = kBH((l- s)x+sf(x) ,(1- s)y+sf(y))
is decreasing.
Theorem 12.14 [60]. Let f : BlI -> BJI be a kBH-nonexpansive mapping. Then, for each 0 ::; t < 1, the mapping gt : BlI -> BlI (hi: BlI -> BlI) implicitly defined by
gdx) = (1 - t) x E9 tf(gdx)) (ht (x) = (1 - t) x + tf(ht (x)))
for all x in B 1I is a firmly k B H -nonexpansive mapping of the first kind (a firmly k B H -nonexpansive mapping of the second kind).
Proof. Let z = (1 - s)x E9 sgt( x) and w = (1 - s)y E9 sgt (y). Noting that there exists o ::; p ::; 1 such that
gt (x) = (1 - p)z EEl pf(gt (x)) and gt (y) = (1 - p)z EEl pf(gt (y)),
we obtain
Since ip is convex, the first result follows.
Now denote ht = h (t, f) by h. The mapping h is kBH-nonexpansive. Let 0::; 'r < s::; 1,
u = (1 - r) x + rh (x) , v = (1 - s) x + sh (x), w = (1 - r) y + rh (y) ,
and
z = (1 - s) y + rh (y).
490
We have to show that kBH(V,Z):::: kBH(U,W). If 3 = 1, then
where
Therefore
v = h(x) = h(p,!) (u) and z = h(y) = h(p,!) (w),
p=p(r)= t(l-r). 1- tr
as required. If 3 < 1, then
We also have
with
Hence
h (x) = hlp (3),!) (v) and h(y) = hlp (3),!) (z).
v=(l-q)u+qh(x) and z=(l-q)w+qh(y),
s-r q=--.
1 - r
v = h(q, h(p(s),J))(u) and z = h(q, h(p(3),J))(W),
so that once again, kBH(V,Z):::: kBH(U,W). • Remark 12.15 If f: EH ---t Ell is holomorphic, then the mappings {ht}oS;t<1 defined in Theorem 12.14 (see also (5.1)) are holomorphic too. More information on firmly nonexpansive mappings of both kinds can be found, for example, in [59J, [60], [153], [159] and [160J. Here we only mention the following result.
Theorem 12.16 [59J, [60], [153J, [159J. If f: Ell ---t Ell is a firmly kBH-nonexpansive mapping of the first or the second kind and Fix(f) = 0, then fk(x) ---t ~f for each xE EH.
We conclude this section with the kBH-nonexpansive version of the Chu-Mellon theorem [36].
Theorem 12.17 [115J. For each compact, kBH -nonexpansive and fixed-point-free mapping f : EH ---t Ell there exists ~f E {JEll such that the sequence Uk} of the iterates of f converges to the constant map taking the value ~f' uniformly on each ball strictly inside EH.
13. The Denjoy-Wolff theorem in Banach spaces
We begin by recalling a useful result of P. Yang [215J concerning a characterization of the horocycle in terms of the Poincare distance on fl.. To wit, he established the following formula:
1 -12 1 1- z~ lim [kD. (z, w) - k", (0, w)] = -2 log 2 '
~3w~~ 1-lzl (13.1)
Fixed points of holomorphic mappings 491
where Z E ~ and ~ E a~. In terms of the horocycle E(~, R) in ~ this provides the following equivalent definition:
E(~,R) = {Z E ~: lim [k~(z,w) - k~(O,w)J < -2110gR}. ~3w ..... e
Since the Kobayashi distance can be defined on each bounded domain in en, one can try to extend this formula and use it as a definition of the horosphere in a domain in en. Unfortunately, the limit analogous to the one in (13.1) does not exist in general. Therefore, the new idea of M. Abate [IJ was to study two kinds of horospheres. More precisely, for xED, ~ E aD, R > 0, Xn E D, n = 1,2, ... , and limn ..... ooxn =~, the small horosphere Ex(~, R) and the big horosphere Fx(~, R) of center x and radius R are defined by
Ex (~, R) = {Y ED: lim sup [kD (y,w) - kD (x, w)J < ~ IOgR} w ..... e
and
It is obvious that
Ex (~, R) c Fx (~, R)
and the horospheres Ex(~, R) and Fx(~, R) are convex. Moreover, for each Y E D there exists R > 0 such that Y E Ex(~, R). Thus every assertion which proclaims for a bounded convex domain D in en and every holomorphic self-mapping f : D --> D the existence of a point ~ E aD such that
for all XED, R> 0 and k = 1,2, ... , and
Ex (~,R) naD = Fx (~,R) naD = {O,
is a generalization of the Denjoy-Wolff theorem. This is true, for example, for a bounded strongly convex 0 2 domain D in en.
Theorem 13.1 [IJ. Let D be a bounded strongly convex 0 2 domain in en and let f : D --> D be a holomorphic mapping without fixed points. Then the sequence of iterates of f converges to a unique point ~f on the boundary aD.
Although Abate's proof does not have a metric character, an application of Theorem 2.5 yields the following result.
Theorem 13.2 [135J Let D be a bounded strongly convex 0 2 domain in en and let f : D --> D be a kD-nonexpansive mapping without fixed points. Then the sequence of iterates of f converges to a unique ~f on the boundary aD.
The proof of the latter theorem included the following simple calculation. Since the kD-nonexpansive mapping f is fixed-point-free, the Earle-Hamilton theorem (Theorem 5.1) and Calka's theorem (Theorem 2.5) show that there is a sequence
492
(Zm E D and 0 < 8 m < 1 for m = 1,2, ... , and limm->oo 8 m = 1) which is convergent to ef E aD. Let us observe that for arbitrary y E D and kEN,
liminf [kDUk(y),w) - kD (x,w)] D3W-->~f
::; liminf [kDUk(Y),Xm) - kD (x,Xm)] m->oo
= l~i~,f [kD (!k (y) ,f:~.z~ (y)) + kD (!:~.z~ (y) ,f:~,z~ (Xm)) - kD (x,Xm)]
::; lim inf [kD (y, xm) - kD (x, xm)] m->oo
::; limsup [kD (y,w) - kD (x,w)J. D3W->~f
The above calculation guarantees that
for all xED, R > 0 and k = 1,2,.... This calculation is also the basis for our subsequent studies of the behavior of the sequence of iterates in the open unit ball of a strictly convex Banach space.
Remark 13.3 [84J. If X is a Hilbert space H, then the limit
. 1 11- (z,~)12 hm [kBH (z, w) - kBH (0, w)J = -log ( ) w->{ 2 1 _ IIz// 2
exists for every Z E BH and e E EJBH, and the ellipsoid
E(C R)={ZEB : /1_(z,~)12 <R} ~, H 1 _ I/zi/2
coincides with the Abate horospheres Eo (e, R) = Fo (~, R) , where limm Wm = ~.
Note that even in the case of a strictly convex Banach space X and its open unit ball B we do not know if
Therefore the inclusions fk (Ex (~, R)) c Fx (e, R), k = 1,2, ... , do not necessarily imply the convergence of the sequence of iterates {Jk(y)} even if we know that
So we need to introduce a new kind of horospheres. To this end, let B be the open unit ball of a complex Banach space X. Let x E B, ~ E aB, R > 0, Xm ED, m = 1,2, ... , and limm ..... oo Xm = e. Fix a Banach limit LIM E (loo)* ([39], [41]). The new horosphere H(x,~, R, {xm}) in B is defined as follows:
H (x,~, R, {xm}) = {y E B : LIM [kB (y,xm) - kB (x,xm)J < ~ 10gR}
[85J. Since
IkB (y, xm) - kB (x, xm)1 ::; kB (y, x)
Fixed points of holomorphic mappings 493
for each m = 1,2, ... , the horospheres H(x,~,R,{xm}) are well defined and have the following properties.
Theorem 13.4 [85]. Let X be a Banach space with the open unit ball B. Let ~ and {xm} be fixed. Then the horospheres H(~, R, x, {x'm}) have the following properties:
(i) H(~,R,x,{x'm}) is convex and nonempty;
(ii) for every 0 < Rl < R2 we have
[H(~,Rl,X,{xm})nB] CH(~,R2,X,{Xm});
(iii) UH(~,R,x,{xm})=Band nH(~,R,x,{xm})=0; R>O R>O
(iv) ~ E n H(~,R,x,{xm}) CaB; R>O
(v) if X is strictly convex, then n H (~,R,x, {xm}) = {O. R>O
Proof. (i) Observe that by Theorem 3.13,
LIM [kB(S~, xm) - kB(X, xm)]
= LIM [kB(S~, xm) - kB(O, xm)] + LIM [kB(O, xm) - kB(X,Xm)]
= lim [kB(S~,Xm) - kB(O, xm)] + LIM [kB(O,Xm ) - kB(X, x'm)] m
::; -kB(O, s~) + kB(O, x)
for each 0 < s < 1. This shows that the horosphere H(~, R, x, {xm}) is nonempty. Its convexity follows from Lemma 3.3.
(ii) and (iii) are obvious.
(iv) See the proof of (i).
(v) This assertion follows directly from (i) and (iv). • Theorem 13.5 [84]. Let X be a strictly convex Banach space with the open unit ball B. Let f : B -> B be a compact kB-nonexpansive map with no fixed point in B. Then there exists ~f E aB such that the sequence Uk} of iterates of f converges to the constant map taking the value ~f' uniformly on each ball strictly inside B.
Proof. Since f is fixed-point-free, the Earle-Hamilton theorem (Theorem 5.1) and the compactness assumption yield a sequence
(Zm E Band 0 < Sm < 1 for m = 1,2, ... , and limm->oo Sm = 1) which is convergent to ~f E aBo
Let us observe that for arbitrary y E B,
LI M [kB (f (y), Xm) - kB (0, xm)]
= LIM [kB (f (y) , fSm,zm (y)) + kB (fsm,zm (y) '/Sm,Zm (xm)) - kB (0, xm)]
::; LIM [kB (y,xm ) - kB (O,xm)].
494
The above inequality implies that
I(H(~,R,O,{xm})) C H(~,R,O,{xm})
for arbitrary R > O. Fix Z E B. Using Calka's theorem (Theorem 2.5) and the convexity of kB-balls we can apply the asymptotic center method to conclude that
(13.2)
Let A c fJB denote the set of all accumulation points of the sequence {fk(z)}. By the compactness of I and (13.2) the set A is nonempty and by Lemma 3.11 it is independent of the choice of z. Theorem 13.4 now implies that
0#Ac nH(~,R,O,{xm})nfJB={O. R>O
Hence limk Ik(z) = ~ and by Lemma 3.11 and the compactness of I the sequence {fk} converges to the constant map ~f' uniformly on each ball strictly inside B. •
Now we examine mappings I : B --+ B which are kB-nonexpansive and condensing with respect to the Kuratowski measure of noncompactness all.II' We begin with the following simple lemma.
Lemma 13.6 [85]. Let D be a nonempty bounded subset of a metric space (M,d). If I : D --+ D is condensing with respect to the Kuratowski measure 01 noncompactness ad and {Xm} is a sequence of elements of D such that d(xm, I(xm)) --+ 0, then the set of all elements of the sequence {xm} is totally bounded in (M,d).
Proof. Since I is condensing with respect to ad and
ad ({Xm : m = 1,2, ... }) = ad ({f (xm) : m = 1,2, ... }) ,
the set {xm : m = 1,2, ... } must be totally bounded in (M, d). • Corollary 13.7 [85]. Let D be a bounded convex domain in a Banach space (X, 11·11). II I : D --+ D is kD-nonexpansive and condensing with respect to all.lI; C is a nonempty, kD-closed, and I-invariant subset 01 D; {sm} is a sequence such that limm 8m = 1, 0< sm < 1; and {zm} is a sequence of elements olC, then the sequence {Xm} implicitly defined by
for each m contains a norm-convergent subsequence.
Proof. It is obvious that I(xm) - Xm --+ O. Thus it is sufficient to apply Lemma 13.6 .
• We now quote a convergence theorem for fixed-point-free condensing mappings. In addition to our previous arguments its proof also requires an application of Theorem 2.7 and Corollary 13.7.
Theorem 13.8 [85]. If B is the open unit ball of a strictly convex Banach space (X, 11·11) and f : B --+ B is kB-nonexpansive, condensing with respect to the Kuratowski
Fixed points of holomorphic mappings 495
measure of noncompactness all.lI, and fixed-point-free, then there exists ~f E BB such that the sequence {fk} of iterates of f converges in the compact-open topology to the constant map taking the value ~f'
In some Banach spaces we are able to show that the convergence provided by Theorem 13.8 is actually uniform on each ball strictly inside B.
Theorem 13.9 [83]. Let N be a norming set for a strictly convex Banach space (X, II· II), B the open unit ball in (X, II . 11), and assume that B is sequentially compact in the topology u(X,N). If (X, 11·11) has the Kadec-Klee property with respect to u(X,N) and f : B -+ B is kB-nonexpansive, condensing with respect to all.1I and fixed-point-free, then there exists ~f E BB such that the sequence {fk} of iterates of f converges to the constant map taking the value ~f' uniformly on each ball strictly inside B.
Proof. By Theorem 13.8 the sequence of iterates {fk(O)} converges to ~f EBB. Let us choose an arbitrary sequence {fkm(Xmn with supm IIxmll = Cl < 1. Since
SUpkB (tkm(O),jkm(Xm)) ::; supkB (O,xm) m J
= sup [argtanh Ilxmlll = argtanhcl = c < 00, m
Lemma 3.10 yields the strong convergence of {fkm(xm)} to ~f' • Remark 13.10 [29]. In the case of separable Banach spaces one can get analogous results to Theorems 3.4 and 13.6 by using total sets [15] instead of norming sets.
Corollary 13.11 [85]. Let X be a uniformly convex Banach space with the open unit ball B. Let f : B -> B be k B -nonexpansive, condensing with respect to a 11.11' and fixedpoint-free. Then there exists ~f E BB such that the sequence Uk} of iterates of f converges on B to the constant map taking the value ~f> uniformly on each ball strictly inside B.
Let f : B -+ B be a kB-nonexpansive mapping and 0 < s < 1. As before, we define hf(s,.) : B -+ B by
hf (s, z) = fs,z(hf (s, z)) = (1 - s) z + sf(hf (s, z)) = !if f:'z (0)
for z E B. This mapping is kB-nonexpansive and holomorphic if f is holomorphic by Corollary 2.11. The method of proof of Theorem 13.9 combined with Corollary 13.7 and Lemma 3.11 yield the following proposition.
Proposition 13.12 [85]. Let (X, 11·11) be a strictly convex Banach space with the open unit ball B. If f : B -+ B is kB-nonexpansive, condensing with respect to all.1I and fixed-point-free, then there exists ~f E BB such that {hf(s,·n converges uniformly on B as s -+ 1 - to the constant map taking the value ~f' This ~f is the unique limit point of the sequence of iterates {fk}.
Remark 13.13 In view of A. Stachura's example (Example 12.5) and Theorem 6.4, the approximating curves {hf(·,xn of fixed-point-free kB-nonexpansive self-mappings of the Hilbert ball BH behave much better than the sequences of iterates {fk(xn·
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Namely, they always converge to the so-called "sink point" ~f on the boundary of BH even when the sequences of iterates do not.
We now present two examples of nonreflexive Banach spaces to which Theorem 13.9 can be applied.
Example 13.14 [83]. Let [1 be furnished with the norm
00 x2 ( )2
Ilxll = Ilxlli + f; 2~ ,
where II . 111 is the usual norm of [1 and x = {Xj} E [1. If we take
(i.e., u(x) = 2::~1 UjXj for U = {Uj} EN = Co and x = {Xj} E [1), then we can observe that N is a norming set for the strictly convex Banach space ([1, II . II), the closed unit ball B in (11, II . II) is sequentially compact in the topology a([I,N), and ([1, II . II) has the Kadec-Klee property with respect to a([l,N).
Example 13.15 The James quasi-reflexive space J consists of all null sequences x = {xi} = 2:::1 xiei ( {ei} is the standard basis in CO) for which the squared variation
1
sup [f IxP; - XP;-112] 2
m,Pl<"'<Pm j=2 (13.3)
is finite, with the norm II . IIJ given by (13.3) (see [80] and [81]). A predual Banach space i to (J, II . II J) is generated by the biorthogonal functionals Ui to the basis {e;} = {el + ... +ei} (see [81] for details).The space (J, II·IIJ) has the uniform Opial property (see [147] for the definition) [125] and the uniform Kadec-Klee property, both with respect to the w*-topology ([125J combined with [88]). Let J be furnished with the new norm:
( 00 2)2 Ilxll = IlxlI~ + 8 (Ui ;~))
If we take N = i, then we can observe that N is a norming set for the strictly convex Banach space (J, 11·11), the closed unit ball Bin (J, 11·11) is sequentially compact in the topology a(J,N), and (J, II . II) has the Kadec-Klee property with respect to a(J,N) [29J.
14. Retraction, onto fixed point set
In order to formulate an extension of the Denjoy-Wolff theorem (Theorem 12.1) in the case where the mapping has a fixed point we need the following two definitions.
Definition 14.1 [53], [79]. Let D be a domain in a complex Banach space (X, 11·11). A net {fj}jEJ C H(D,X) is said to converge to a mapping f E H(D,X) in the topology
Fixed points of holomorphic mappings 497
of locally uniform convergence over D (or briefly, T-converge) if for every ball BII'II strictly inside D,
lim sup llii (x) - f (x)11 = O. JEJ XEBII'II
In this case we write f = T-limjEJ Ii.
If D is a bounded domain in a finite-dimensional X, then its T-topology is equivalent to the compact open topology on D, i.e., the topology of uniform convergence on compact subsets of D.
Definition 14.2 A mapping f E H(D) is said to be power convergent on D (with respect to the T-topology) if the sequence of iterates {fk: fO = I, fk+l = f 0 fk, k = 0,1,2, ... } T-converges to a mapping j E H(D,X).
So, in these terms the classical Denjoy-Wolff theorem (Theorem 12.1) states that a holomorphic self-mapping of the unit disc .a. is power convergent if and only if it is not an elliptic automorphism of .a.. In other words, if f E H(.a.) is not the identity and has a fixed point 0 E .a., then it is power convergent if and only if 11'(0)1 < 1.
Theorem 14.3 [198J, [200J . Let D be a bounded convex domain in a complex Banach space X and let f E H(D) have a fixed point. Then f is power convergent if and only if there exists 0 E Fix(f) such that the linear operator A = f' ( 0) satisfies the following conditions:
(i) the spectrum u(A) c .a. U {I}, and
(ii) 1 is a pole of at most the first order for the resolvent of A.
Note that in this case the limit mapping r = T-limk->oo fk is a holomorphic retraction of D onto Fix(f), i.e., r2 = rand Fix(f) = r(D). Therefore Fix(f) cD is a holomorphic retract of D. Note also that once the existence of a holomorphic retraction onto Fix(f) is established, it follows that Fix(f) is a complex-analytic submanifold of D [33J. Thus it is natural to ask if Fix(f) is a holomorphic retract of D even if f is not power convergent. If it is, then it would be of interest to construct a retraction of D onto Fix(F).
To formulate some results in this direction we will use the following definitions.
Definition 14.4 Let h be a holomorphic mapping from a domain D C X into X, and suppose that Fix(h) =I 0. Set g = I-h. A point 0 E Fix(h) is said to be quasi-regular if the following condition holds:
Ker g' (0) EB 1m 9' (0) = X. (14.1)
If, in addition, Ker 9'(0) = 0, then we say that 0 is a regular fixed point of h.
Note that condition (14.1) is, in fact, equivalent to condition (ii) of Theorem 14.3 ([62], [191]). However, if condition (i) of this theorem does not hold, then f is not power convergent. Nevertheless, it is shown in [141J that if f f' H(D) has at least one quasiregular fixed point, then there is another mapping cp E H(D) with Fix(cp) = Fix(f)
498
which is power convergent. More precisely, P. Mazet and J.-P. Vigue considered the Cesaro averages
(14.2)
and proved the following result.
Theorem 14.5 [141]. Let D be a bounded convex domain in X, and let f E H(D) have a fixed point. IfFix(f) contains at least one quasi-regular point, then there is an integer p such that the mapping Cp defined by (14.2) is power convergent to a holomorphic retraction of D onto Fix(f).
The following result shows, in particular, that in Theorem 14.5 the Cesaro averages defined by (14.2) are, in fact, power convergent for all p = 2,3, ....
Theorem 14.6 [166]. Let D be a bounded convex domain in X, and let f E H(D) have a fixed point. If Fix(f) contains a quasi-regular point a ED, then each mapping of the form
m
'P = L{3k!k, k=O
where 2:::;;'=0 {3k = 1 and 0 ::; (3k 1'1 for all 0 ::; k ::; m, is power convergent.
Now we tackle the retraction problem using metric methods.
Theorem 14.7 [44], [126]. Let D C X be a bounded convex domain in X. If f : D --+
D is a compact holomorphic (kv-nonexpansive) mapping and Fix(f) I' 0, then Fix(f) is relatively compact in X and a holomorphic (kv-nonexpansive) retract of D.
Proof. Since f is compact, for each xED the sequence {hn(x)} defined by
hm(x) = ~x+ (1-~) f(hm(x))
for m = 1,2, ... , is relatively compact in X by Corollary 13.7, and since Fix(f) I' 0, it lies strictly inside D. Let {m",}aEl be an ultranet in N ([14], [50], [87]). Then the ultranet {hma}'" is pointwise convergent to a compact holomorphic (kv-nonexpansive) retraction r : D --+ Fix(f). •
A more complicated proof is required in the case of all'lI-condensing mappings. However, by using Bruck's method [27] (which can be applied by Theorem 3.7), we can obtain the following result.
Theorem 14.8 [126]. Let D be a bounded convex domain in a reflexive Banach space X. If f : D --+ D is a condensing holomorphic (k v -nonexpansive) mapping and Fix(f) I' 0, then Fix(f) is relatively compact in X and a holomorphic (kvnonexpansive) retract of D.
To prove Theorem 14.8 we need three auxiliary lemmas. In all these lemmas we assume that X is a reflexive Banach space and that D c X is a bounded convex domain.
Fixed points of holomorphic mappings
Lemma 14.9 [126]. Suppose A is a nonempty subset of D and let
N(A) = {g : D --t D :g is holomorphic (kD-nonexpansive)
and g(x) = x for all x E A}.
Then N (A) is compact in the topology of weak pointwise convergence.
Proof. Let Xo be a point in A. Then
N (A) c II BkD (xo, kD (x, xo)) , xED
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where BkD (y, r) denotes a closed ball in the metric space (D, kD)' Since each BkD (y,,,.) is convex (see Lemma 3.3), hence weakly compact, the set
II BkD (xo, kD (x, xo)) xED
is compact in the topology of weak pointwise convergence. Now to obtain our claim it is sufficient to observe that N(A) is closed in this topology (see Corollary 2.11 and Theorem 3.7). •
Lemma 14.10 [126]. If A is a nonempty subset of D, then there exists an r in N(A) such that each 9 E N(A) acts as an isometric mapping on the range of r. In other words,
kD (g(". (x)),g(r(y))) = kD (r(x) ,r(y))
for all x,y E D.
Proof. We introduce the following order::; on N(A):
g::; h == ( V kD(g(x),g(y))::; kD(h(x),h(y))). x,YED
It is easy to observe (see the proof of Lemma 14.9) that the initial segment
I (h) = {g E N (A) : 9 ::; h}
is compact in the topology of weak pointwise convergence. By the Kuratowski-Zorn Lemma, (N(A),::;) contains a minimal element r and for this element we have
V go r EN (A). gEN(A)
The minimality of". in N(A) now implies that
V V kD(g(r(x)),g(".(y))) = kD(r(x),r(y)). gEN(A) x,YED
• Lemma 14.11 [126]. If A is a nonempty subset of D and for every y ED the".e exists 9 in N(A) such that g(y) E A, then A is a holomorphic (kD-nonexpansive) retract of D.
500
Proof. We will show that the mapping r obtained in Lemma 14.10 is a retraction of D onto A. To this end, it suffices to prove that r(x) E A for each xED. Indeed, by the assumption of the lemma applied to the point r(x), there exists 9 E N(A) such that g(r(x)) EA. Applying now Lemma 14.10 to x and z = g(r(x)) we get
kD(g(r(x)),g(r(z))) = kD(r(x),r(z)).
However, since z E A we have z = g(r(z)) = r(z), and therefore g(r(x)) = g(r(z)) and r(x) = r(z) = z E A. •
We are now ready to prove Theorem 14.8.
Proof of Theorem 14.8. It is obvious that Fix(f) is relatively compact in X. Let A = Fix(f). To apply Lemma 14.11 we have to show that for each xED there exists 9 E N(A) such that g(x) E A. When we fix xED, the x-th coordinate projection of N(A) is equal to
C={h(x):hEN(A)}
and C is nonempty, weakly compact and convex. Since f 0 h E N(A) whenever h E N(A), we also have f(C) C C. Using the assumption that f is all.1I -condensing we see that f has a fixed point in C. This means that there exists 9 E N(A) with g(x) E A = Fix(f), as required. •
Remark 14.12 In [141J and [142J P. Mazet and J.-P. Vigue prove that in a reflexive Banach space X, if Dc X is a bounded convex domain and f : D ---+ D is holomorphic with Fix(f) # 0, then Fix(f) is a holomorphic retract of D. However, their methods of proof use complex-analytic arguments and are not applicable to kD-nonexpansive mappings.
For our next result we need another lemma.
Lemma 14.13 [126J. Let X be a reflexive Banach space and let D C X be a bounded convex domain. If f : D ---+ D is a holomorphic (kD-nonexpansive) mapping with Fix(f) # 0, and the set A is relatively compact in X, invariant under f, and a holomorphic (kD-nonexpansive) retract of D, then Fix(f) n A is a nonempty, relatively compact in X and holomorphic (kD-nonexpansive) retract of D.
Proof. Let r be a holomorphic (kD-nonexpansive) compact retraction of D onto A. Since for: D -----> A, (f 0 r)IA = flA, Fix(f) # 0, and f : D ---+ D is a holomorphic (kD-nonexpansive) mapping, there exists a kD-bounded, convex and II . II-closed subset C C D which is (f 0 r)-invariant. Next, we observe that A is relatively compact in X and
(for)(C)cCnA.
This implies that the set Fix(f 0 r) is nonempty. It is easy to see that
Fix (f) n A = Fix (f 0 r)
and therefore Fix(f 0 r) is relatively compact in X. Now we show that Fix(f 0 r) is a holomorphic (kD-nonexpansive) retract. Let Al = Fix(f) n A. To apply Lemma 14.11
Fixed points of holomorphic mappings 501
we need to prove that for each xED, there exists g E N(A1) such that g(x) E A 1. Indeed, when we fix xED, the x -th coordinate projection of N(A1) is equal to
C1 = {h(x): h EN(A1)}
and C1 is nonempty, weakly compact and convex. Since (j 0 r) 0 h E N(A1 ) whenever hE N(A1), we also have (j 0 r)(C1) C C1. We now observe that
(jor)(C1)cC1nA
and therefore for has a fixed point in C1. This means that there exists g E N(A1) with g(x) E A1 = Fix(j 0 r). •
Theorem 14.14 [126]. Let X be a reflexive Banach space and let D c X be a bounded convex domain. If for each j = 1,2, ... , n, fj : D -> D is a holomorphic (kD-nonexpansive) mapping with Fix(/j) i' 0, at least one of these mappings is condensing and all the mappings fj, j = 1,2, ... , n, commute, then 0 i' nj=l Fix(jj) is relatively compact in X and a holomorphic ( kD-nonexpansive) retract of D.
Proof. We apply mathematical induction with respect to n, Theorem 14.8 and Lemma 14.13. •
We conclude this section with another existence result for a possibly infinite family of commuting mappings.
Theorem 14.15 [126]. Let X be a reflexive strictly convex Banach space and let B be the open unit ball in X. If fj : 13 -> 13, j E J, is a family of norm continuous condensing mappings which are holomorphic (each fj E N(B») in B, Fix(jj) i' 0 for each j, and all the mappings /j commute, then njEJ Fix(jj) is nonempty.
Proof. We consider three cases.
Case 1. There exists jo E J such that fjo (B) c aB. Then /jD is a constant mapping (see Corollary 2.17, the definition of N(B) in Section 7 and Lemma 3.11) and fio == Xo for some Xo E aBo Hence we obtain
for each j E J. In other words,
Xo E n Fix(jj). jEJ
Case 2. Fix(jjIB) i' 0 for each j E J. Then by Theorem 14.14 the set
n~lFix(jjkIB) c n;.:'=l Fix(jjk)
is nonempty and compact in X for every choice {j1, ... , jm} of indices. Hence we obtain
jEJ jEJ
Case 3. fj(B) c B for each j E J and Fix(jjoIB) = 0 for some jo E J. Then there exists ~fjo E aB such that
502
for every x E B (see Theorem 13.8). Hence we get
for each j E J, and therefore
E,fjo E n Fix(/j). jEJ
• 15. Fixed points of continuous semigroups
In this section we examine continuous semigroups, their fixed points and their orbits. Continuous semigroups of holomorphic mappings arise in several diverse fields, including, for instance, Markov stochastic branching processes, Krein spaces, and the geometry of complex Banach spaces (see [161] and [164] for relevant references). We begin with the following definitions.
Definition 15.1 A family S = {It}, where either t E R+ = {s E R : s 2: O} or tEN U {OJ, of self-mappings of a metric space (M,d) is called a (one-parameter) semigroup if fsH = fs 0 ft, s, t E R+ (s, tEN U {O}), and fa = I. A semigroup S = {It}, t E R+, is said to be (strongly) continuous if the function fo(x) : R+ -t M is continuous in t for each x E M. If t E NU {OJ we say that the semigroup S is discrete . In other words, a discrete semigroup S = {It}, tEN U {OJ, is the family of iterates of a self-mapping f = iI : M -t M.
Definition 15.2 Let S = {fd, t E R+ = {s E R : s 2: O} (or tEN U {OJ) be a continuous (or discrete) semi group acting on a metric space M. A point x E M is said to be a fixed (or a stationary) point of S if It (x) = x for all t E R+ (or tEN). A point x E M is said to be a periodic point of S if there is to E R+ (or to EN), such that fto(x) = X.
Note that in the discrete case a stationary point x E M of S is simply a fixed point of iI, and a periodic point x E M of S is a fixed point of f1', where mEN. In the sequel, the mappings it, t E R+ (or tEN), will often be denoted by f(t).
Theorem 15.3 [126]. Let S = {f(t)} be a continuous or discrete semigroup of kB'Hnon expansive self-mappings of B'H. Then the following statements are equivalent:
(i) there is to > 0 such that f(to) has a fixed point in B'H;
(ii) f(t) has a fixed point for all t;
(iii) there is a stationary point of the semigroup S.
Proof. It is clear that (iii)=>(ii)=>(i). Therefore it is enough to show that (i)=>(iii). Indeed, let x E B'H be a fixed point of f(to) for some to. Then x is a periodic point of S, i.e.,
f (t + to) (x) = f (t) (j (to) (x)) = f (t) (x)
Fixed points of holomorphic mappings 503
for all t. But the set {f(t)(x) : 0 :::; t :::; to} is a compact subset of B"H and therefore it is kBl}-bounded. By Theorem 7.6 this implies the existence of a common fixed point ~~ . Corollary 15.4 [126J. Let S be a continuous semigroup of holomorphic mappings of BH such that for some to E (0,00), fto = I. Then S is a group of automorphisms of BH of elliptic type, i.e. f (t) = M'::-~ 0 etA 0 M_a , where A is a linear conservative operator (Re (Ax,x) = 0 for all x in H), a E BH and M_a is a Mobius transformation of BH·
Proof. Since fto = I, Theorem 15.3 shows that S has a common fixed point a E BH· Consider the semigroup J(t) = M-aoftoM'::-~ defined on BH. It is clear that J(t)(O) = 0
for all t :::: 0 and that J(to) = I. Since
1(t) 0 1(to - t) = j(to) = I
for all 0 :::; t :::; to, we see that for all 0 :::; t :::; to, j-l(t) = j(to - t) and J(t) is an automorphism of BH. Hence J(t) is the restriction to BH of a linear isometry of H onto itself for each t :::: 0 (see Theorem 8.1). This implies our assertion (see [114J, [204]) .
• Definition 15.5 We will say that a class g of kwnonexpansive mappings on BH has the Denjoy- Wolff property if whenever f E g has no fixed point in BH the sequence of iterates {fk} strongly converges to a point ~f on the boundary 8BH of BH, uniformly on each compact subset of BH.
Proposition 15.6 [126J. Let g be a class of kBH-nonexpansive mappings on BH which has the Denjoy- Wolff property. Let S = {f(t)h;:::o be a one-parameter continuous semigroup of kBH-nonexpansive mappings on BH which has no common fixed point in BH. If f(to) E g for at least one to > 0, then there is a point ~ E 8BH such that S converges to ~, as t tends to infinity, uniformly on each compact subset of BH.
Proof. First we note that by Theorem 15.3, f(to) = fto has no fixed point in BH· Therefore there is a point ~!to E 8BH such that Ato = ft~ converges to efto' uniformly
on each compact subset of BH. Let C be a compact subset of BH. Since the semigroup S = {f(t)h;:::o is continuous, the set
C = {Zt = f (t) z: Z E C, 0 :::; t :::; to}
is also a compact subset of BH. Therefore, using the notation Zt = f(t)z, for each to> 0 one can find ko E N such that
sup Ilfta (z) - ~fto II = su~ sup IIAto (Zt) - efto II = su~ sup IIAto+t (Z) - ~fto II < to zEC zEC 0990 zEC 0990
for all k :::: ko. It now follows that
11ft (z) - ~fto II < to
for each t :::: koto and z E C. •
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Remark 15.7 [126]. The following classes of self-mappings of BH are known to enjoy the Denjoy-Wolff property:
(1) the class {h consisting of the condensing kBH-nonexpansive mappings (see Theorem 13.8);
(2) the class {h consisting of the firmly kBH-nonexpansive mappings of the first kind (see Theorem 12.16);
(3) the class 93 consisting of the firmly kBH-nonexpansive mappings of the second kind (see Theorem 12.16);
(4) the class 94 consisting of the averaged mappings of the first kind, i.e., h,>. = (1 - ),,)1 Ell )..f, where f is kBH-nonexpansive and)" E (0,1) (see Theorem 12.9);
(5) the class 95 consisting of the averaged mappings of the second kind, i.e., 12,>. = (1 - ),,)1 +)..j, where f is kBH-nonexpansive and)" E (0,1) (see Theorem 12.9).
Thus if a semigroup S = {f(t)}t>o contains at least one element f(to), to > 0, of one of the classes 9i, 1 ::; i ::; 5, then -;'e have the conclusion of Proposition 15.6.
Corollary 15.8 [126]. Let 1 be a kBH-nonexpansive self-mapping of BH.
1) The Cauchy problem
{ au't~' -=tiF+u(t,x)-f(u(t,x)) =0
u(O,x)=x
has a unique solution {u(t,x) : t E R+} C BH for each x E BH.
2) The semigroup {f(t)}t;:.o defined by the formula
f(t)x=u(t,x)
consists of kBH-nonexpansive mappings f(t), t 2': O.
3) If f has no fixed point in BH , then f(t) is also fixed-point-free for all t > 0, and f(t)x converges, as t -+ 00, to a point ~ on the boundary of BH for all x E BH.
Proof. First we note that the curve gt = (1 - t)1 + t1 converges to 1 as t -+ 0+ and
1 -t (I - gt) = 1 - f·
Since
kBH (gt (x) ,gt(Y))::; max [kBn (x,y) ,kBH (J(x)'!(y))],
each gt is kB-nonexpansive. Hence it follows by the product formula (see [167]) that for each t> 0 the mapping f(t) given by
f(t) = lim g1 k~oo k
is the solution operator for the Cauchy problem in 1) and is also a kBn-nonexpansive
mapping on BH. If now 1 has no fixed point in B H , then it follows that
n Fix (f (t)) = 0. t>O
Fixed points of holomorphic mappings 505
By Theorem 15.3 we now see that each It has no fixed point in BH.
At the same time direct calculations show that the following formula holds
I (t) (x) = e-tx + l e-(t-8) 1(1 (8) (x) )d8.
Since
111(1 (8)x)11 < 1
for all 8 E [0, t] and x E BH, we can write
where 9 = ~ r e-(t-8) (101 (8») d8 1- e t 10
is a kBH-nonexpansive mapping on BH (cf. [18]). Hence It is an averaged mapping of the second kind and the result follows by Proposition 15.6 and Remark 15.7. •
Remark 15.9 Similar results for other kBH-nonexpansive semigroups can be found in [154] and [160]. The paper [154] also contains a continuous analog of Theorem 10.5.
A continuous one-dimensional analog of the classical Denjoy-Wolff Theorem (Theorem 12.1) was established by E. Berkson and H. Porta [26] in their study of composition operators on Hardy spaces of the unit disc. M. Abate [2] generalized their result to the finite-dimensional case. In order to consider the infinite-dimensional case, we need the following definition.
Definition 15.10 Let D be a domain in a Banach space X. A one-parameter semigroup S = Uth~o c H(D) is said to be locally unilormly continuous (or briefly, T -continuous) if
T- lim It = 1. t--+o+
Proposition 15.11 [167] Let D be a bounded domain in X and let S = Uth~o C
H(D) be a strongly continuous semigroup. The lollowing conditions are equivalent:
(a) S is aT-continuous semigroup;
(b) the differences
1 gt = t (I - It)
are unilormly bounded on each subset strictly inside Dj
(c) lor each xED, there exists the strong limit
lim ! (1 - It) (x) = g (x) t--+o+ t
which is bounded on each subset strictly inside D.
Since it is clear that if (c) holds, then the mapping g belongs to H(D,X) (see Corollary 2.11), it is called the holomorphic infinitesimal generator of the semigroup S. As a matter of fact,
g = T- lim ! (I - It) , t--+O+ t
506
i.e., the convergence in (c) is actually locally uniform convergence over D [161].
Moreover, condition (c) means that aT-continuous semigroup is right-differentiable with respect to the parameter t at the origin. This implies, in turn, that for each xED, ft(x) is differentiable at each point t E lR.+ and that the function u : lR. -+ X defined by
u (t) = ft(x)
is the solution of the Cauchy problem ([161])
{ u'(t)+g(u(t)) =0, u (0) = x.
Since in the finite-dimensional case a continuous semigroup of holomorphic selfmappings is T-continuous, it follows that such a semigroup always has a holomorphic infinitesimal generator (see also [22] and [6]).
The uniqueness of the solution to the Cauchy problem shows that
F = nFix(ft) = Null(g) = {x ED: g(x) = O}, t~O
i.e., the stationary point set of aT-continuous semigroup S = {ft}t~O on D coincides with the null point set of its generator g. This property of T-continuous semigroups allows us to study the structure of the stationary point set F as well as the asymptotic behavior of the semigroup S by using the local and global properties of the analytic set Null(g). It turns out that under certain conditions this set is, in fact, a complex-analytic submanifold of D.
Theorem 15.12 [161], [167]. Let D be a bounded convex domain in X, 9 E H(D,X) the generator of a one-parameter continuous semigroup S of holomoryhic self-mappings of D and h = I-g. If Fix(h) contains a quasi-regular point a E D, then it is a holomoryhic retract of D, hence a complex-analytic submanifold of D tangent to K er g' (a). In particular, if a E Fix( h) is regular, then it is the unique stationary point of S.
Remark 15.13 Note that since condition (14.1) always holds in the finite-dimensional case, so does Theorem 15.12.
However, simple examples show that even in the finite-dimensional case, and even if F = nt>oFix(ft) is not empty, the net {ft(x)h>o might not converge, as t -+ 00, to a point ofF for all xED. Take, for example, the s-;;migroup (even group) ft = eit I, where I is the identity mapping on the open unit ball B C X. Nevertheless, regarding the T-convergence of a generated (T-continuous) semi group of holomorphic self-mappings of D, the situation can be completely described by using the spectral properties of the linear operator obtained by differentiation at a stationary point. We again denote the spectrum of a linear operator A : X -+ X by O"(A).
Theorem 15.14 [99] Let D be a bounded convex domain in a complex Banach space X and let S = {ft(x)h>o C H(D) be a one-parameter T-continuous semigroup such that F = nt~O Fix(ft) is-not empty. Then the following assertions are equivalent:
Fixed points of holomorphic mappings 507
(1) the semigroup {ft(x)h~o is T-convergent, as t --+ 00, to a holomorphic mapping r which is a retmction 0/ D onto Fi
(2) there exists a quasi-regular point a E F 0/ 9 = limt--->o+ t(I - It) such that
{IT(gl(a))}npE<C: Re'\=O, '\;f0}=0.
Note that even in the one-dimensional case, when there is a fixed point, both versions of the Denjoy-Wolff theorem (discrete and continuous) establish the convergence of the corresponding semigroup in all cases except for automorphisms with exactly one fixed point. For higher dimensions, the scope of counterexamples is wider.
However, it turns out that for an arbitrary holomorphic self-mapping, one can construct an associated continuous semigroup which has the same fixed point set and which converges to a retraction onto this set provided a rather simple condition (which always holds in the finite-dimensional case) is satisfied.
More precisely, let D be a bounded convex domain in X and let / E H(D). It was shown in [161] that 9 = 1 - / generates a T-continuous one-parameter semigroup S = {Jt(x)h~o C H(D). We refer to this semigroup as the semigroup associated with /.
Theorem 15.15 [99] Let D be a bounded convex domain in a complex Banach space X and let / E H(D). Suppose that / has a fixed point a in D such that 1m (1 - !,(a)) is closed. Then .the associated semigroup {fth~o is T-convergent, as t --+ 00, to a holomorphic mapping r E H(D) which is a retmction 0/ D onto the fixed point set Fix(f).
This theorem is, in fact, a consequence of Theorem 15.14 because the fixed point a is quasi-regular and the semigroup associated with a holomorphic self-mapping always satisfies the spectral condition (2) of that theorem.
16. Final notes and remarks
For more information on Kobayashi distances and Schwarz-Pick systems see [42], [68], [78], [110], [112], [194], [195], [196] and [i97].
Applications of the metric approach to the study of holomorphic inequivalence of domains can be found in [60] and [179] (see also [63] and [86]).
Different approaches to the study of kBH can be found in [65], [178], [182] and [183].
More information on fixed and null points of holomorphic mappings and their applications is in [8], [9]' [12], [42], [44], [72], [90], [93], [94], [95], [100], [102], [127], [140], [141], [155], [157], [163], [186], [196], [199], [205], [206], [207], [208], [209] and [211].
The approximate fixed point property in the Hilbert ball and in more general hyperbolic spaces is investigated in [60], [154] and [180].
For results on commuting families of holomorphic mappings see [3], [7], [51], [134], [185], [189] and [210].
Iterates of holomorphic and p-nonexpansive mappings, approximating sequences, approximating curves and ergodic theory are also studied in [4], [5], [21], [23], [31], [35], [97], [120], [122], [136], [143], [151]' [158], [159], [160], [165], [166], [169], [184], [190], [198] and [200].
508
Semigroups of holomorphic and p-nonexpansive mappings are examined, for example, in [2], [6], [10], [11], [89], [90], [92], [94]' [95], [96], [98]' [101]' [152], [156], [162], [164], [167], [168]' [201]' [202] and [203]. We note, in particular, that the related concept of kBH-monotonicity is studied in [10] and [164].
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Chapter 15
FIXED POINT AND NON-LINEAR ERGODIC THEOREMS FOR SEMIGROUPS OF NON-LINEAR MAPPINGS
Anthony To-Ming Lau
Department of Mathematical Sciences
University of Alberta
Edmonton, Alberta, Canada T6G-2G1
Wataru Takahashi
Department of Mathematical and Computing Sciences
Tokyo Institute of Technology
Ohokayama, Meguro-ku, Tokyo 152-8552, Japan
1. Introduction
Let 8 be a semigroup, £00(8) be the Banach space of bounded real valued functions on 8 with the supremum norm. There is a strong relation between the existence of an invariant mean (or submean) on an invariant subspace of £00(8) and fixed point or ergodic properties of S when S is represented as a semigroup of nonexpansive mappings on a closed convex subset of a Banach space. It is the purpose of this chapter of the Handbook to exhibit on some recent results on such relations. Since this handbook is intended for researchers and graduate students, detailed proofs for central results, historical remarks, open problems and many references will be included. It is our hope that our effort will generate further research in this direction of non-linear analysis which depends on the ideal theory of 8, and existence of an invariant mean (or submean) on a subspace of £00(8) of a semigroup 8.
This chapter is organized as follows: In section 3, we study the notion of- non-linear "mean" (called submean) on a subspace of £00(8), and its relation with left reversibility of 8 (i.e. 8 has the finite intersection property with respect to closed right ideals of 8). In section 4, we prove among other things that (Theorem 4.4) if £00(8) has a left invariant submean J.L, and S = {T. : s E 8} is a representation of 8 as nonexpansive maps on a weakly compact convex subset C of a Banach space E with more than one point, and C has normal structure, then the set {z E C;J.LtllTtx - zll = inf{J.LtIlTtxyll;y E C}} is a proper subset of C. Theorem 4.4 is then used to obtain an improvement
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WA. KirkandB. Sims (eds.), Handbook o/Metric Fixed Point Theory, 517-555. © 2001 Kluwer Academic Publishers.
518
of a classical fixed point theorem of Lim [66] in section 5 (Theorem 5.1) for left reversible semigroups of nonexpansive maps. In section 6, we study the relation between left ideal orbit and the fixed point set F(S) when S is a representation of S as nonexpansive mappings on a closed convex subset G of E into G. The most important result of this section is Theorem 6.7 which shows that if the norm of E is Frechet differentiable, then for each x E G, F(S) n Q(x) consists of at most one point, where Q(x) is the intersection of all left ideal orbits of x. Theorem 6.7 is crucial for some of the ergodic theorems (Theorem 7.4 and Theorem 7.10) which we study in section 7. In section 8, we shall discuss some related results.
2. Some preliminaries
All topologies in this paper are assumed to be Hausdorff. If E is a Banach space and A ~ E, then A and coA will denote the closure of A and the closed convex hull of A in E, respectively.
Given a non-empty set S, we denote by floo(S) the Banach space of bounded real-valued functions on S with the supremum norm. Let S be a semigroup. Then a subspace X of floo(S) is left (resp. right) translation invariant if fla(X) ~ X(resp. ra(X) ~ X) for all a E S, where (flaf)(s) = f(as) and (raf)(s) = f(sa), s E S.
A semitopological semigroup S is a semigroup with Hausdorff topology such that for each a E S, the mappings s ...... a . sand s ...... s . a from S into S is continuous. Examples of semi topological semigroups include all topological groups, the set M( n, IC) of all n x n matrices with complex entries, with matrix multiplication and the usual topology, the unit ball of floo with weak*-topology and pointwise multiplication, or 8(H) ( = the space of bounded linear operators on a Hilbert space H) with the weak*-topology and composition.
If S is a semi topological semigroup, we denote by GB(S) the closed sub algebra of floo(S) consisting of continuous functions. Let LUG(S) (resp. RUG(S)) be the space of left (resp. right) uniformly continuous functions on S, i.e. all f E GB(S) such that the mapping from S into GB(S) defined by s -> flsf (resp. s -> rBf) is continuous when GB(S) has the sup norm topology. Then as is known (see [12]), LUC(S) and RUG(S) arc left and right translation invariant closed subalgebras of GB(S) containing constants. Note that when S is a topological group, then LUG(S) is precisely the space of right uniformly continuous functions on S defined in [36]. Also let AP(S) (resp. W AP(S)) denote the space of almost periodic (resp. weakly almost periodic) functions f in GB(S), i.e. all f E GB(S) such that {flaf; a E S} is relatively compact in the norm (resp. weak) topology of GB(S), or equivalently {raf; a E S} is relatively compact in the norm (resp. weak) topology of GB(S). Then as is known [12, p.164]' AP(S) ~ LUG(S) n RUG(S), and AP(S) ~ W AP(S). When S is a group, then W AP(S) ~ LUC(S) n RUG(S) (see [12, p.167]).
Let E be a real Banach space and let E* be its topological dual. Then, the value of x' E E* at x E E will be denoted by (x,x') or x*(x). With each x E E, we associate the set
J(x) = {x' E E*; (x,x') = IIxl1 2 = Ilx*ln. Using the Hahn-Banach theorem, it is immediately clear that J(x) =1= ¢ for each x E E. The multivalued operator J: E --t E* is called the duality mapping of E. Let See) = {x E E; Ilxll = I} be the unit sphere of E. Then the norm of E is said to be Gdteaux
Theorems for semigroups of non-linear mappings 519
differentiable (and E is said to be smooth) if for each x, y E S(E),
r Ilx + Ayll - Ilxll ,\~ A
exists. It is said to be Fnichet differentiable if for each x E S(E), this limit is attained uniformly for y E S(E). It is said to be uniformly Giiteaux differentiable if for each y E S(E), this limit is attained uniformly for x E S(E). It is well known that if E has a Frechet differentiable norm, then J is single value and norm to norm continuous; see [14] or [23] for more details.
3. Submean and reversibility
Let S be a non-empty set and X be a subspace of £oo(S) containing constants. Then fL E X' is called a mean on X if IIfLll = fL(l) = 1. As is well known, fL is a mean on X if and only if
inf f(s) S fL(f) S supf(s) sES sES
for each f E X.
By a submean on X, we shall mean a real-valued function fL on X satisfying the following properties :
(1) fL(f+9)SfL(f)+fL(9) for every f,gEX;
(2) fL(aJ) = afL(f) for every f E X and a 2: 0;
(3) for f, 9 EX, f S q implies fL(f) S fL(9);
(4) fL(C) = c for every constant function c.
If S is a semigroup, and X <::: £oo(S) is a left translation invariant subspace of £oo(S) containing constants, a submean fL on X is left [right] invariant if fL(£aJ) = fL(f) [fl(raJ) = fl(f)] for each a E S, f EX.
We abbreviate left invariant submean to LISM and left invariant mean to LIM.
Depending on time and circumstances, the value of a submean (or mean) fL at f E X will also be denoted by fL(f), (fL, J) or fLtf (t).
A semitopological semigroup S is left reversible if any two closed right ideals of Shave non-void intersection.
Lemma 3.1 Let S be a semitopological semigroup and X be a left translation invariant subspace ofCB(S) containing constants and which separates closed subsets of S. If X has a LISM, then S is left reversible.
Proof. Let fL be a LISM of X, hand h be disjoint nonempty closed right ideals of S. By assumption, there exists f E X such that f == 1 on hand f == 0 on h. Now if al E h, then £a,! = 1. So,
fl(f) = P·(£a,!) = 1.
But if a2 E 12 , then £a2f == 0 . So fL(f) = fL(£a2J) = 0, which is impossible. •
520
Corollary 3.2 If S is normal and CB(S) has a LISM, then S is left reversible.
Corollary 3.3 If S is normal and CB(S) has a LISM, then AP(S) has a LIM.
Proof. This follows from Corollary 3.2 and [47, Corollary 3.3J. • If S is a left reversible semitopological semigroup, then (S,:S) is a directed system when the binary relation ":s" on S is defined by a :S b if and only if {a} U as;;:> {b} U bS, a,b E S.
Lemma 3.4 Let S be a semitopological semigroup, J be a non-empty subset of Sand f E LUC(S). If sup{f(t); t ~ u} 2 (3 for each U E J, then sup{f(t); t ~ p} 2 (3 for each p E J.
Proof. Let p E] and sup{J(t); t ~ p} S; (3 - 8,8> O. Then
f (ps) S; {3 - 8 for each s E S U {e },
where xe = ex = e. Let Un E J be a net such that U a --t p. Hence lIt'uaf - t'pfll --t O. Consequently there exists ao such that
8 f(uas)S;{3-'2 foreach sESU{e}, a2ao.
Hence for a 2 ao, we have sup{f(t); t ~ un} S; (3 - 8/2, which contradicts the assumption. •
A semitopological semigroup S is left amenable [resp. subamenableJ if LUC(S) has a LIM [resp. LISMJ.
Proposition 3.5 Let S be a semitopological semigroup. If S is left reversible, then S is left subamenable.
Proof. For each f E CB(S), define
M(f) = inf sup f(t). S tts
Then M is a submean on CB(S). Indeed, if f, g E CB(S), and E > 0, choose a, b E S such that
sup f(t) S; M(f) + E and sup g(t) S; M(g) + E. t?:::a t?:::b
Let c E as n bS (which is non-empty by left reversibility). Then c t a, and c t b. Hence SUPt?:::c f( t) S; M(f) + E and SUPt?:::c g(t) S; M(g) + E. SO
sup(f(t) + g(t)) S; sup f(t) + sup g(t) t~c ttc ttc
S; M(f) + M(g) + 2E.
Consequently
M(f + g) S; M(f) + M(g) + 2E.
Theorems for semigroups of non-linear mappings 521
Since f > 0 is arbitrary, condition (1) for submean holds. Proofs of conditions (2), (3) and (4) are routine.
To see that 1-£ is left invariant, let f E LUC(S) and a E S. Then
I-£(£af) = inf sup f(at) 8 tts
= inf {sup{f(at)jt E sSU is}}} 8
= inf {sup{f(at)j tEsS U {s}}} 8
(by continuity of f and multiplication in S) = in£{ sup{f(ast)jt E S U {e}}}
s
(where se = s)
= inf {sup{f(t)jt E asS U {as}}} s .
= inf sup f(t) s ttas
~ I-£(f).
To prove the reverse inequality, let a = I-£(f) and f3 = I-£(£af), f E LUC(S). Then for each s E S, SUPttas f(t) ~ f3. Hence, by Lemma 3.4, .
sup f(t) ~ f3 for all p E as. ttv
(3.1)
If a < f3, let E = (f3 - a)/2. Choose So such that SUPttso f(t) < a + f. Then for each s t so, SUPtts f(t) < a + E. Let
P E soS naS.
Then p t So j so SUPttv f(t) < a + E, contradicting (3.1). • Corollary 3.6 Let S be a discrete semigroup. Then S is left reversible if and only if S is left subamenable. In this case W AP(S) has a LIM.
Proof. The first statement follows from Corollary 3.2 and Proposition 3.5, and the last statement follows from [39] (see also [48]). •
Notes and remarks.
(i) When S is a discrete semigroup, the following implications hold:
A(S) left amenable
S left subamenable ¢::::::} A(S) left reversible ~
il W(S) has LIM
A(S) has LIM
522
The implication: S is left reversible =} A(S) has a LIM for any semitopological semigroup was established in [48]. During the 1984 Richmond, Virginia, conference on analysis on semigroup, T. Mitchell [72] gave two examples to show that for discrete semigroups A(S) has a LIM =} S left reversible (see [50]). The implication: S is left reversible =} W(S) has a LIM for discrete semigroups was proved by Hsu [39]. Results and their proofs presented in this section are essentially taken from our work in [61].
(ii) Clearly every mean is a submean. The notion of submean was first introduced by Mizoguchi and Takahashi in [74].
Let SM denote the set of submeans on X. For each ¢ E SM, -lIfll s ¢(I) s Ilfll by (3) and (4). Hence SM may be identified as a subset of the product space llfEX[-lIfll, Ilfll], which is compact by Tychonoff's Theorem. Hence SM is a compact convex subset of the product topological vector space llfEx lRf' where each lRf = lR.
(iii) Corollary 3.2 is false without normality. Indeed, let S be the topological space which is regular and Hausdorff and GB(S) consists of constant functions only ([35]). Define on S the multiplication ·st = s for s, t E S. Let a E S be fixed. Define
fl(l) = f(a) for all f E GB(S).
Then fL is a LISM on GB(S), but S is not left reversible.
(iv) The theory of amenability began in 1904 when Lebesgue asked if the uniqueness of the Lebesgue integral is still preserved if the Monotone Convergence Theorem (which is equivalent to countable additivity) is replaced by finite additivity. Banach in 1923 [6] showed that there exists a mean on the bounded real-valued functions on the integers which is invariant under all translations, i.e. the group of integers under addition is amenable. This stands in contrast to the result of Hausdorff [34] who showed that there does not exist any mean on the bounded real-valued functions on the sphere in three dimensions, which is invariant under all rotations. Von Neumann [106] introduced and studied the class of discrete amenable groups. The term "amenable" is due to Day [19]. Since then the theory has grown in many directions to include amenability of discrete and topological semigroups, locally compact groups and Banach algebras (in particular G*-algebras and von Neumann algebras). Interested readers should consult the books [32], [77] and [76], and the survey article [51].
(v) The class 8 of all left reversible semitopological semigroups includes trivially all topological semigroups which are algebraically groups, and all commuting topological semigroups.
The class § is closed under the following operations.
(a) If S E 8 and S' is a continuous homomorphic image of S, then S' E 8.
(b) Let So. E §, a E I and S be the topological semigroup consisting of the set of all functions f on I such that f(a) E Sa, a E I, the binary operation defined by fg(a) = f(a)g(a) for all a E I and f,g E S, and the product topology. Then S E S.
(c) Let S be a topological semigroup and Sex, a E I, topological sub-semi groups of S with the property that S = USex and, if aI, a2 E I, then there exists a3 E I such that Sex3 :2 Sexl U Sex2. If So. E § for each a E I, S E 8.
Theorems for semigroups of non-linear mappings 523
4. Submean and normal structure
A convex subset C of a Banach space is said to have normal structure [13] if every bounded convex subset D of C with IDI > 1 contains a point x such that
sup{llx - yll : y E D} < diam(D),
where diam(D) = sup{lIx - yll : x, y E D} (the diameter of D).
In their original paper [13], Brodskii and Milman characterized normal structure as follows: A convex subset of a Banach space has normal structure if and only if it contains no diametral sequences. (A diametral sequence is a nonconstant bounded sequence {xn };:O=1 such that d(xn+1,co(x1, ... ,xn)) -> diam({xn };:O=1)') The following lemma is a simple variation of the above characterization.
Lemma 4.1 A convex subset C of a Banach space has normal structure if and only if it does not contain a sequence {xn } such that for some c > 0, IIxn - xmll :s; c, Ilxn+1 - xnll ~ c - ~ for all n ~ 1, m ~ 1, where xn = ~ I:~=1 Xi·
Proof. If C contains a bounded convex subset D such that IDI > 1 and
sup{llx - yll : y E D} = diam(D)
for every xED, then it is easy to choose, by induction, a nonconstant sequence {xn} ~ D satisfying the condition in the Lemma with c = diam(D). On the other hand, assume that {xn } ~ C is a sequence satisfying the condition in the Lemma. If x E CO(X1, .. . , x n ), it is not difficult to show that x = A1Xn + A2Xi2 + ... + AnXin for some i2, ... ,in E {I, ... ,n} and Ai, 1:S; i:S; n, with I:~=1Ai = 1, 0 < A1:S; n, and Aj :s; 0 for 2 :s; j :s; n. It follows that
1 c ~ Ilxn+1 - xii ~ c - -
n
for every n ~ 1 and every x E CO(X1, ... ,xn). Hence d(xn+1,co(x1, ... ,xn)) -> C and c is necessarily equal to diam({xn}). •
Let 8 be a non-empty set and f be a function from 8 into C such that {J(s)j s E 8} is bounded, where C is a closed convex subset of a Banach space E with more than one point. For each x E C, define fx(s) = IIf(s) - xII, s E 8. Let X be a closed subspace of Coo (8) containing constants such that fx E X for each x E X and let J1. be a submean on X. Define r : C -> lR by r(x) = J1.(fx).
Lemma 4.2 The function r : C -> lR is continuous, convex on C and if IIxnll -> 00,
then r(xn) -> 00.
Proof. Let Xn -> x, and X n , x be in C. Then since
Ilf(t) - xnll- IIf(t) - xII :s; Ilxn - xii and Ilf(t) - xii - Ilf(t) - xnll :s; Ilxn - xiI,
it follows that
Hence 1J1.(fxn) - J1.(fx) I :s; Ilxn - xii -> 0, which implies that r is continuous.
524
If 0i,(:J ~ 0, Oi + (:J = 1 and x,y E C, then since
IIf(t) - (OiX + (:Jy)1I ::; Oillf(t) - xII + (:Jllf(t) - yll,
we have r(Oix + (:Jy) ::; Oir(x) + (:Jr(y).
Finally, if Ilxnll -+ 00, then since Ilxnll ::; Ilxn - f(t) II + Ilf(t)1I for each t E S, we have
IIxnll ::; r(xn) + M -+ 00, where M = sup{lIf(t)II}. tES
• Let Po = inf{r(x);x E C}. Then we have the following:
Lemma 4.3 If C is non-empty weakly compact and convex, then the set
A = {x;r(x) = po}
is non-empty, closed and convex. Furthermore, if C = A, then
Po = inf sup{lIx - YII}· YECxEC
Proof. Since r : C -+ lR is continuous and convex (Lemma 4.2), r is weakly lower semicontinuous. So A is non-empty, closed and convex. Suppose C = A and let E > 0 and a = {Xl,... , xn} 0;;; C. Consider a system {h.;} f=l of convex weakly lower semicontinuous functions on C defined by
h;(x) = IIx-x;ll, forxEC and i=I, ... ,n.
We shall show that the set D" = {z E C; h;(z) ::; Po + E,i = 1, ... , n} is non-empty. In fact, from A = C, we have r(x;) = Po for i = 1,2, ... , n. So for any {Oii}f=l 0;;; lR with Oil, 0i2,··· , Oin ~ 0 and I:~=l Oii = 1, we have
J.Lt (~Oiillf(t) - XiII) ::; ~ Oiir(Xi) = Po < Po + E.
Then, there exists to E S such that
n
E Oiillf(to) - xiII ::; Po + E
i=l
and hence I:~=l Oiih.;(J(tO)) ::; po + E. Hence by Fan's Theorem for convex inequalities on a topological vector space [26], there exists z E C such that
h.;(z) = liz - XiII ::; Po + E for each i = 1, ... , n.
This implies that the set D" is non-empty. Hence by weak compactness of C, the set D = {z E C; IIx - zll ::; po + E for all X E C} is also non-empty. So we have
Po = min{r(y);y E C}
::; min {sup IIf(t) - yll} yEC tES
::; min {sup IIx - yll} ::; Po + E. yEC xEC
Theorems for semigroups of non-linear mappings
Since to > 0 is arbitrary, we have
pO = inf {sup \Ix - YII}. yEO xEO
525
• If 8 is a semigroup and X is left translation invariant, a submean ft on X is left sub invariant if
ft(la!) :::: ft(f) for each f E X and a E 8.
A representation S = {Tsi s E S} as mappings from a subset C of a Banach space into C is called X -admissible iffor each x, y E C, the function t ...... IlTtx - yll belongs to X.
Theorem 4.4 Let C be a non-empty weakly compact convex subset of a Banach space E. If C has more than one point and normal structure, then C satisfies:
(P) Whenever 8 is a semigroup, cI> is a closed left translation invariant subspace of £00(8) containing constants with a left sub invariant submean ft, S = {Tsi s E 8} is a cI>-admissible representation of S as nonexpansive mappings from C into C, then the set Ax = {y E CiJLtIlTtx - y\l = Px} is a proper subset of C for some x E C, where Px = inf{fttllTtx - y\li y E C}. Furthermore, for each x E C the set Ax is non-empty, closed, convex and Ts-invariant.
Proof. If Ax = C for all x E C, then by Lemma 4.3,
Px = Po = inf {sup IIx - Y\l} = fttllTt x - yll xEO yEO
for all x, y E C. So let
Ao = {Z E CiSUP IITtx - zil ~ Po, "Ix E c}. tEB
By weak compactness of C and (4.1), there exists Zo E C such that
sup \lzo - yll = Po· yEO
Hence Ao is a non-empty set. Let Zo E Ao and 8 E 8. Then for any x E C,
and
Po = ftt\lTtx - zoll ~ ftt\lTstx - zo\l ~ sup \lTstx - zoll tES
~ sup IITtx - zoll ~ pO tES
Po = ftt\lTtx - Tszoll ~ ftt\lTstx - Tszo\l ~ sup \lTstx - Tszo\l tES
~ sup IITtx - zoll ~ Po tES
(4.1)
by left subinvariance of ft. To apply Lemma 4.1, fix Zo E Ao. Then since pO
fttllTtzo - zoll, there exists 81 E 8 such that \IT.1zo - zo\l :::: Po - 1. Let Xl = Zo, X2 = T81 ZO and
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Since Po = JLtllTtzo - X211 ::::: JLtilTsltZO - x211, there exists 82 E S such that
1 IITslS2Z0 - x211 :::: Po - 22.
So, let X3 = TSlS2Z0. Then, we have
and
IlxI - x211 = IIzo - TSI zoll ::::: sup IIzo - TtZo11 = Po, tES
IIx2 - x311 = IITslzo - TSlS2Zoli ::::: Ilzo - TSlzol1 ::::: po,
II x3 - xIII = IITslS2Z0 - zoll ::::: sup IITtzo - zoll = po· tES
Similarly, let
1 1 1 X3 = 3XI + 3X2 + 3X3.
Then, Po = JLtllTtzo - x311 ::::: JLtilTsls2tZO - x311, there exists 83 E S such that
1 IITslS2S3Z0 - x311 :::: po - 32 .
So, let X4 = TSlS2S3Z0. Then, we have
and
IIX4 - xIII = IITslS2S3Z0 - zoll ::::: sup IITtzo - zoll = Po, tES
IIx4 - x211 = IITslS2S3Z0 - TSI zoll ::::: IITs2S3Z0 - zoll ::::: sup IITtzo - zoll = Po, tES
IIX4 - x311 = IITslS2S3Z0 - TSlS2zoli ::::: II Ts3 zo - zoll ::::: sup IlTtzo - zoll = Po· tES
By mathematical induction, let X5 = TSlS2S3S4Z0, X6 = TSlS2S3S4S5Z0,· ... Then we havc
1 Ilxn-xmll:S:po, Vn,m and IIXn+I-Xn ll::::po-2".
n
Using Lemma 4.1, C does not have normal structure. This is a contradiction. To see that Ax is Ts-invariant, x E C, 8 E S, let y E Ax. Then
Hence JLt(IITtx - TsYII) = Px, i.e., TsY E Ax· • Proposition 4.5 Let K be a closed convex subset of a Banach space E. If K satisfies the following property (Q), then K has normal structure:
(Q) Whenever S is a non-empty set and f is a function from S into C, where C is a bounded closed convex subset of K with more than one point, and for each x E C, the function fx(s) = Ilf(s)-xll belongs to a closed subspace X ojeOO(S) containing constants, and JL be a submean on X, r(x) = JL(fx), Po = inf{r(x);x E C}, then the set A = {x E C; r(x) = po} is a proper subset of C.
Theorems for semigroups of non-linear mappings 527
Proof. Suppose K does not have normal structure. Then there exists a bounded closed convex subset C of K such that diam(C) > 0 and SUPyEC lIy - xII = diam(C) for every x E C. Let S = C, X = £oo(S), f : C ..... C and f(s) = s. Then for each x E C, fx(s) = lis - xii, SEC is in £OO(C) (since C is bounded).
Let fL(h) = suph, hE £OO(C), and r(x) = flUx), x E C, r = inf{r(x);x E C}. Then since r(x) = diam(C) for every x E C, we have r = diam(C), that is,
A = {x E C: r(x) = r} = C.
This completes the proof. • Notes and remarks.
It is an immediate consequence of Brodskii and Milman's characterization [13] of normal structure that all compact convex set have normal structure. Using a straight forward transfinite induction argument, they showed that a weakly compact convex set K which has normal structure always contains a point z which is fixed under every linear isometry. In [28] (see also [27]), Kirk proved the following fundamental existence theorem for nonexpansive mappings.
Theorem 4.6 Let K be a nonempty, weakly compact, convex subset of a Banach space, and suppose K has normal structure. Then every nonexpansive mapping T : K ..... K has a fixed point.
In [1] Alspach gave the first example of a weakly compact convex set (in Ll[O, 1]) without normal structure.
Lemma 4.1 is duc to Lim [67]. The other results presented in this section are takcn from [62].
5. Fixed point theorem
We are now ready to apply Theorem 4.4 to prove a fixed point theorem for a semigroup of nonexpansive mappings on weakly compact convex subsets of a Banach space with normal structure. This is the first known application of Fan's Theorem for convex inequalities on a topological vector space (see proof of Theorem 4.4) in fixed point theory for semigroup of nonexpansive mappings.
Theorem 5.1 Let S be a semitopological semigroup, let D be a nonempty weakly compact convex subset of a Banach space E which has normal structure and let
S = {Ts; s E S}
be a continuous representation of S as non expansive self mappings on D. Suppose RUC(S) has a left subinvariant submean. Then S has a common fixed point in D.
Proof. We first prove that for any xED and y E E, a function h defined by h(t) = IITtx - yll for all t E S is in RUC(S). In fact, we have, for s,u E S,
Ilrsh - ruhll = sup I(rsh)(t) - (ruh)(t)1 = sup Ih(ts) - h(tu) I tES tES
= sup IllTtsx - yll-IITtux - ylll tES
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Let
::; sup IlTtsX - Ttuxll ::; IITsx - Tuxll. tES
U = {K CD: K is nonempty, closed, convex, and Ts-invariant }.
Then by Zorn's Lemma, there exists a minimal element C of U. Let diam(C) > 0 and let Jl be a left subinvariant submean. Then, for any x E C,
Ax = {z E C: JltllTtX - zll = min JltllTtx - Yll} yEC
is nonempty, closed, convex, and Ts-invariant (see Theorem 4.4). So, we have Ax = C by minimality of C. Hence by Theorem 4.4, C cannot have normal structure which is a contradiction. •
Corollary 5.2 Let S be a left reversible semitopological semigroup. Let D be a nonempty weakly compact convex subset of a Banach space E which has normal structureand let S = {Tsi s E S} be a continuous representation of S as nonexpansive self mappings on D. Then S has a fixed point in D.
Proof. If S is left reversible, define Jl(J) = infs SUPtEsS f(t). Then the proof of Lemma 3.6 in [12] shows that Jl is a submean on CB(S) such that Jl(£aJ) 2: Jl(f) for all f E CB(S) and a E S, i.e., Jl is left subinvariant. •
Notes and remarks.
(i) Belluce and Kirk [11] first proved that if K is a nonempty weakly compact convex subset of a Banach space and if K has complete normal structure, then every family of commuting nonexpansive self-maps on K has a common fixed point. Later Lim [66, Theorem 3] extended this theorem to a continuous representation of a left reversible semi topological semigroup S as nonexpansive mappings on a weakly compact convex set K with normal structure (Corollary 5.2).
Theorem 5.1 was proved by us in [62]. It answers affirmatively a problem posed during the Conference on Fixed Point Theory and Applications held at ClRM, MarseilleLuminy, 1989 (see [51, p.307, Problem 5].)
(ii) It has been proved by Hsu [39] (see also [9]) that if S is discrete and left reversible, and S = {Tsi s E S} is a representation of S as weakly continuous nonexpansive mappings on a weakly compact convex subset C of a Banach space, then C has a common fixed point for S. Note that it follows from Alspach's example [1] that there exists a commutative semigroup of nonexpansive mappings on a weakly compact convex subset of LI[O, 1] with no common fixed point. But as is well known, £OO(S) always has an invariant mean when S is commutative. Also Schechtman [80] has shown that there exists a weakly compact convex subset W of LI[O, 1] and a sequence TI, T2, ... of commuting nonexpansive operators of W into itself such that any finite number of them have a common fixed point but there is no common fixed point for the entire sequence.
(iii) (see [51]). Consider on a pair (S, E), where S is a semitopological semi group and E is a Banach space, the following fixed point properties:
Theorems for semigroups of non-linear mappings 529
(P) Whenever S = {Ts; S E S} is a representation of S as nonexpansive mappings from a non-empty closed convex subset C of E and for each x E C, the orbit O(x) = {Tsx; S E S} is relatively compact, then closed convex hull of O(x) contains a common fixed point of S.
(Q) Whenever S = {Ts; S E S} is a continuous representation of S as nonexpansive mappings on a non-empty closed convex subset C of E into C and C contains an element with bounded orbit, then C contains a common fixed point for S.
Of course any pair (S,E) satisfying (Q) must satisfy (P). However as shown by R. Sine [86], (P) is not true for arbitrary finite dimensional space E and commutative S (see [49, Problem 2]).
Theorem 5.3 ([9]) If S is left reversible (or more generally A(S) has a LIM) and E is strictly convex, then (P) holds.
Theorem 5.4 ([59] [60]) If RUC(S) has a LIM or S is left reversible and E is uniformly convex and uniformly smooth, then (Q) holds.
Problem 1. Does (Q) hold when W(S) has a LIM and E is a Hilbert space (see [49], Problem I)?
A representation S = {Ts; S E S} of S as self-maps on a subset C of a Banach space is called uniformly k-Lipschitzian, k > 0, if
IITsx - T.yll ~ kllx - yll for all S E S, x, Y E C. Unif6rmly k-Lipschitzian semigroups were introduced by Goebel, Kirk and Thele in [28] who proved the following (in slightly more general form) basic result:
Theorem 5.5 If E us a uniformly convex Banach space, then there is a ko > 1 such that whenever K ~ E is a closed, bounded, convex set and S = {T.; s E S} is a representation from K into K with k < ko, then K contains a common fixed point for s.
Precisely how large ko may be taken remains, even in Hilbert space, an open question. The estimate provided for the Hilbert space case in [28] is ../5/2 with an upper bound of 2. Downing and Ray [24], by refining the technique in [28] shows that ko may be taken to be V2 in this case (see [41, Theorem 2] for a different proof). Ishihara and Takahashi [41, Theorem 1] shows that Downing and Ray's result [24] remains valid when S is a semitopological semigroup such that RUC(S) has a LIM and the action of Son C is jointly continuous (see also [40]).
(iv) Locally convex spaces (see [48] and [51])
Let S be a semitopological semigroup and (E, Q), or simply E, be a separated locally convex space. Let S = {Ts; XES} be a representation of S as mapping on a subset X into X. We assume (throughout this discussion) that the map 'I{; : S x X -+ X, (s,x) ...... Tsx, s E S and x E X is separately continuous, i.e. 'I{; is continuous in each of the two variables when the other is kept fixed. Then S is jointly continuous if the map 'I{; is continuous when S x X has the product topology; S is Q-nonexpansive if p(Tsx - TsY) ~ p(x - y) for all s E S, P E Q and X,y E X.
530
Consider on S the following fixed point properties:
(G1 ) Whenever S is a jointly continuous representation of S on a non-empty weak'compact convex subset X of a dual Banach space with the weak* -topology and S is norm nonexpansive, then X has a common fixed point in S.
(G2 ) Whenever S is a representation of S as weakly continuous Q-nonexpansive mappings on a non-empty weakly compact convex set X of a separated locally convex space (E, Q), then X contains a common fixed point for S.
(G3) Whenever S is a representation of S as nonexpansive mappings on a non-empty compact convex subset X of a separated locally convex space (E, Q), then X contains a common fixed point for S.
DeMarr [22J proved that any commuting semigroup S has fixed point property (G3 ).
Later, W. Takahashi [87J proved that any discrete left amenable semigroup S has property (G3). T. Mitchell [70J generalized both DeMarr and Takahashi's result by showing that any discrete left reversible semigroup S has property (G3). Finally, Lau proved in [47J the following:
Theorem 5.6 A(S) has LIM iff S has fixed point property (G3 ).
Since the action of S on the weak*-compact convex set X in LUC(S)*:
X = {m E LUC(S)*;m 2: 0 and IImll = I},
defined by 'IjJ(a,m) = £~m, a E S, mE X, is jointly continuous (see [71]) and norm nonexpansive, it follows readily that
(Gd =;. S is left amenable.
However the converse is unknown even for S commutative.
Problem 2. Does S left amenable =;. (G1 )?
A similar argument as in the proof of the theorem in [48, p.123J shows that
However, the following problems are still open (see [48]):
Problem 3. Does S left reversible =;. (G2 ) for semitopological semigroups?
Notice that weak*-continuity of the representation in (G1 ) is important; (G1 ) is not true even for a commutative semigroup when the mappings {Ts; s E S} are not weak*weak* continuous. In fact, Alspach [lJ has shown that there is a weakly compact convex subset K of L1[0, 1J and a nonexpansive mapping T : K -t K without a fixed point. Since Ll [0, 1J may be embedded isometrically into (£OO(N), II . 1100), K regarded as a subset of £OO(N) is still weakly compact and convex. Hence K is weak*-compact. Let S = N, the positive integers with addition. Then S = {Tn; n E N} is a representative of S as nonexpansive mappings on K without fixed point.
(v) Fixed point property and Radon Nikodym property (see [52]).
Theorems for semigroups of non-linear mappings 531
A Banach space E is said to have the fixed point property (f.p.p.) if for each non-empty weakly compact convex subset X ~ E and a nonexpansive mapping T : X ---+ X, X contains a fixed point for T. As is well known [27], a weakly compact convex subset of a Banach space E with normal structure has the f.p.p. On the other hand, L1[0, 1J does not have the f.p.p.
Bruck [15J shows that if a Banach space E has the f.p.p., then E has the f.p.p. for commutative semigroups, i.e. whenever C is a non-empty weakly compact convex subset of E, and S is a commutative semigroup of nonexpansive mappings from C into C, then C contains a common fixed point for S.
Let G be a locally compact group, and let A(G) denote the Fourier algebra of G, i.e. A(G) is the subalgebra of Co(G) (complex-valued continuous functions on G vanishing at infinity) consisting of all functions ¢ of the form
¢(x) = (p(s)h, k), h, k E L2(G),
p(x)h(t) = p(x-1t), x, t E G. Then A(G) is a semi-simple commutative Banach algebra with norm
and A(G)* = VN(G), the von Neumann algebra in B(L2(G)) generated by {p(x)jX E
G} (see [25J for details). It follows from Alspach [lJ that if G = (Z, +), then A(Z) does not have the f.p.p.
A Banach space E is said to have the Radon-Nikodym property (RN P) if each closed bounded convex subset D of E is dentable, i.e. for any e > 0, there exists xED such that x fJ- co{D\B.(x)} where B.(x) = {y E Ej Ily - xii < e}.
Lemma 5.7 ([56]) Let G be a locally compact group. If A(G) has the RNP, then A(G) has the j.p.p.
The following problem is still open:
Problem 4. Does the fixed point property for A(G)imply RNP?
A locally compact group G is called an [INJ-group if there is a compact neighbourhood U of the identity e such that x-lUx = U for all x E G. Examples of [INJ-groups include all compact groups, discrete groups and abelian groups.
Theorem 5.8 ([56]) If G is either an abelian group or a connected IN-group, then the following are equivalent:
(a) G compact.
(b) A(G) has the j.p.p.
(c) A(G) has RNP.
Theorem 5.9 ([56]) If G is discrete and A(G) has the f.p.p., then G cannot contain an infinite abelian subgroup. In particular, elements of G must have finite order.
532
Note that there are noncompact groups G (e.g. the "ax + b" group) such that A(G) has the RN P and hence the f.p.p. (see [105] or [65] for details). Also in the case G is compact, then A(G) = B(G) (the Fourier Stieltjes algebra of G) is the dual Banach space of C*(G), the group CO-algebra of G. In this case, A(G) even has the weak*f.p.p., i.e. for every non-empty weak*-compact convex subset X of A(G) and every nonexpansive mapping T : X -> X, X contains a fixed point for T (see [13]).
When G is a locally compact abelian group, then the following are equivalent:
(i) G is compact;
(ii) B(G) has the weak*-f.p.p.;
(iii) The weak* and norm topologies agree on {¢ E B(G); II¢II = ¢(e) = I}.
When G is not abelian, we still have (i) ==} (ii) ~ (iii). However (ii) does not imply (i) in general (see [55], [56] and [10] for details).
6. Fixed point sets and left ideal orbits
Let 8 be a semitopological semigroup, E be a uniformly convex Banach space and S = {Ts; s E 8} be a continuous representation of 8 as nonexpansive mappings on a closed convex subset C of E into C. In this section, we shall discuss the relation between the left ideal orbits and the fixed point set F(S). We begin with a simple lemma:
Lemma 6.1 Let E be a strictly convex Banach space and let C be a convex subset of E. Let T be a nonexpansive mapping of C into itself. Then the set F(T) of fixed points of T is convex.
Proof. Let x, y E F(T) and 0 :::; a :::; 1. Then putting z = ax + (1 - a)y, we have
IIx - Tzil = IITx - Tzil ::; IIx - zll and Ily - Tzil = IITy - Tzil ::; lIy - zII. Hence,
Ilx - yll :::; Ilx - Tzil + IITz - yll ::; IIx - zll + liz - yll = IIx - YII·
This implies that IIx - zll = IIx - Tzil and lIy - zll = lIy - TzlI· Since E is strictly convex, we have z = Tz. Therefore F(T) is convex. •
Let £(8) denote the collection of closed left ideals in 8. Assume that F(S) i= 0. For each x E C and L E £(8), define the real-valued function qX,L on F(S) by
qx,LCf) = inf{IITtx - fl12 : tEL}
and let
qx(J) = sUp{qx,L : L E £(8)}.
Then qx(J) = supinf IITtsx - fll2 as readily checked. s t
Lemma 6.2 Let C be a nonempty closed convex subset of a Banach space E. If F(S) i= 0, then for each x E C, qx is a continuous real-valued function on F(S) such that
Theorems for semigroups of non-linear mappings 533
o :::: qx(f) :::: Ilx - fl12 for each f E F(S) and qx(fn) --t 00 if Ilfnll --t 00. Further, if F(S) is convex, then qx is a convex function on F(S).
Proof. Since 0 :::: IITtx - fll2 = IITtx - Tdl1 2 :::: Ilx - fll2 for every f E F(S) and t E 8, it follows readily that 0 :::: qx(f) :::: Ilx - f112. Also if f E F(S) and t E 8, then IITtx - fll :::: Ilx - fII. Hence I ITt x II :::: IITtx - fll + IIfll :::: IIx - fll + IIfll, i.e., M = sup{IITtXII : t E 8} < 00. Let Un} be a sequence in F(S) such that Ilfnll --t 00.
Then we have for each t E 8,
and hence for each L E £(8),
So we have qx(fn) --t 00.
To see that qx is continuous, let {fn} be a sequence in F(S) converging to some f E F(S) and
M' = sup{IITtX - fnll + IITtx - fll : n = 1,2,··· and t E 8}.
Then since
IITtx - fnl1 2 - IITtx - fl12 :::: (11Ttx - fnll + IITtx - fll) IIITtx - fnll - IITtx - fill
:::: M/llfn - !II,
we have for each L E £(8),
Similarly, we have
So we obtain Iqx(fn) -qx(f)1 :::: M/llfn - fll. This implies that qx is continuous on F(S).
If F(S) is convex, for each f,g E F(S) and a,f3 2: 0 with a + 13 = 1, af + f3g E F(8). Let c: > O. Then there exists Lo E £(8) such that
sup inf(allTtx - fl12 + f3l1Ttx - g112) < inf (allTtx - fll2 + f3l1 TtX _ g1l2) + ~. LEL(S) tEL tELa 2
Let U E Lo. Then 8u ~ Lo and hence
Moreover, there exist v, w E 8 such that
IITvux - fll2 < inf IITtux - fl12 + ~ and IITwvux - flf < inf IITtvux - fl12 + ~2· tES 2 tES
534
Therefore we obtain
qx(af + (3g) = sup inf IITtx - (af + (3g)1I2 LE£(S) tEL
~ sup inf(allTtx - fll2 + (3llTt x _ g112) LE£(S) tEL
< inf(aliTtux - fl12 + (3II Ttux - g112) + ~2 tES
II 2 2 £ ::; a Twvux - fll + (3I1 Twvux - gil +"2
2 2 £ ::; allTvux - fll + (3 II Twvu x - gil +"2
. 2· II 112 a£ (3£ £ <amfIiTtux-fll +(3mf Ttvux-g +-2 +-2 +-2 tES tES
= a inf IITtx - fl12 + (3 inf IITtx _ gll2 + £ tELl tEL2
(where Ll = 5u and L2 = 5vu)
::; aqx(f) + (3qx(g) + £.
Since £ > 0 is arbitrary, we have qx(af + (3g) ::; aqxU) + (3qx(g). • Theorem 6.3 Let C be a nonempty closed convex subset of a uniformly convex Banach space E. Assume that F(S) f- 0. Then for any x E C, there exists a unique element hE F(S) such that
qx(h) = inf{qxU) : f E F(S)}.
Proof. Since E is uniformly convex, the fixed point set F(S) in C is closed and convex (see Lemma 6.1). Hence it follows from Lemma 6.2 and [8] that there exists h E F(S) such that
qx(h) = inf{qxU) : f E F(S)}.
To see that h is unique, let k E F(S). Then by [107], there exists a strictly increasing and convex function (depending on hand k) 9 : [0,(0) -t [0,00) such that g(O) = 0 and
IITtx - (Ah+ (1- A)k)112 = 11.\(TtX - h) + (1- A)(Ttx - k)112
::; AIITtx - hl1 2 + (1 - A) IITtx - kl1 2 - A(l - A)g(lIh - kll)
for each t E 5 and A with 0 ::; A ::; 1. So we have for each A with 0 ::; A ~ 1,
qx(h) ::; qx(Ah + (1 - A)k)
::; Aqx(h) + (1 - A)qx(k) - A(l - .\)g(lIh - kll)
and hence qx(h)::; qx(k) - Ag(llh - kll). It follows that
qx(h) ::; qx(k) - g(llh - kill as A -t 1.
Since 9 is strictly increasing, it follows that if qx(h) = qx(k), then h = k.
We call the unique element h E F(S) in Theorem 6.3 the minimizer of qx in F(S).
For each x E C, let
Q(x) = n co{Ttx: tEL} (= nSEsco{Ttsx: t E 5}). LEL(S)
•
Theorems for semigroups of non-linear mappings 535
Theorem 6.4 Let C be a nonempty closed convex subset of a Hilbert space H. Let S = {Ta : s E S} be a continuous representation of S as nonexpansive mappings from C into C. Then for any x E C, any element in Q(x) n F(S) is the unique minimizer of qx in F(S). In particular, Q(x) n F(S) contains at most one point.
Proof. Let z E F(S) be the minimizer of qx in F(S) and y E Q(x) nF(S). Then for some e > 0, there exists u E S such that
sup inf(IITtax - zl12 + 2(Ttax - z, z - y) + liz _ Y112) a t
< iljf(IITtux - zll2 + 2(Ttux - z, z - y) + liz _ y1l2) + ~.
Moreover there exist v, W E S such that
and
(Twvux - z, z - y) < iljf(Ttvux - z, z - y) + ~.
Therefore we obtain
This implies
qx(Y) = sup inf IITtsx _ Yll2 s t
= sup inf(IITtsx - zll2 + 2(Ttsx - z, z - y) + liz _ Y1l2) s t
< iljf(IITtux - zll2 + 2(Ttux - z, z - y) + liz _ Y112) + ~ 2 2 e
~ IITwvux - zll + 2(Twvux - z, z - y) + liz - yll + 4
~ IITvux - zlf + 2(Twvux - z,z - y) + liz _ Yll2 + ~ < inf IITtux - zll2 + 2 inf(Ttvux - z, z - y)
t t
+ liz - Yl12 + ~ + ~ + ~ 424
~ supinfllTt8x - zll2 +2supinf(Ttsx - z,z -y) s tat
+ IIz-yIl2+ e
= qx(z) + 2 sup inf(Ttsx - z,z - y) + liz - Yl12 + e. s t
2 sup inf(Ttsx - z, z - y) > qx(Y) - qx(z) - liz - Yl12 - e s t
~ -liz - Yll2 - e.
So, there exists a E S such that
2(Ttax - z,z - y) > -liz - Yll2 - e
for every t E S. From y E co{Ttax : t E S}, we have
2(y - z, z - y) ~ -liz - Yll2 - e.
This inequality implies liz - Yl12 ~e. Since e > 0 is arbitrary, we have z = y. •
536
Lemma 6.5 For each x E C, Y, z E F{S) and A with 0 ~ A ~ 1,
inf sup IIATtsx + {I - A)Y - zll ~ sup inf IIATtsx + (I - A)Y - zII. sES tES sES tES
Further, if S is semitopological and right reversible and t t-+ Ttx is continuous for each x E C, then equality holds.
Proof. Let x E C, let Y, z E F{S) and let 0 ~ A ~ 1. Let
D = {u E C : lIu - yll ~ IIx - YII}·
Then by Lemma 1.1 in [16], there exists a strictly increasing, continuous, convex nmction 'Y from [0,00) to [0,00) such that 'Y{O) = 0 and
'Y(II1)Ttu + {I -1))7tv - Tt(1)u + (I -1))v)ll) ~ Ilu - vll-IiTtu - Ttvll (6.1)
for each t E S, 1.1, V E D and 1) with 0 ~ 1) ~ 1. Let e > O. Since each Tt is nonexpansive,
inf sup IITtsx - YII ~ inf IITsx - YII· s t s
So there exists So E S such that IITtsox - yll ~ IITsx - yll + e for each t, s E S. Putting 1) = A, 1.1 = Twsox and v = yin (6.1), we get
IIATtwsox + {I - A)Y - zll
~ IIATtwsox + {I - A)Y - Tt{ATwsox + {I - A)Y)II + IITt{ATwsox + (I - A)Y) - zll
~ 'Y- 1{IITwso x - yll - IITtwsox - yin + IIATwsox + (I - A)Y - zll
~ 'Y-1{e) + IIATwsox + {I - A)Y - zll
for each t, w E S. Hence we have
inf sup IIATtsx + (I - A)Y - zll ~ 'Y-1{e) + inf IIATwsox + (I - A)Y - zll SEStES wES
~ 'Y-1{e) + sup inf IIATwsx + {1 - A)Y - zll. sEswES
Since e > 0 is arbitrary, we obtain the desired inequality.
Suppose now S is semitopological and right reversible. Set
a = inf sup IIATtsx + {1 - A)Y - zll and b = sup inf IIATtsx + {1 - A)Y - zll. sES tES sES tES
Let e > O. Then there exist so, Sl E S such that
sup IIATtsox + {1- A)Y - zll ~ a + e and inf II ATts l X + {1- A)Y - zll ~ b - e. tES tES
Since Sso n SSl is nonempty, there exists S2 E Sso n SSl. Let t E S. From S2 E Sso, there is a net {ta} of S such that taso --+ S2. Since IIATttasoX + {1- A)Y - zll ~ a+e, we have IIATts2 X+{1-A)y-zll ~ a+e. Similarly, we have IIATts2x+{I-A)y-zll ~ b-e. Hence
a + e ~ sup IIATts2 x + {I - A)Y - zll ~ inf IIATts2 x + {I - A)Y - zll ~ b - e. tES tES
From the arbitrariness of e > 0, we have a ~ b and so a = b by above. •
Theorems for semigroups of non-linear mappings 537
The following inequality will be useful for us:
Lemma 6.6 Assume that the norm of E is Jilrechet differentiable. Then for each x E C, Y E Q(x) n F(S) and z E F(S),
sup inf (Ttsx - y, J(y - z)) 2: O . • ES tES
Proof. Let x E C, let y E Q(x) n F(S) and let z E F(S). Put
g(A) = inf sup IIATt.X + (1 - A)y - zll BES tES
for 0 ::; A::; 1. Let 0 ::; A::; 1 and let c > O. Then there exists So E S such that
IIATtsox + (1 - A)y - zll ::; g(A) + c for each t E S.
From y E co{TtBox : t E S}, we have lIy - zll ::; g(A) + c. Hence we get lIy - zll ::; g(A) for each>' with 0 ::; >. ::; 1. Set
8(>') = sup (1/2I1Y - z + >'k1l2 -1/2I1Y - zl12 _ (k, J(y _ z))) Ilkll=l >.
for a < A::; 1. Then 8 is an increasing function and 8(A) -+ 0 as >. -+ 0 from (6.1). Let M = SUPt IlTtx - yll. By lIy - zll ::; g(A) for 0 < >. ::; 1 and Lemma 6.5, we have
0::; -21 inf sup II>'Ttsx + (1 - >.)y - zll2 - ~IIY - zl12 sES tES 2
::; ~ sup inf II>.Ttsx + (1 - >.)y - zl12 - ~ lIy - zll2 2 sEStES 2
::; sup inf(A(Ttsx - y, J(y - z)) + >.IITtsx - YIl8(>.llTtsx - yll)) sES tES
::; A sup inf (TtsX - y, J(y - z)) + >.M8(>.M) sES tES
for a < A::; 1, and hence the assertion follows. • Theorem 6.7 Assume that the norm of E is Jilrechet differentiable. Then for each x E C, F(S) n Q(x) consists of at most one point.
Proof. Let y, z E Q(x) n F(S) and let c > O. From Lemma 6.6, there exists So E S such that (Ttsox - y, J(y - z)) > -c for each t E S. Since z E co{Ttsox : t E S}, we have (z - y, J(y - z)) 2: -c, Le., lIy - zll2 ::; c. From the arbitrariness of c > 0, we obtain y = z. This completes the proof. •
Notes and remarks.
(i) Theorem 6.7 was originally proved by us in [59, Theorem 1] when S is right reversible (Le. S has finite intersection property for closed left ideals). This was improved to arbitrary semitopological semigroups by Lau, Nishiura and Takahashi in [57, Theorem 3.11]. The proof given here is taken from [58]. A study of the relation between the left ideal orbits and the fixed point set was first done [57] where Theorem 6.3 and Theorem 6.4 were proved.
(ii) In view of theorem 6.3, it is natural to ask the following questions.
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Problem 5. If E is a uniformly convex Banach space, x E G and y E Q(x) n F(S), is y always the minimizer of qx in F(S)?
Problem 6. If E is a uniformly convex Banach space, does Q(x) n F(S) contain at most one point for each x E G?
Clearly, by Theorem 6.3, an affirmative answer for problem 5 gives an affirmative answer to problem 6; Theorem 6.7 gives an affirmative answer for problem 6 when E has a Fnlchet differentiable norm.
(iii) It should be noted that in general, nsEsco{Ttsx; t E S} cannot be replaced by nsEsco{Tstx;t E S} in Theorem 6.7. Indeed, let n > 1 and let S = {al,'" ,an} with multiplication defined by s·t = t for each t, s E S. Let E = £2(S), i.e., the n-dimensional Hilbert space. For t E S, define a mapping Tt from E into itself by (Tt/)(s) = fest) for each fEE and s E S. Then S = {Tti t E S} is a nonexpansive semigroup on E such that each Tt is linear. Let f E £2(S) such that f(ai) = i for i = 1, ... ,n. In this case, nsEsco{Tst/; t E S} = co{Tt/; t E S} and {Tt/; t E S} has n common fixed points.
7. Ergodic theorems
Let S be a semi topological semigroup, p. be an element in GE(S)*. Let E be a reflexive Banach space. Let f be a bounded and continuous function on S into E. Then the function cp on E* into the set IR of real number given by
cp(x*) = 11s(f(S),x*) for every x* E E*
is linear and continuous. Hence there exists an element f(p.) in E such that
(f(p.),x*) = p.s(f(s),x*) for every x* E E*.
If S = {Ts; s E S} be a continuous representation of S as nonexpansive mappings on a closed convex subset C of E, {Tsx; s E S} is bounded for some x E G and f(x) = Tsx for s E S, we shall denote f(p.) by TJ1.x.
Lemma 7.1 Let S be a semitopological semigroup having a right invariant mean p. on GE(S), let E be a reflexive Banach space and let f be a bounded and continuous function from S to E. Then f(p.) E nSEsCO{f(ts); t E S}.
Proof. Let p. be a right invariant mean on GE(S). Then there exists a net {CPo,} of finite means such that
(h, epa - p.) --> 0 for every h E GE(S);
see [19]. So putting epa = 2::~1 Af8(tf) with 2::~:1 Af = 1 and Af 2: 0 (i 1,2, ... ,na ), we have, for each s E Sand y* E E*.
/ na ) na \t;Aif(tiS),Y* = t;Ai(f(tis),y*)
= (cp,,)t(f(ts),y*)
--> p.t(f(ts), yO) = p.t(f(t), y*) = (f(p.) , y*).
This implies that for each s E S, 2::~:1 Ai f(tfs) converge weakly to f(p.) and hence f(p.) E co{f(ts); t E S}. Therefore we have f(p.) E nSEScO{J(ts); t E S}. •
Theorems for semigroups of non-linear mappings 539
Lemma 7.2 Let S be a semitopological semigroup, let C be a nonempty bounded closed convex subset of a uniformly convex Banach space E, and let S = {T.; S E S} be a non expansive semigroup on C. Let /-L be a mean on CB(S). Then:
(a) T" is nonexpansive on C.
(b) IfT"x E F(S) for each x E C, then T; = TIL' i.e. TIL is a retraction on C.
(c) If S is right reversible and /-L is a right invariant mean, then
TILTs = TIL for each S E S.
Proof. (a) Let J: E --> E* be the duality map, and let x* E J(T"x - T"y). Then
IITllx - TIlyll2 = (Tllx - T"y, x*) = /-Lt(Ttx - Tty, x*)
::; sup IITtx - TtylillTllx - Tilyll t
::; Ilx - ylillTllx - Tilyll,
so that TIL is nonexpansive.
(b) From (T;x,x*) = /-Lt(TtTllx,x*) = /-Lt(Tllx,x*) = (T"x,x*), we obtain T; = Til'
(c) Since /-L is right invariant, we have, for any x* E E* and s E S,
(TIlT.x, x*) = /-Lt(TtT.x, x*) = /-Lt(Tt.x, x*)
= /-Lt(Ttx,x*) = (Tllx,x*),
and hence TILTs = T" for every s E S. Also, if x E C, then Tllx E nsESCO{Ttx; t ~ s} by Lemma 7.1. •
Theorem 7.3 Let C be a closed, convex subset of a uniformly convex Banach space E, let S be a semigroup, let S = {Tt : t E S} be a nonexpansive semigroup on C with F(S) =f. 0, and let X be a subspace of (oo(S) such that 1 EX, the mapping t I--> (Ttx, x*) is an element of X for each x E C and x* E E* and X is I.-and r.-invariant for each s E S. If X is amenable, then there exists a nonexpansive retraction P from C onto F(S) such that PTt = TtP = P for each t E Sand Px E co{Ttx : t E S} for each xEC.
Proof. Assume that X is amenable. Then there is an invariant mean J1. on X. Let t E S and let x E C. Let z be an arbitrary element of F(S). Set
D = {y E C : lIy - zll ::; IIx - zll}· We remark that xED and Tt(D) CD. Let e: > O. By [17, Theorem 1.2], there exists {j> 0 such that COF6(Tt ;D) C Fe (Tt ; D). By [17, Corollary 1.1], there also exists a natural number N such that
for all y E D. So we have
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for all s E S. Since f-l is left invariant, we get
where tOs represents s. From the arbitrariness of c > 0, we have TtTJLx = TJLx. Hence we obtain TtTJL = TJL for each t E S. The other assertions now follow from Lemma 7.2 w~P=~. •
Theorem 7.4 Let C, E, S, S and X be as in Theorem 1. Assume that the norm of E is Prechet differentiable and X is amenable. Then there exists a unique nonexpansive retraction P from C onto F(S) such that PTt = TtP = P for each t E Sand Px E co{Ttx : t E S} for each x E C. Further, if {f-lo,} is an asymptotically invariant net of means on X, then for each x E C, {TJLa x} converges weakly to Px.
Proof. From Theorem 7.3, there exists a nonexpansive retraction P from Canto F(S) such that PTt = TtP = P for each t E Sand Px E co{Ttx : t E S} for each x E C. For each x E C and s E S, we have Px = PTsx E co{Ttsx : t E S}. So by Theorem 6.7, we get {Px} = F(S) nQ(x) for each x E C. Hence such a retraction P is unique. Let {f-lo,} be an asymptotically invariant net of means on X. We shall show {Tl'ax} converges weakly to Px for each x E C. Let x E C and let {!Lal'} be a subnet of {!La} such that {f-la,a} converges to an invariant mean f-l on X with respect to the weak star topology on X*. Then for each x* E E*,
lim(TJLa x,x*) = lim(!L",")t(Ttx,x*) = !Lt(Ttx,x*) /3 I' /3 ~
= (TJLx,x*) = (Px,x*).
Hence {Tl'ax} converges weakly to Px. • In the case where S is commutative, we have the following:
Corollary 7.5 Let E, C, S, S = {Tt : t E S} and X be as in Theorem 7.3. Assume S is commutative and the norm of E is Prechet differentiable. Then the same conclusion in Theorem 7.4 holds. Moreover, if {!La} is a strongly regular net of means on X, then for each x E C, {TJLaTtx} converges weakly to Px uniformly in t E S.
Proof. We remark that Is = rs for each s E S. Let {f-la} be a net of means on X such that lima Ilr;!La - !Lall = 0 for each s E S. We shall show that for each x E C, {TJlaTtx} converges weakly to Px uniformly in t E S. Suppose not. Then there exist x E C, x* E E* and c > 0 such that for each a, there exist f3a 2: a and ta E S satisfying I (Tl'l'a Tta x - Px,x*)I2: c. Since S is commutative, we have
liml (r;r;a!L/3a - r;a!L/3a)(f) I = lim I (r;a r;!L/3a - r;af-l/3JU) I a a
::; lim Ilr;!L/3a - !L/3a 1IIIfil = 0 a
Theorems for semigroups of non-linear mappings 541
for each s E Sand f E X. So {r;"tL,B,,} is an asymptotically invariant net of means on X. By Theorem 7.4, {T!'.a"Tt"x} converges weakly to Px, which contradicts that I(T!'.a"Tt"x - Px,x*)1 ~ e for all a. •
The following three results are analogues of Day's ergodic theorem [18) and [21] for an amenable semigroup of nonexpansive mappings on a uniformly convex Banach space.
Theorem 7.6 Let C, E, S, S and X be as in Theorem 7.3. Let {tL,,} be a strongly regular net of means on X. Then for each t E S and for each bounded subset B of C, lim" IIT!,,,Ttx - T/,,,xll = 0 and lim" IITtT!,,,x - T!,,,xll = 0 uniformly for x E B.
Proof. Let t E S and let B be a bounded subset of C. Let e > 0 and set
M = sup{IITsxlliS E S,X E B}.
Since {J.!,,} is strongly regular, there exists ao such that IIr;tL,,-tL,,11 ~ elM for a ~ ao. Then for a ~ ao, we have
sup IIT/,,,Ttx - T!,,,xll = sup sup 1(J.!,,)s(TsTtx,x*) - (tL,,)s(Tsx,x*)1 xEB xEB Ilx*II=1
~ IIr;tL" - tL"IIM ~ e.
Hence we obtain lim" IIT!'"Ttx-T!'"xll = 0 uniformly for x E B. It has been established in [84, Lemma 1) that lim" IITtT!,,,x - T!,,,xll = 0 uniformly for x E B. For the sake of completeness, we give here a sketch of its proof. Let e > 0 and let M be as above. From F(S) f 0, there exists a bounded, convex subset D of C satisfying {Tsx : s E S} c D. We can choose {j > 0 and a natural number N such that
(COFc5(Tti D) + {y E E : lIyll ~ (j}) n C c Fe(Tti C) (7.1)
and
liN ~ 1 ~TtiSX -Tt(N ~ 1 ~TtiSX) II ~ {j for all x E Band s E S. (7.2)
Since {tL,,} is strongly regular, there exists 0<1 such that IltL,,-I;,tL,,1I ::; (jIM for 0< ~ 0<1
and i = 1, ... ,N. Then we have
sup liT!'" - J N 1 1 t Ttisx dtL,,(s) II xEB + i=O
= sup sup I (tL,,)s(Tsx,x*) - (tL")S\-N1 tTti.X,x*)1 xEB IIx*II=1 + 1 i=O
N
~ N 1 1 I: sup sup l(tL,,)s(T.x,x*) - (/;iJ.!,,).(Tsx,x*)1 + i=O xEB IIx*II=1
::; . max IItL" - l;itL,,11 . M ~ {j 1.=l, ... ,N
for 0< ~ 0<1. From (7.1), (7.2) and the inequality above, we have
542
for a 2: a1. Hence we obtain lim", IITtTl'ax - T/"axll = 0 uniformly for x E B. •
As a direct consequence of Theorem 7.6, we have the following by using [19J:
Theorem 7.7 Let C, E, 8 and S be as in Theorem 1. Assume that £00(8) is amenable. Then there exists a net {A",} of finite averoges of S such that for each t E 8 and for each bounded subset B of C, lim", IIA",Ttx - A",xll = 0 and lima II1IAax - A",xll = 0 uniformly for x E B.
Let (F, G) be a dual pair of vector spaces, and let a(F, G) be the weak topology on F generated by G.
Lemma 7.8 Let X be a tronslation invariant closed subspace of £00(8) containing constants, and assume that X has an invariant mean. Then there exists a net {<p",} of finite means such that (£;<p", - <p",)(f) ....... 0 for each s E 8 uniformly on a(X, £1(8))_ compact absolutely convex subsets of x.
Proof. We follow ideas of Day [19J and N amioka [75J. Consider the dual pair W (8), X) and the Mackey topology T on £1(8) with respect to this dual pair. By the MackeyArens theorem, the continuous dual of (£1(8), T) is also X. Let E = (£1(8), T)8. Define a map T : <I> ....... E given by
T(<p)(8) = r;<p - <p for every s E 8,
where <I> is the set of finite means. Then T( <I» is a convex subset of E. Now since X has a left invariant mean, there exists a net {>'a} C <I> such that (r;>.",->'",)(f) ....... 0 for each f E X. Hence T(>'",) ....... 0 in the weak topology of E, i.e., 0 E the weak closure of T(<I». Hence by Mazur's theorem, 0 E the strong closure of T(<I», that is, there exists a net {<p",} of finite means such that (r;<p", - <p",)(f) ....... 0 uniformly on a(X, £1 (8))-compact absolutely convex subsets of X. •
Theorem 7.9 Let C and E be as in Theorem 7.3. Let 8 be asemitopological semigroup such that CB(8) is amenable and CB(8) separotes each distinct two points of 8, and let S = {Ts; s E 8} be a nonexpansive semigroup on C such that F(S) =f. 0 and t 1--+ Ttx is continuous for each x E C. Then there exists a net {Aa} of finite averoges of S such that for each t E 8 and x E C, lim", IIA",Ttx - A",xll = 0 and lim", IITtA",x - A",xll = o.
Proof. By Lemma 7.8, there exists a net {fL",} of finite means on CB(8) such that for each s E 8, both £;fL", - fL", and r;fL", - fL", converge to 0 uniformly on absolutely convex, a(CB(8), £1 (8))-compact subsets of CB(8). We know from [42, Lemma 3.2] that for each x E C, {t 1--+ (Ttx,x'); Ilx'lI = I} is contained in an absolutely convex, a(CB(8), £1 (8))-compact subset of CB(8). Hence by the similar argument in the proof of Theorem 7.6, we obtain the conclusion. •
Theorem 7.10 Let C, E, 8 and S be as in Theorem 7.3. Assume that the norm of E is Frechet differentiable and for each x E C, Q(x) n F(S) is nonempty. Then there exists a nonexpansive retroction P from C onto F(S) such that PTt = TtP = P for each t E 8 and Px E co{Ttx; t E 8} for each x E C.
Proof. From Theorem 6.7 and the assumption, Q(x) n F(S) consists of exactly one point for each x E C. So we can define a retraction P from C onto F(S) such that
Theorems for semigroups of non-linear mappings 543
PTt = TtP = P for each t E Sand Px E co{Ttx : t E S} for each x E C. We show that P is nonexpansive. Let c: > 0 and let x, y E C. By Lemma 6.6, there exists s E S such that
(Ttsx - Px, J(Px - Py)} > -c:
for each t E S. For such an element 8, we have
sup inf(TtuTsY - PTsY, J(PTsY - Px)} 2: 0 UEStES
from Lemma 6.6. So there also exists u E S such that
(TtusY - PTsY, J(PTsY - Px)} > -c:
for each t E S. Then from PTsY = Py, we have
-2c: < (Tsusx - Px, J(Px - Py)} + (TsusY - Py, J(Py - Px)}
= (Tsusx - TsusY, J(Px - Py)} - IIpx _ Pyll2
::; IIx - yllllPx - Pyll-IIPx _ pY1l2.
Since c: > 0 is arbitrary, this implies IIPx - Pyll ::; IIx - yll.
Notes and remarks.
• (i)In 1975, Baillon [5] originally studied the nonlinear ergodic theorem in the framework of Hilbert spaces: If C is a closed, convex subset of a Hilbert space and T is a nonexpansive mapping from C into itself such that the set F(T) of fixed points of T is nonempty, then for each x E C, the Cesaro mean
converges weakly to some y E F(T). In this case, putting y = Px for each x E C, P is a nonexpansive retraction from C onto F(T) such that PT = T P = P and Px E co{ynx : n = 1,2, ... } for each x E C. In [88], Takahashi proved the existence of such an retraction for an amenable semigroup of nonexpansive mappings on a Hilbert space: If S is an amenable semigroup, C is a closed, convex subset of a Hilbert space Hand S = {Tt : t E S} is a nonexpansive semigroup on C such that the set F(S) of common fixed points of S is nonempty, then there exists a nonexpansive retraction P from C onto F(S) such that PTt = TtP = P for each t E Sand Px E co{Ttx : t E S} for each x E C. Rode [79] also found a sequence of means on a semigroup, generalizing the Cesaro means, and extended Baillon's theorem as follows: If S, C, Hand S are as above and {J.La } is an asymptotically invariant net of means, then for each x E C, {Tl'ax} converges weakly to an element of F(S). Further, for each x E C, the limit point of {Tl'ax} is the same for all asymptotically invariant nets {/La} of means. From their results, we know that for each x E C, {Tl'ax} converges weakly to Px for all asymptotically invariant nets {/La} of meanSj see [93]. These results were extended to a uniformly convex Banach space whose norm is Frechet differentiable in the case when S is commutative by Hirano, Kido and Takahashi [37]. However, it has been an open problem whether Takahashi's result and Rode's result can be fully extended to such a Banach space for an amenable semigroupj see [94]. On the other hand, Day [18] proved the following ergodic theorem for an amenable semigroup of bounded linear operators on a Banach space: If S is an amenable semigroup and S = {Tt : t E S} is
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a bounded representation of S as bounded linear operators on a Banach space E, then there exists a net {A",} of finite averages of S such that lim", IIA",(Tt - 1)11 = 0 and lim", II(Tt - I) A", II = 0 for each t E S. In this case, there is also a projection P from E onto F(S) such that PTt = TiP = P for each t E Sand Px E co{Tix : t E S} for each x E E; see also [21] and [45].
Results in this section are essentially taken from [58]. Theorems 7.3 and 7.4 give an answer to the open problem mentioned above by extending Takahashi's result and Rhode's result. The proofs of our ergodic theorems depend on a deep convex approximation property for nonexpansive mappings due to Bruck [17]. Theorem 7.7 was proved in [57, Theorem 4.2] when E is a Hilbert space, and [65, Theorem 2] where S is assumed to be right reversible.
(ii) Ergodic Sequences (see [52])
Let S be a semigroup and £l(S) denote the Banach space of all f : S --> R such that IIfll1 = 2: If(x)1 < 00. Let (£l(S))t = all 0 E £l(S) such that 0 2 0 and 1I0lit = 1 (countable means).
Let H be a Hilbert space, C be a closed convex subset of H, and S = {T.; s E S} be a representation of S as nonexpansive mappings from C into C such F(S) -# ¢.
Let x E C. For each y E H, consider the bounded real-valued function on S defined by s ...... (Tsx, y). Let 0 be a mean on £oo(S), and define
(To(x),y) = O.({T.x,y))
= ~)(T.x,y)O(s);s E S} if () E £l(S)t.
Then To is a nonexpansive mapping of C into C (see [79]).
Call a sequence (net) {(}n} of means on S an ergodic sequence (net) for nonexpansive mappings if for any representation S = {Ts; s E S} of S as nonexpansive mappings on a closed convex subset C of a Hilbert space into C such that F(S) -# ¢, then for each x E C, the sequence (net) TOn (x) converges weakly to a fixed point of S.
A net of means {",,,,} on £oo(S) is called "asymptotically invariant" if
lim(",,,,(£.f) - ",,,,U)) = 0 and '"
lim(",,,,(rsf) - ",,,,U)) = 0 for all s E S. '"
Theorem 7.11 ([79]) Let S be an amenable semigroup. Then any "asymptotically invariant net" of means is an ergodic net for nonexpansive mappings.
Notes:
1 Every invariant mean on £OO(S) is asymptotically invariant.
2 If m is an invariant mean on £OO(S), then there is a net 0", E (£l(S))t such that (}", has finite support, i.e., 0", = 2:?=1 AiO.; (convex combination) such that
0", ~ m. In particular the net {(}",} is asymptotically invariant. Hence {(}"'} is an ergodic net of finite means on S for nonexpansive mappings.
Example([79]): If S = ({O, 1, 2, ... }, +) and
1 n-1
(}n=-LOk, n k=O
Theorems for semigroups of non-linear mappings 545
then {lin} is an asymptotically invariant sequence of finite means on S. Consequently, {lin} is an ergodic sequence of finite means on S for nonexpansive mappings.
Problem 7. Given an amenable semigroup S, when does there exist an ergodic sequence of countable (or finite) means on S for nonexpansive mappings?
Problem 8. When can the net {Aa} of finite averages of S in Theorem 7.9 be chosen to be a sequence dependent on the semigroup S?
8. Related results
Let C be a nonempty closed convex subset of a Banach space E. Let
S = {Set) : t :2: O}
be a family of nonexpansive mappings of C into itself such that S(O) = I, Set + s) = S(t)S(s) for all t,s E [0,00) and S(t)x is continuous in t E [0,00) for each x E C. Then S is said to be a one-parameter nonexpansive semigroup on C. Miyadera and Kobayasi[73] introduced the notion of an almost-orbit of a one-parameter nonexpansive semigroup on C. A continuous function u : [0,00) -> C is called an almost-orbit of S = {Set) : t :2: O} if
lim (sup lIu(t + s) - S(t)u(s) II) = O. S-HXl t~O
(8.1)
For example, consider the initial value problem:
duet) dt:" + Au(t) :;) J(t), t> 0, (8.2)
u(O) = x,
where A is an m-accretive operator in E, J E L1(0, 00; E) and x E D(A). Then it is well known that (8.2) has a unique integral solution [7] and the integral solution u(t) of (8.2) is an almost-orbit of a one-parameter nonexpansive semigroup on D(A) generated by -A. In fact, let ACE x E be an m-accretive operator and let u(t) be an integral solution of (8.2). Then we know that
holds for each (u,v) E A and 0::; s::; t < +00, where for x and y in E,
(y,X)8 = max{(y,j) : j E J(x)}.
We also know that if vet) is another integral solution of (8.2) corresponding to g E L1(0,00;E) and y E D(A), then
Ilu(t) - v(t)II-lIu(s) - v(s)1I ::; [IIJ(T) - g(T)lldT, (8.3)
whenever 0::; s::; t < +00. If A-10 = {x: X E D(A),Ax:;) O} is nonempty, we can take z E A-10 and get) = 0 in (8.3) to obtain
lIu(t) - zll- lIu(s) - zll ::; [IJ(T)ldT. (8.4)
546
Consequently, u(t) is bounded on [0,00) and the function t f-> Ilu(t) -zll- J~ IIf(r)lIdr is nonincreasing on [0,00). Then, since f E Ll(O,oojE), we deduce that
lim lIu(t) - zll = p(z) t~oo
exists for each z E A-10. Finally, let {Set) : t ;::: O} be a one-parameter nonexpansive semigroup generated by -A and let u(t) be the integral solution of (8.2). Then we obtain that 1t+s
IIS(t)u(s)-u(t+s)IIS s IIf(r) II dr, (8.5)
whenever 0 S s, t < +00. In fact, the function t f-> vet) = u(t + s) is an integral solution of (8.2) corresponding to get) = f(t + s) and u(s) E D(A). From (8.5), we obtain that
lim sup IIS(t)u(s) - u(t + s)11 = o. s~oo t~O
This implies that the integral solution of (8.2) is an almost-orbit of the one-parameter nonexpansive semigroup generated by -A. Using a fixed point theorem for a commutative family of nonexpansive mappings, we can prove the following lemma.
Lemma 8.1 Let C be a closed convex subset of a real reflexive Banach space which has normal structure and let {u(t) : t ;::: O} be an almost-orbit of S = {Set) : t ;::: oJ. Then F(S) '" <p if and only if {u(t) : t ;::: O} is bounded.
Proof. Suppose that {u(t) : t ;::: O} is bounded. Then, since {u(t) : t ;::: O} is an almost-orbit of S = {Set) : t ;::: OJ, there exists tl such that {S(t)U(tl) : t ;::: O} is bounded. Therefore by [90], F(S) is nonempty. Conversely, if F(S) '" <p, then since the limit of lIu(t) - zll exists for each z E F(S), it follows that {u(t) : t ;::: O} is bounded .
• As in the proof of Theorem 6.7, we can prove the following theorem. See [73, 100] for details.
Theorem 8.2 Let E be a uniformly convex Banach space E with a F1rechet differentiable norm and let {u(t) : t ;::: O} be an almost-orbit of S = {Set) : t ;::: oJ. Suppose F( S) '" <p. Then the set
n co{u(t + s) : t;::: O} nF(S) s~O
consists of at most one point.
Using Theorems 7.4 and 8.2, we also have the following theorems.
Theorem 8.3 Let E be a uniformly convex Banach space E with a F1rechet differentiable norm and let {u(t) : t ;::: O} be an almost-orbit of S = {Set) : t ;::: oJ. Suppose F(S) '" <p. Then, there exists an element y E F(S) such that
w - lim ~ t u(h + t)dt = y uniformly in h ;::: O. 8--+00 8 io
Theorems for semigroups of non-linear mappings 547
Proof. Let S = [0,00), S = {Set) : t ~ O} and let X be the Banach space GE(S) of all bounded continuous functions on S with the supremum norm. Define
11' A,(!) = - f(t)dt s 0
for every s > 0 and f E X. Then, we obtain
IIA, - rkA, II = sup II~ r f(t)dt - ~ r f(t + k)dtll 11/119 s io s io
= ~ sup II r f(t)dt - rH f(t)dtll
s 11/119 io i k
= ~ sup II rk f(t)dt _iSH f(t)dtll
s 11/119 io ,
1 (l k 1'H: ) 2k ::; - sup IIf(t)lIdt + IIf(t)lIdt = - --> 0, s 11/119 0 , s
as s --> 00. Therefore, using Theorem 7.4, we obtain Theorem 8.3. • Theorem 8.4 Let E, G, Sand u be as in Theorem B.3. Then, r Iooo e-rtu(t + k)dt converges weakly to some y E F(S), as r --> 0, uniformly in k ~ O.
Proof. Let S = [0,00), S = {Set) : t ~ O} and X = GE(S). Define
ArC!) = r 100 e-rt f( t)dt
for each r > 0 and f E X. Then, for each s E [0,00), we have
liAr - r;Arll = sup Ir roo e-rt f(t)dt - r roo e-rt f(s + t)dtl 11/119 io io
= sup Ir r e-rt f(t)dt + r(1 _ erS ) roo e-rt f(t)dtl 11/119 io i,
::; rs + 11 - er'l --> 0,
as r --> O. Therefore, using Theorem 7.4, we obtain Theorem 8.4. • Using some ideas in the nonlinear ergodic theory, we can also study the iteration procedure of Mann's type [69] for a family of nonexpansive mappings in a Banach space. The following proposition is crucial to prove our result; see [78, 99].
Proposition 8.5 Let G be a nonempty, closed, convex subset of a uniformly convex Banach space E whose norm is Frechet differentiable and let {Ti; i E N} be a sequence of nonexpansive mappings from G into itself such that n;:"=oF(Tn) is not empty. Let SnY = TnTn-l ... Toy for each y E G and n E No Then for each x E G, the set
00 00 n co{Smx;m ~ n} n n F(Tn) n=O n=O
consists of at most one point.
548
Let C be a nonempty closed, convex subset of a Banach space E. A family
s = m; t E S}
is said to be a nonexpansive semigroup on C if
(i) for each t E S, Tt is a nonexpansive mapping from C into itself;
(ii) Tts = TtTs for each t, s E S.
Theorem 8.6 Let C be a nonempty, closed, convex subset of a Banach space E. Let S be a semigroup and let S = {Tt ; t E 5} be a nonexpansive semigroup on C such that F(S) is not empty. Let X be a subspace of £00(5) such that 1 E X, it is Is-invariant for each s E 5, and the function t r-t (Ttx, x*) is an element of X for each x E C and x* E E*. Let {Jln} be a sequence of means on X such that IIJln - e:/Lnll -t 0 for every s E 5 and let {an} be a real sequence such that 0 :s: an :s: 1 for each n E Nand L~=o(l - an) = 00. Let x E C and let {Xn} be the sequence defined by
Xo = x, Xn+! = anxn + (1 - an)Tpnxn for each n E N. (8.6)
If E is a uniformly convex Banach space whose norm is Fh5.chet differentiable, then {xn} converges weakly to an element of F(S).
For proving Theorem 8.6, we need the following lemmas.
Lemma 8.7 Let D be a bounded closed convex subset of C such that D is Tt -invariant for each t E 5. Then
lim sup IITpnY - TtT,'nyll = 0 for each t E 5. n-ooyED
Proof. Let t E 5 and e > O. Set M = sup{IITsYII;s E 5,y ED}. As in the proof of Theorem 7.6, we can choose 8 > 0 and N E N such that
(CO(F6(Tt) n D) + {y E E; lIyll :s: 8}) nee F,m) for all t E 5 (8.7)
and
liN ~ 1 ~TtiSY - -Tt(N ~ 1 ~TtiSY) II :s: 8 for all y E D and s E 5. (8.8)
Since {Jln} is strongly left regular, there exists no E N such that IIJln - l;i/Lnll :s: 81M for n ~ no and i = 1,. .. , N. Then we have
SUP!!Tpny - J -N1 "tTtisYd/Ln(S)!! yED + 1 ;=0
= sup sup !(Jln)s(TsY,y*) - -(Jln)/ N ~ 1 "tTtisY,y*)! yED lIy'll=l \ ;=0
N
:s: N~lLSUP sup 1(/Ln)s(Tsy,y*)--(I;i/Ln)s(Tsy,y*)1 ;=0 yED Ily'lI=l
:s: . max IIJln - 1;.Jlnll . M :s: 8 7.=l" .. ,N
Theorems for semigroups of non-linear mappings
for n 2: no. From (8.7), (8.8) and the inequality above, we have
lim sup IITtTPnY - TPnYIl :::; c. n yED
Since c > 0 is arbitrary, we obtain the conclusion.
549
• Lemma 8.8 Let {On} be a real sequence such that 0 :::; On :::; 1 for each n E Nand 2:~=o(l - on) = 00. Let x be an element of C and let {xn} be the sequence defined by (8.6). Then
lim Ilxn - Ttxnll = 0 for each t E S. n--+oo
Proof. Fix c > 0 and t E S. Set r = max{suPnllxnll,suPnSUPtEsIITtXnll}. Then there exists 6> 0 which satisfies (8.7). From Lemma 8.7, there exists n E N such that IITpn+kY - TtTpn+kYIl :::; 6 for each y E D and kEN. From 2:~o(l - Ok) = 00, there exists mEN such that 0non+!··· On+k-l < 6j(2r) for all k > m. From (3.1), it follows that for each k 2: 1, there exists Yk E co{TPnxn, ... , Tpn+k_l xn+k-d such that
Xn+k = On+k-l ... OnXn + (1 - On+k-l ... On)Yk.
So for each k > m, Tp,Xi E Fo(Tt) for i = n, ... , n + k - 1, and
6 IIXn+k - Ykll :::; On+k-l ... On· IIxn - Ykll < 2r . 2r :::; 6.
Since Yk E co(Fo(Tt ) n D)g and Xn+k - Yk E {y E E; lIyll :::; 6}, (8.7) yields
IIxn+k - Ttxn+kll :::; c
for each k 2: m. Hence we get limn Ilxn -Ttxnll :::; c. Since c > 0 is arbitrary, we obtain the conclusion. •
Proof of Theorem 8.6 Let W E F(S). Since
IIxn+! - wll :::; onllxn - wil + (1- on)IITpnxn - wll :::; Ilxn - wll, limn IIxn - wll exists. Assume that the norm of E is Frechet differentiable. Set
TnY = OnY + (1 - on)TJJnY and SnY = TnTn-lTn-2··· Toy
for each y E D and n E N. From Lemma 8.8 and Browder [14] , the set W of weak limit points of {xn} is contained in n~=oco{xm; m 2: n}nF(S). Since x n +! = Snx and F(S) C n~=oF(Tn)' the set W is contained in n:'oco{Smx;m 2: n} n n:'oF(Tn). Hence, from Proposition 8.5, {xn} converges weakly to an element of F(S). •
Using Theorem 8.6, we can prove the following.
Theorem 8.9 Let T be a non expansive mapping from C into itself such that F(T) is not empty. Let {on} be a real sequence such that 0 :::; On :::; 1 for each n E Nand 2:~=o(l- on) = 00. Let x E C and let {Xn} be the sequence defined by
Xo =x, 1 n .
xn+! = OnXn + (1- on)-- '"'T'xn for each n E N. n+ 1 L..J i=O
If E is a uniformly convex Banach space whose norm is Prechet differentiable, then {xn} converges weakly to a fixed point ofT.
550
Theorem 8.10 Let S = {T(t); t E [O,oo)} be a one-pammeter nonexpansive semigroup on C such that F(S) is not empty and the mapping t I---> (T(t)u, u*) is measumble for each U E C and u* E E*. Let {O!n} be a real sequence such that ° :S O!n :S 1 for each n E N and L~=o(1 - O!n) = 00 and let {sn} be a positive real sequence with Sn --+ 00. Let x be an element of C and let {xn} be the sequence defined by
1 lsn Xo = x, Xn+1 = O!nXn + (1 - O!n) - T(t)xn dt for each n E N. Sn 0
If E is a uniformly convex Banach space whose norm is Fh~chet differentiable, then {xn} converges weakly to a common fixed point of S.
Proof. Let X be the space of all bounded measurable functions from [0,00) into the set of real numbers. We remark that an element f in X is not an equivalence class with the usual equivalence relation, where the usual equivalence relation 9 ~ h means the Lebesgue measure of the set {t E [O,oo);g(t) =f. h(t)} is zero. The reason is that we consider that X is a subspace of B([O, 00)) with the supremum norm. For each n E N, define a mean Pn on X by /-In(J) = f.;- J;n f(t) dt for f EX. We see that {/-In} is strongly regular and
T/-'nY = ~ ['n T(t)ydt sn io
for y E C and n E N; see [37]. Hence by Theorem 8.6, we obtain the conclusion. •
Notes and Remarks.
(i) The notion of almost-orbit was introduced by Miyadera and Kobayasi [73] and Theorem 8.2 was proved by them. A similar result for commutative semigroups of nonexpansive mappings was also proved by Takahashi and Park [100].
(ii) Mann [69] introduced an iteration procedure for approximating fixed points of a mapping T as follows:
Xo = x E C, Xn+1 = O!nXn + (1 - O!n)Txn for each n E N,
where {O!n} is a sequence in [0,1]. Later, Shimizu and Takahashi [82] considered iterative method for approximation of common fixed points; see also [81]. Motivated by Shimizu and Takahashi [82], Atsushiba and Takahashi [3] studied the iteration procedures of Mann's type for a family of nonexpansive mappings in Hilbert space. Theorem 8.6 was proved by Atsushiba, Shioji Takahashi [2].
(iii) Shioji and Takahashi [85] studied the iterative procedure of Halpern's type [33] and proved the following theorem.
Theorem 8.11 Let E be a uniformly convex Banach space E with a uniformly Gateaux differentiable norm. Let C be a nonempty closed convex subset of E and let
S={Tt;tES}
be a nonexpansive semigroup on C such that F(S) =f. c{>. Let X be a subspace of footS) such that 1 EX, X is fs-invariant for each s E Sand t I---> (Ttx, x*) is an element of X for each x E C and x* E E*. Let {Pn} be a sequence of means on X such that II/-In - f:/-Inll = ° for every s E S. Suppose X,Yl E C and {Yn} is given by
Yn+1 = !3nx + (1 - !3n)T/-'nYn for every n 2: 1,
REFERENCES 551
where {!1n} is a sequence in [0,1]. If {!1n} is chosen so that limn->co!1n = ° and L:::"=l!1n = 00, then {Yn} converges strongly to the element of F(S) which is nearest to x in F(S).
Using Theorem 8.11, we can prove the following theorem.
Theorem 8.12 Let E be a uniformly convex Banach space E with a uniformly Gateaux differentiable norm. Let C be a nonempty closed convex subset of E and let
S = {Set); t ~ o}
be a one-parameter nonexpansive semigroup on C such that F(S) of cpo Suppose x, Yl E C and {Yn} is given by
1 t·,. Yn+l = !1nx + (1 - !1n) An 10 S(t)Yndt for every n ~ 1,
where {!1n} is a sequence in [0,1]. If {!1n} and {An} are chosen so that limn_ co !1n = 0, L:::"=l!1n = 00 and An ...... 00, then {Yn} converges strongly to the element of F(S) which is nearest to x in F(S).
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Chapter 16
GENERIC ASPECTS OF METRIC FIXED POINT THEORY
Simeon Reich
Alexander J. Zaslavski
Department of Mathematics
The Technion-Israel Institute of Technology
32000 Haifa, Israel
1. Introduction
Let X be a complete metric space. According to Baire's theorem, the intersection of every countable collection of open dense subsets of X is dense in X. This rather simple, yet powerful result has found many applications. In particular, given a property which elements of X may have, it is of interest to determine whether this property is generic, that is, whether the set of elements which do enjoy this property contains a countable intersection of open dense sets. Such an approach, when a certain property is investigated for the whole space X and not just for a single point in X, has already been successfully applied in many areas of Analysis. We mention, for instance, the theory of dynamical systems [12, 18, 24, 35, 33, 52]' optimization [22, 44], variational analysis [2,9], [20,21]' the calculus of variations [4, 14,55] and optimal control [56, 57].
In this exposition we present several recent results in metric fixed point theory which exhibit these generic phenomena.
2. Hyperbolic spaces
It turns out that the class of hyperbolic spaces is a natural setting for our generic results. In this section we briefly review this concept.
Let (X, p) be a metric space and let Rl denote the real line. We say that a mapping c : Rl -t X is a metric embedding of Rl into X if
p(c(s), c(t)) = Is - tl
557
W.A. Kirk and B. Sims (eds.), Handbook of Metric Fixed Point Theory, 557-575. © 2001 Kluwer Academic Publishers.
558
for all real sand t. The image of Rl under a metric embedding will be called a metric line. The image of a real interval [a, b] = {t E Rl: a ::; t ::; b} under such a mapping will be called a metric segment.
Assume that (X, p) contains a family M of metric lines such that for each pair of distinct points x and y in X there is a unique metric line in M which passes through x and y. This metric line determines a unique metric segment joining x and y. We denote this segment by [x,y]. For each 0::; t::; 1 there is a unique point z in [x,y] such that
p(x, z) = tp(x, y) and p(z, y) = (1 - t)p(x, y).
This point will be denoted by (1 - t)x EB ty.
We will say that X, or more precisely (X,p,M), is a hyperbolic space if
for all x, y and z in X.
An equivalent requirement is that
for all x,y,z and w in X. A set K c X is called p-convex if [x,y] C K for all x and y inK.
It is clear that all normed linear spaces are hyperbolic. A discussion of more examples of hyperbolic spaces and in, particular, of the Hilbert ball can be found, for instance, in [28, 3D, 32, 42].
In the sequel we will repeatedly use the following fact (cf. [3D, pp. 77,104]' and [42]): If (X, p, M) is a hyperbolic space, then
p((1 - t)x EB tz, (1 - t)y EB tw) ::; (1 - t)p(x, y) + tp(z, w) (2.1)
for all x,y,z and w in X and 0::; t::; 1.
3. Successive approximations
Let (X, p, M) be a complete hyperbolic space and let K be a closed p-convex subset of X. Denote by A the set of all operators A : K ~ K such that
p(Ax, Ay) ::; p(x, y) for all x, y E K.
In other words, the set A consists of all nonexpansive self-mappings of K.
Fix some B E K and for each s > 0 set
B(B, s) = B(s) = {x E K: p(x,B) ::; s}.
For the set A we consider the uniformity determined by the following base:
E(n,E) = ((A, B) E A x A: p(Ax,Bx)::; E, x E B(n)},
(3.1)
(3.2)
(3.3)
where f > 0 and n is a natural number. Clearly the space A with this uniformity is metrizable and complete [31]. We equip the space A with the topology induced by this uniformity.
Generic aspects 559
A mapping A : K -7 K is called regular if there exists a necessarily unique XA E K such that
lim Anx = XA for all x E K. n .... oo
A mapping A : K -7 K is called super-regular if there exists a necessarily unique XA E K such that for each s > 0,
Anx -7 XA as n -7 00, uniformly on B(s).
Denote by I the identity operator. For each pair of operators A, B : K -7 K and each t E [0,1] define an operator tA Ell (1 - t)B by
(tA Ell (1 - t)B)(x) = tAx Ell (1 - t)Bx, x E K.
Note that if A and B belong to A, then so does tA Ell (1 - t)B.
The first result in this section shows that in addition to (locally uniform) power convergence,power convergence, locally uniform super-regular mappings also provide stability. The second result will show that most mappings in A are, in fact, super-regular. This is an improvement of the classical result of De Blasi and Myjak [17] who established power convergence (to a unique fixed point) for a generic nonexpansive self-mapping of a bounded closed convex. subset of a Banach space. In this connection see also [23, 53] and [3, p. 41], where the existence of a (unique) fixed point for certain classes of self-mappings is shown to be generic.
Theorem 3.1 ([45]) Let A: K -7 K be super-regular and let E, s be positive numbers. Then there exist a neighborhood U of A in A and an integer no 2': 2 such that for each BE U, each x E B(s) and each integer n 2': no, we have p(xA,Bnx) ::; E.
Proof. We may assume that E E (0,1). Recall that XA is the unique fixed point of A. There exists an integer no 2': 4 such that for each x E B(2s + 2 + 2p(XA, e)) and each integer n 2': no,
(3.4)
Set
U = {B E A: p(Ax,Bx) ::; (8nO)-lE, x E B(8s + 8 + 8p(XA, e))}. (3.5)
Let B E U. It is easy to see that for each x E K and all integers n 2': 1,
and
p(Anx, Bnx) ::; p(Anx, ABn-1x) + p(ABn-1x, Bnx)
::; p(An-1x,Bn-1x) + p(ABn-1x,Bnx)
p(Bnx, XA) ::; p(Bnx, Anx) + p(Anx, XA)
::; p(Bnx, Anx) + p(X,XA)
::; p(Bnx, Anx) + p(x, e) + pre, XA).
(3.6)
(3.7)
Using (3.5), (3.6) and (3.7) we can show by induction that for all x E B(4s + 4 + 4p(XA, e)), and for all n = 1,2, ... , no,
(3.8)
560
and 1
p(Bnx, e) ::; 2p(XA, e) + p(x, e) + 2.
Let y E B(s). We intend to show that p(xA,Bny) ::; E for all integers n ~ no. Indeed, by (3.8),
pee, Bmy) ::; ~ + 2p(XA, e) + s, m = 1, ... , no. (3.9)
By (3.8) and (3.4),
(3.10)
Now we are ready to show by induction that for all integers m ~ no,
p(xA,Bmy) ::; E. (3.11)
By (3.10), the inequality (3.11) is valid for m = no.
Assume that an integer k ~ no and that (3.11) is valid for all integers m E [no, kJ. Together with (3.9) this implies that
. 1 p(e,B'y)::; 2 + 2p(XA,e) +s, i = 1, ... ,k. (3.12)
Set
j = 1 + k - no and x = Biy. (3.13)
By (3.12), (3.13), (3.4) and (3.8),
p(Anox,BnOx) ::; E18, p(xA,Anox )::; EI8 and p(xA,Bk+1y)::; E/4.
This completes the proof of Theorem 3.1. • Theorem 3.2 ([45]) There exists a set :Fo c A which is a countable intersection of open everywhere dense sets in A such that each A E :Fo is super-regular.
Proof. For each A E A and -y E (0,1) define A-y : K -> K by
A-yx = (1 - -y)Ax ED -ye, x E K.
Let A E A and -y E (0,1). Clearly,
p(A-yx, A-yy) ::; (1 - -y)p(Ax, Ay) ::; (1 - -y)p(x, y), x, Y E K.
Therefore there exists a unique x(A, -y) E K such that
Evidently A-y is super-regular and the set {A-y : A E A, -y E (0, I)} is everywhere dense in A. By Theorem 3.1, for each A E A, each -y E (0,1) and each integer i ~ 1, there exist an open neighborhood U(A,-y,i) of A-y in A and an integer n(A,-y,i) ~ 2 such that the following property holds:
(i) for each BE U(A,-y,i), each x E B(4i+1) and each n ~ n(A,-y,i),
p(x(A,-y),Bnx)::; 4-i - 1 .
Generic aspects 561
Define Fa = n~1 U {U(A, ,,(, i) : A E U, "( E (0,1), i = q, q + 1, ... }.
Clearly Fa is a countable intersection of open everywhere dense sets in A.
Let A E Fa. There exist sequences {Aq}~1 C A, bq}~1 C (0,1) and a strictly increasing sequence of natural numbers {iq}~1 such that
A E U(Aq, "(q, iq), q = 1,2, ... (3.14)
By property (i) and (3.14), for each x E B(4iq +1) and each integer n;::: n(Aq,"(q,iq),
p(x(Aq,"(q),Anx) ::; 4-iq - 1.
This implies that A is super-regular. Theorem 3.2 is proved. • 4. Contractive mappings
Let K be a bounded closed p-convex subset of a complete hyperbolic space (X, p, M) and set
diam(K) = sup{p(x,y) : X,y E K}.
We equip the set A with the metric h(.,.) defined by
h(A,B) = sup{p(Ax,Bx) : x E K}, A,B E A.
Clearly the metric space (A, h) is complete.
We say that a mapping A E A is contractive if there exists a decreasing function ¢A: [O,diam(K)]--+ [0,1] such that
¢A(t) < 1 for all t E (O,diam(K)] (4.1)
and
p(Ax,Ay) ::; ¢A(p(x,y))p(x,y) for all X,y E K. (4.2)
The notion of a contractive mapping as well as its modifications and applications were studied by many authors. See, for example, [1, 10, 39] and the references mentioned there. We now quote a convergence result which is valid in all complete metric spaces [38].
Theorem 4.1 Assume that A E A is contractive. Then there exists a unique XA E K such that Anx -+ XA as n -+ 00, uniformly on K.
Next, we note the following fact.
Proposition 4.2 If A E A is contractive, B E A and a E (0,1), then the operators AB, BA and aA EI7 (1 - a)B are also contractive.
Now we show that most of the mappings in A (in the sense of Baire's categories) are, in fact, contractive. In view of Theorem 4.1 this implies the main result of [17].
Theorem 4.3 ([46]) There exists a set F which is a countable intersection of open everywhere dense sets in A such that each A E F is contractive.
562
Proof. Fix () E K. For each A E A and each 'Y E (0,1) define AI' E A by
A,.x = (1 - 'Y)Ax EB 'Y(), x E K.
Clearly the set {A,,: A E A, 'Y E (0, In is everywhere dense in A.
Let A E A and 'Y E (0,1). The inequality (2.1) implies that
p(A,.x, A"y) ::; (1 - 'Y)p(x, y)
for all x, y E K. For each integer i 2: 1 define
U(A,'Y, i) = {B E A: h(A,.,B) < 4-;i-1'Ydiam(Kn.
(4.3)
(4.4)
(4.5)
We will show that for each A E A, 'Y E (0,1) and each integer i 2: 1, the following property holds:
pel) For each BE U(A,'Y,i) and each x,y E K satisfying p(x,y) 2: 4-i diam(K), the inequality p(Bx,By) ::; (1- 2-1'Y)p(x,y) is valid.
Indeed, let A E A, 'Y E (0,1) and let i 2: 1 be an integer. Assume that
B E U(A, 'Y, i), x, y E K and p(x, y) 2: 4-i diam(K).
By (4.5), (4.6) and (4.4),
p(Bx, By) ::; p(A"x, A,.y) + 2-1 . 4-i 'Y diam(K)
::; (1- 'Y)p(x,y) + T14-i'Ydiam(K)
::; (1 - 'Y)p(x, y) + 2-1'YP(x, y)
= (1 - T1'Y)p(x, y).
Thus property pel) holds. Now define
F = n~l U {U(A,'Y,i): A E A, 'Y E (0,1), i 2: q}.
(4.6)
Clearly F is a countable intersection of open everywhere dense sets in A. We claim that any B E F is contractive. To see this, assume that q is a natural number. There exist A E A, 'Y E (0,1) and an integer i 2: q such that B E U(A, 'Y, i). By property pel), for each x, y E K satisfying p(x, y) 2: 4-q diam(K) we have p(Bx, By) ::; (1- 2-1'Y)p(x, y). Since q is an arbitrary natural number we conclude that B is contractive. Theorem 4.3 is established. •
Note that at least in Hilbert space the set of strict contractions is only of the first category in A [17] (cf. [3, p. 43] and [25]).
We continue with a discussion of nonexpansive mappings which are contractive with respect to a given subset of their domain.
Let K be a closed (not necessarily bounded) p-convex subset of the complete hyperbolic space (X,p,M). For each x E K and each subset E c K, let p(x,E) = inf{p(x,y): y E E}. For each x E K and each r > 0, set
B(x,r) = {y E K: p(x,y)::; r}.
Fix () E K. We equip the set A with the same uniformity and topology as in Section 3 (see (3.3».
Generic aspects 563
Let F be a nonempty closed p-convex subset of K. Denote by A(F) the set of all A E A
such that Ax = x for all x E F. Clearly A(F) is a closed subset of A. We consider the topological subspace A(F) c A with the relative topology.
An operator A E A(F) is said to be contractive with respect to F if for any natural number n there exists a decreasing function ¢~: [0,00) -+ [0,1] such that
¢~(t) < 1 for all t > 0
and p(Ax,F) S; ¢~(p(x,F»p(x,F) for all x E B(O,n).
Clearly this definition does not depend on our choice of 0 E K.
The following result, which was established in [46], shows that the iterates of an operator in A(F) converge to a retraction of K onto F.
Theorem 4.4 Let A E A (F) be contractive with respect to F. Then there exists B E A(F) such that B(K) = F and Anx -> Bx as n -> 00, uniformly on B(O, m) for any natural number m.
Finally, we present the following theorem of [46] which shows that if A(F) contains a retraction, then almost all the mappings in A(F) are contractive with respect to F.
Theorem 4.5 Assume that there exists
Q E A(F) such that Q(K) = F.
Then there exists a set F c A(F) which is a countable intersection of open everywhere dense sets in A (F) such that each B E F is contractive with respect to F.
Proof. We may assume that () E F. Then for each real r > 0,
C(B(O,r» C B(O,r) for all C E A(F).
For each A E A(F) and each "( E (0,1), define Ay E A(F) by
Ayx = (1 - "()Ax EEl "(Qx, x E K.
(4.7)
(4.8)
The inequality (2.1) implies that for each A E A(F), Ay -> A as "( -> 0 in A(F).
Therefore the set {Ay: A E A(F), "( E (0, I)) is everywhere dense in A(F). Let A E A(F) and "( E (0,1). Evidently,
p(Ayx,F) = inf {p((1- "()Ax EEl "(Qx,y)) yEF
for all x E K. Thus
S; inf {p((1 - "()Ax EEl "(Qx, (1 - "()y EEl "(Qx)) yEF
S; inf {(I - "()p(Ax,y)) S; (1 - "()p(x, F) yEF
p(Ayx, F) S; (1 - "()p(x, F) for all x E K. (4.9)
For each integer i ~ 1, denote by U(A, "(, i) an open neighborhood of A-y in A(F) for which
(4.10)
564
(see (3.3)). We will show that for each A E A(F), each'"( E (0,1) and each integer i 2: 1, the following property holds:
P(2) For each B E U(A,'"(,i) and each x E B(8,2i ) satisfying p(x,F) 2: 4-i , the inequality p(Bx, F) ~ (1 - 2-1'"()p(X, F) is true.
Indeed, let A E A(F), '"( E (0,1) and let i 2: 1 be an integer. Assume that
BE U(A,'"(,i), x E B(8,2i ) and p(x,F) 2: 4-i .
Using (4.9), (4.10) and (4.11) we see that
p(Bx,F) ~ p(A-yx,F) +8-i1'
~ (1 - '"()p(x, F) + 8-i1'
~ (1 - '"()p(x, F) + 2-11'P(x, F)
~ (1 - 2-11')p(x,F).
(4.11)
Thus property P(2) holds for each A E A(F), each l' E (0,1) and each integer i 2: 1. Define
F = n~1 U {U(A, 1', i) : A E A(F), l' E (0,1), i 2: q}.
Clearly F is a countable intersection of open everywhere dense sets in A (F). Let B E F. To show that B is contractive with respect to F it is sufficient to show that for each r > 0 and each 10 E (0,1) there is K E (0,1) such that p(Bx,F) ~ Kp(X, F) for each x E B(8, r) satisfying p(x, F) 2: 10. Let r > ~ and 10 E (0,1). Choose a natu~al num~er q such that 2q > 8r and 2-q < 8-110. There eXIst A E A{F), l' E (0,1) and an mteger z 2: q such that B E U(A,'"(,i). By property P(2), for each x E B(8,r) C B(8,2i ) satisfying p(x,F) 2: 10 > 2-i , the following inequality holds: p(Bx,F) ~ (1- 2-11')p(x,F). Thus B is contractive with respect to F. This completes the proof of Theorem 4.5. •
5. Infinite products
In this section we present several recent results concerning the asymptotic behavior of (random) infinite products of generic sequences of nonexpansive as well as uniformly continuous operators on closed convex subsets of a complete hyperbolic space. Such infinite products find application in many areas of mathematics (see, for example, [5, 6, 7, 26, 27, 41, 47J and the references mentioned therein). Instead of considering a certain convergence property for a single sequence of operators, we investigate it for a space of all such sequences equipped with some natural metric, and show that this property holds for most of these sequences. This allows us to establish convergence without restrictive assumptions on the space and on the operators themselves.
Let (X, p, M) be a complete hyperbolic space and let K be a nonempty bounded closed p-convex subset of X with the topology induced by the metric p.
Denote by M the set of all sequences {AtJ~I' where each At: K -+ K is a continuous operator, t = 1,2, .... Such a sequence will occasionally be denoted by a boldface A.
For the set M we consider the metric hs : M x M -+ [0,00) defined by
hs ( {At}~I' {Bt}~I) = sup{p(Atx,Btx) : x E K, t = 1,2, ... },
{At}~I' {Bt}~1 EM.
Generic aspects 565
It is easy to see that the metric space (M, hs ) is complete. The topology generated in M by the metric hs will be called the strong topology.
In addition to this topology on M we will also consider the uniformity determined by the base
where N is a natural number and f > O. It is easy to see that the space M with this uniformity is metrizable (by a metric hw : M x M -+ [0,(0)) and complete. The topology generated by hw will be called the weak topology.
Define Mne = {{AtH~'l EM: At is nonexpansive for t = 1,2, ... }.
Clearly Mne is a closed subset of M in the weak topology. We will consider the topological subspace Mne C M with both the weak and strong relative topologies.
In Theorem 2.1 of [43] we showed that for a generic sequence {Ct}~l in the space Mne with the weak topology,
p(CT· .... C1X, CT· .... C1Y) -+ 0 as T -+ 00,
uniformly for all x, y E K. (Such results are usually called weak ergodic theorems in the population biology literature; see [15, 37].)
Here is the precise formulation of this weak ergodic theorem.
Theorem 5.1 There exists a set :F C Mne which is a countable intersection of open (in the weak topology) everywhere dense (in the strong topology) subsets of Mne such that for each {Bt}~l E:F and each f > 0 there exist a neighborhood U of {Bt}~l in Mne with the weak topology and a natuml number N such that;
For each {Ct}~l E U, each x, y E K, and each integer T 2: N,
P(CT· .... C1X, CT· .... ClY) ::; f.
Note that in [43] we also proved a random version of this theorem.
We will say that a set E of operators A : K -+ K is uniformly equicontinuous (ue) if for any f > 0 there exists 8 > 0 such that p(Ax,Ay)::; f for all A E E and all x,y E K satisfying p(x, y) ::; 8. Define
Mue = {{At}~l EM: {Atl~l is a (ue) set}.
Clearly Mue is a closed subset of M in the strong topology.
We will consider the topological subspace Mue C M with the weak and strong relative topologies.
Denote by M~e the set of all {At}~l E Mne which have a common fixed point and denote by M~e the closure of M~e in the strong topology of the space Mne.
Let M~e be the set of all A= {At}~l E Mue for which there exists x(A) E K such that for each integer t 2: 1,
Atx(A) = x(A) and p(Aty,x(A)) ::; p(y,x(A)) for all y E K,
and denote by M:e the closure of M~e in the strong topology of the space Mue.
566
We consider the topological subspaces M~e and M:e with the relative strong topologies. In Theorem 2.4 of [43] we showed that a generic sequence {Ct}~l in the space M:e has a unique common fixed point x. and all random products of the operators {Ct}~l converge to x., uniformly for all x E K. We now quote this theorem.
Theorem 5.2 There exists a set :F c M:e which is a countable intersection of open everywhere dense (in the strong topology) subsets of-x;t;.e such that for each {Btl ~l E :F there exists x. E K for which the following assertions hold:
1. Btx. = x., t = 1,2, ... , and
p(Bty, x.) ::; p(y, x.), y E K, t = 1,2, ....
2. For each E > 0 there exist a neighborhood U of {Bt}~l in M:e with the strong topology and a natural number N such that for each {Cd ~l E U, each integer T 2: N, each mapping r : {I, ... , T} -> {I, 2, ... }, and each x E K,
P(Cr(T)····· Cr(l)X,X.) ::; E.
In [43] we also proved an analog of this theorem for the space M~.
We remark in passing that one can easily construct an example of a sequence of operators {Atl~l E M~e for which the convergence properties described in Theorems 5.1 and 5.2 do not hold. Namely, they do not hold for the sequence each term of which is the identity operator.
Now assume that F is a nonempty closed p-convex subset of K and that Q : K -> F is a nonexpansive operator such that
Qx=x, x EF.
Such an operator Q is usually called a nonexpansive retraction of K onto F (see [30]).
Denote by M~) the set of all {Atl~l E Mne such that
Atx = x, x E F, t = 1,2, ....
Clearly M~) is a closed subset of Mne in the weak topology. We equip the topological
subspace M~) c Mne with both the weak and strong relative topologies.
In Theorem 3.1 of [43] we showed that for a generic sequence of operators {Btl~l in the
space M~) with the weak topology there exists a nonexpansive retraction p. : K ...... F such that
uniformly for all x E K. We end this section with the precise statement of this convergence theorem.
Theorem 5.3 There exists a set :F c M~) which is a countable intersection of open
(in the weak topology) everywhere dense (in the strong topology) subsets of M~) such that for each {Bt}~l E:F the following assertions hold:
1. There exists an operator p. : K ...... F such that
lim B t .•... BIX = p.x for each x E K. t~oo
Generic aspects 567
2. For each € > ° there exist a neighborhood U of {Btl~l in M~) with the weak topology and a natural number N such that for each {Ct}~l E U, each integer T 2: N, and each x E K,
p(CT' .... CIX, P.x) :::; E.
Theorem 3.2 of [43] is a random version of this theorem.
6. (F)-attracting mappings
In this section we discuss the class of (F)-attracting mappings.
Assume that (X, p, M) is a complete hyperbolic space and K is a closed p-convex subset of X. Let B be the set of all continuous mappings A : K --t K. We denote by Bu the set of all A E B which are uniformly continuous on bounded subsets of K and by Bne the set of all A E B sucl! that
p(Ax,Ay):::; p(x,y) for all x,y E K.
Note that Bne = A (see Section 3).
Fix 0 E K. For the set B we consider the uniformity determined by the following base:
E(n, €) = {(A, B) E B x B: p(Ax, Bx) :::; E, X E B(O, n)},
where € > ° and n is a natural number. The space B with this uniformity is metrizable and complete. We endow B with the topology induced by this uniformity. Clearly this topology does not depend on our choice of 0 E K. The sets Bu and Bne are closed subsets of B. We equip the topological subspaces Bu and Bne C B with the relative topologies.
Let F be a nonempty closed p-convex subset of K. Denote by B(F) the set of all A E B such that
Ax = x for all x E F and p(Ax,y) :::; p(x,y) for all x E K and all y E F.
It is obvious that B(F) is a closed subset of B. Set
B~F) = B(F) n Bu and B~) = B(F) n Bne.
Once again, we equip the topological subspaces B(F), B~F) and B~) C B with the relative topologies.
We may assume that 0 E F. Then for each real r > 0,
C(B(O,r)) C B(O,r) for all C E B(F).
A mapping A E B(F) is called weakly (F)-attracting (cf. the definition on p. 372 of [6]) if
p(Ax,y) < p(x,y) for each x E K\F and each y E F.
It is obvious that if A E B(F) is weakly (F)-attracting, then the fixed point set of A, {x E K: Ax = x}, coincides with F.
Let € E F. A set of mappings E C B(F) is called uniformly (F)-attracting with respect to € (w.r.t. € for short) if for eacl! natural number n there exists an increasing function cPn : [0,00) --t [0, 1] such that
cPn(t) > 0, t E (0,00), cPn(O) = 0,
568
and p(x,O - p(Ax,~) 2: rPn(p(x,F»
for each x E B(O, n) \ F and each A E E. Clearly this definition of a uniformly (F)-attracting (with respect to ~) set of mappings does not depend on our choice of o. A mapping A E B{F) is called (F)-attracting with respect to ~ if the singleton {A} is uniformly (F)-attracting with respect to ~.
It is easy to verify that our definition of an (F)-attracting (w.r.t. ~) mapping is equivalent to the notion of a projective (w.r.t. ~) mapping introduced in [5].
For examples of (F)-attracting (w.r.t. 0 mappings in a Hilbert space see [5] and [6, p. 373]. For a class of examples in any uniformly convex Banach space X, consider a nonexpansive retraction P : X -> F. If 0 < t < 1, then the averaged mapping T = (l-t)I +tP is strongly nonexpansive in the sense of [11] and hence (F)-attracting w.r.t. all ~ E F. Other examples of (F)-attracting mappings are given in [48].
In [48] we studied (F)-attracting mappings and uniformly (F)-attracting sets and showed that they have the following important properties:
If there exists an (F)-attracting mapping for a given F c K, then a generic mapping is also (F)-attracting.
An infinite product of a sequence of mappings which has a uniformly (F)-attracting subsequence converges to a retraction.
If each mapping Ai is (Fi)-attracting, i = 1, ... , n, and the intersection F = nf=l Fi is nonempty, then the convex feasibility problem for F can be solved by iterating either products or convex combinations of the mappings Ai, i = 1, ... ,n.
Here we will only state the generic result mentioned above. Let F be a nonempty closed p-convex subset of K.
Theorem 6.1 Assume that ~ E F and that P E B(F) is (F)-attracting w.r.t. ~. Then there exists a set F c B(F) which is a countable intersection of open everywhere dense sets in B(F) such that each A E F is (F)-attracting w.r.t. ~. If P E B?"), then there
is a set Fu C F n B~F) which is a countable intersection of open everywhere dense sets in B~F) and if P E B~), then there is a set Fne C Fu n B~) which is a countable
intersection of open everywhere dense sets in B~) .
7. Contractive set-valued mappings
Assume that (X, 11·11) is a Banach space, K is a nonempty bounded closed subset of X and there exists 0 E K such that for each x E K,
tx + (1 - t)O E K, t E (0,1).
We consider the complete metric space K with the metric Ilx - yll, x, y E K. Denote by S(K) the set of all nonempty closed subsets of K. For x E K and D C K set
p(x,D) = inf{llx - yll : y ED},
and for each C, D E S(K) let
H(C, D) = max {sUPP(X,D), SUPP(y,C)}. xEC yED
Generic aspects 569
We equip the set S(K) with the Hausdorff metric H(·, .). It is well known that the metric space (S(K), H) is complete.
Denote by Ql the set of all nonexpansive operators T : S(K) -> S(K). For the set Ql we consider the metric Pm defined by
Denote by N the set of all mappings T : K -> S(K) such that
H(T(x),T(y)) S; Ilx-yll, x,y E K.
A mapping TEN is called contractive if there exists a decreasing function cjJ [O,diam(K)] -> [0,1] such that
cjJ(t) < 1 for all t E (0, diam(K)]
and H(T(x), T(y)) S; cjJ(llx - yll)llx - yll for all x,y E K.
Assume that TEN. For each D E S(K) denote by T(D) the closure of the set U{T(x) : XED} in the norm topology.
It was shown in [49] that for any TEN, the mapping T belongs to Ql and moreover, the mapping T is contractive (see Section 4) if and only if the mapping T is contractive.
We equip the set N with the metric PN defined by
It is not difficult to verify that the metric space (N, PN) is complete.
For each TEN set peT) = T. It is easy to see that for each T 1 , T2 EN,
Denote 'E = {P(T): TEN}.
Clearly the metric spaces ('E, pm) and (N, PN) are isometric.
In [49] we obtained the following results (cf. Section 4).
Theorem 7.1 Assume that the operator TEN is contractive. Then there exists a unique set AT E S(K) such that T(AT) = AT and (T)n(B) -> AT as n -> 00,
uniformly for all BE S(K).
Theorem 7.2 There exists a set F which is a countable intersection of open everywhere dense subsets of (N, PN) such that each T E F is contractive.
8. Nonexpansive set-valued mappings
Let (X, II· II) be a Banach space and denote by Sco(X) the set of all nonempty closed convex subsets of X. For x E X and D c X set
p(x, D) = inf{llx - yll : y ED},
570
and for each C, D E Sco(X) let
H(C,D) = max {sup p(x,D), SUPP(y,C)}. xEC yED
The interior of a subset D c X will be denoted by int(D). For each x E X and each r > 0 set B(x, r) = {y EX: Ily - xII :-::: r}. For the set Sco(X) we consider the uniformity determined by the following base:
Q(n) = ((C,D) E Sco(X) x Sco(X): H(C,D):-::: n-1},
n = 1,2, .... It is well known that the space Sco(X) with this uniformity is metrizable and complete. We endow the set Sco(X) with the topology induced by this uniformity.
Assume now that K is a nonempty closed convex subset of X and denote by Sco(K) the set of all D E Sco(X) such that D c K. Clearly Sco(K) is a closed subset of Sco(X). We equip the topological subspace Sco(K) C Sco(X) with its relative topology.
Denote by Nco the set of all mappings T : K -+ Sco(K) such that T(x) is bounded for all x E K and
H(T(x),T(y)) :-::: IIx - YII, x,y E K.
In other words, the set Nco consists of those nonexpansive set-valued self-mappings of K which have nonempty bounded closed convex point images.
Fix () E K. For the set Nco we consider the uniformity determined by the following base:
&(n) = {(TI, T2) E Nco x Nco: H(T1(x), T2(X)) :-::: n-1
for all x E K satisfying IIx - ()II :-::: n}, n = 1,2, ....
It is not difficult to verify that the space Nco with this uniformity is metrizable and complete.
The following result is well known [16, 36]; see also [39].
Theorem 8.1 Assume that T : K -+ S(K), 'Y E (0,1), and
H(T(x), T(y)) :-::: 'Yllx - YII, x,y E K.
Then there exists XT E K such that XT E T(XT).
The existence of fixed points for set-valued mappings which are merely nonexpansive is more delicate and was studied by several authors. See, for example, [29, 34, 40] and the references therein. We now state a result established in [50] which shows that if int(K) is nonempty, then a generic nonexpansive mapping does have a fixed point.
Theorem 8.2 Assume that int(K) =f 0. Then there exists an open everywhere dense set F C Nco with the following property: For each S E F there exist x E K and a neighborhood U of S in Nco such that x E S(x) for each S E U.
9. Porosity
In this section we present a refinement of the classical result obtained by De Blasi and Myjak [17J. This refinement, which is also due to De Blasi and Myjak [19], involves
Generic aspects 571
the notion of porosity which we now recall [8, 20, 22, 54]. In this connection, see also [13,51]
Let (Y, d) be a complete metric space. We denote by B(y, r) the closed ball of center y E Y and radius r > 0. A subset E c Y is called porous (with respect to the metric d) if there exist a E (0,1) and ro > ° such that for each r E (O,ro] and each y E Y there exists z E Y for which
B(z,ar) c B(y,r) \ E. (9.1)
A subset of the space Y is called a-porous (with respect to d) if it is a countable union of porous subsets of Y.
Since porous sets are nowhere dense, all a-porous sets are of the first category. If Y is a finite-dimensional Euclidean space, then a-porous sets are of Lebesgue measure 0. In fact, the class of a-porous sets in such a space is much smaller than the class of sets which have measure ° and are of the first category. Also, every Banach space contains a set of the first category which is not a-porous.
To point out the difference between porous and nowhere dense sets note that if E c Y is nowhere dense, y E Y and r > 0, then there is a point z E Y and a number s > ° such that B(z,s) c B(y,r) \E. If, however, E is also porous, then for small enough r we can choose s = ar, where a E (0,1) is a constant which depends only on E.
Let (X, p, M) be a complete hyperbolic space and K C X a nonempty bounded closed p-convex set. Once again we denote by A the set of all nonexpansive self-mappings of K. For each A, B E A we again define
h(A, B) = sup{p(Ax,Bx) : x E K}. (9.2)
It is easy to verify that (A, h) is a complete metric space.
Theorem 9.1 There exists a set:F C A such that the complement A \:F is a-porous in (A, h) and for each A E :F the following property holds:
There exists a unique XA E K for which AxA = XA and Anx -> XA as n -> 00,
uniformly on K.
Proof. Set
diam(K) = sup{p(x,y) : x,y E K}. (9.3)
Fix (J E K. For each integer n 2: 1 denote by An the set of all A E A which have the following property:
(Cl) There exists a natural number peA) such that
p(AP(A)x, AP(A)y) :S lin for all x, y E K. (9.4)
Let n 2: 1 be an integer. We will show that An is porous in (A, h). To this end, let
a = (diam(K) + 1)-1(8n)-1. (9.5)
Assume that A E A and r E (0,1]. Set
1= T 1r(diam(K) + 1)-1 (9.6)
572
and define Ay E A by
Ayx = (1 - -y)Ax EEl -yO, x E K. (9.7)
It is easy to see that
p(Ayx,Ayy) ~ (1- -y)p(x,y), x,y E K, (9.8)
and
h(A,Ay) ~ -ydiam(K). (9.9)
Choose a natural number p for which
(9.10)
Let B E A satisfy
h(Ay, B) ~ ar, (9.11)
and let x,y E K. We will show that p(BPx,BPy) ~ lin. (We use the convention that CO = I, the identity operator.)
Assume the contrary. Then for i = 0, ... ,p,
p(BiX,Biy) > lin. (9.12)
It follows from (9.11), (9.2), (9.8) and (9.12) that for i = 0, ... ,p - 1,
p(Bi+lx, Bi+ly) ~ p(Bi+1x, AyBix) + p(AyBix, AyBiy) + p(AyBiy, B i+1y )
~ ar + p(AyBiX, AyBiy) + ar
~ 2ar+ (1- -y)p(BiX, Biy) ~ p(BiX, Biy) + 2ar - -yIn
and p(BiX, Biy) - p(Bi+1X, B i+1y ) ~ -yIn - 2ar.
Combined with (9.3), (9.6) and (9.5) this latter inequality implies that
and
diam(K) ~ p(x,y) - p(BPx,BPy ) p-l
= L [p(BiX, Biy) - p(Bi+lX, B i+1y )]
~ p{-Yln - 2ar)
~ p [r(diam(K) + 1)-1(2n)-1 - 2r(diam(K) + 1)-1(8n)-1]
~ pr(diam(K) + 1)-1(4n)-1
p ~ r-1 diam(K)(diam(K) + 1) 4n,
a contradiction (see (9.10». Thus p(BPx,BPy ) ~ lin for all x,y E K. This means that
{B E A: h(Ay,B) ~ ar} cAn. (9.13)
It now follows from (9.9), (9.6) and (9.5) that
{BEA: h(Ay,B)::;ar}c{BEA: h(A,B)::;ar+-ydiam(K)}
REFERENCES 573
C{BEA: h(A,B)::;r}.
In view of (9.13) this inclusion implies that A \ An is porous in (A, h). Define F = n~=lAn. Then A \ F is u-porous in (A, h).
Let A E F. It follows from property (Cl) that for each integer n 2: 1 there exists a natural number s such that p(Aix,Ajy)::; lin for all x,y E K and all integers i,j 2: s. Since n is an arbitrary natural number, we conclude that for each x E K, {Aix}~l is a Cauchy sequence which converges to a point x. E K satisfying Ax. = x. and moreover, Aix -4 x. as i -+ 00, uniformly on K. This completes the proof of Theorem 9.1. •
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[34J T.-C. Lim, A fixed point theorem for multivalued nonexpansive mappings in a uniformly convex space, Bull. Amer. Math. Soc. 80(1974), 1123-1126.
[35J J. Myjak, Orlicz type category theorems for functional and differential equations, Dissertationes Math. (Rozprawy Mat.) 206(1983), 1-8l.
[36J S.B. Nadler, Jr., Multi-valued contraction mappings, Pacific J. Math. 30(1969), 475-488.
[37J RD. Nussbaum, Some nonlinear weak ergodic theorems, SlAM J. Math. Anal. 21(1990),436-460.
[38J E. Rakotch, A note On contractive mappings, Proc. Amer. Math. Soc. 13(1962),459-465.
[39J S. Reich, Fixed points of contractive functions, Boll. Un. Mat. Ital. 5(1972), 26-42.
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[43J S. Reich and A.J. Zaslavski, Convergence of generic infinite products of nonexpansive and uniformly continuous operators, Nonlinear Analysis 36(1999), 1049-1065.
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[52J G. Vidossich, Most of the successive approximations do converge, J. Math. Anal. Appl. 45(1974), 127-13l.
[53J G. Vidossich, Existence, uniqueness and approximation of fixed points as a generic property, Bol. Soc. Brasil Mat. 5(1974), 17-29.
[54J L. Zajicek, Porosity and <T-porosity, Real Analysis Exchange 13(1987), 314-350.
[55J A.J. Zaslavski, Turnpike property for extremals of variational problems with vector-valued functions, Trans. Amer. Math. Soc. 351(1999), 211-23l.
[56J A.J. Zaslavski, Existence of solutions of optimal control problems for a generic integrand without convexity assumptions, Nonlinear Analysis, 43(2001), 339-36l.
[57J A.J. Zaslavski, Generic well-posedness of optimal control problems without convexity assumptions, SIAM J. Control Optim., in press.
Chapter 17
METRIC ENVIRONMENT OF THE TOPOLOGICAL FIXED POINT THEOREMS
Kazimierz Goebel
Maria Curie-Sklodowska University
20-031 Lublin, Poland
1. Introduction
In metric fixed point theory the term the fixed point property is usually related to a certain class of mappings described by some metric conditions. In topological part of the theory however, we use this term with respect to the wide class of spaces and families of continuous transformations. Let us begin with recalling the classical definition and facts.
Definition 1.1 A topological space X is said to have the (topological) fixed point property (t.f.p.p. for short) if each continuous mapping f : X -> X has a fixed point.
It is easy to see that the above property is a topological invariant. Suppose X and Y are two homeomorphic spaces with X having the fixed point property. Let h : X -> Y be a homeomorphism with h (X) = Y. Suppose f : Y -> Y is continuous. Then 9 = h- 1 0 f 0 h : X -> X and 9 is continuous. Thus there exists x E X such that gx = x, and consequently, for y = hx we have y = fy.
Other operations which preserve the fixed point property are retractions. Recall that a subspace Y of X is a retract of X if there exists a continuous mapping (a retraction) r : X -> Y such that rx = x for all x E Y. For any given retract Y of X there can be many retractions on it. Suppose Y is a retract of X and let f : Y -> Y be continuous. Then, with the use of any retraction r : X -> Y, such f may be extended to a continuous mapping 9 = for: X -> X. Any fixed point of 9 must be a fixed point of f. Thus if X has the tJ.p.p. so have it all of its retracts.
The most famous and important theorem on the t.f.p.p., Brouwer's Fixed Point Theorem, which was first proved in 1912 by L.E.J. Brouwer. In our terminology it can be formulated as
Theorem 1.2 The closed unit ball En in ]R.n has the t.J.p.p.
Brouwer's Theorem has many applications in analysis, differential equations and generally in proving all kinds of so called existence theorems for many types of equations.
577
WA. Kirk and B. Sims (eds.), Handbook a/Metric Fixed Point Theory, 577-611. © 2001 Kluwer Academic Publishers.
578
Its discovery has had a tremendous influence in the development of several branches of mathematics, especially algebraic topology. All known proofs of the Theorem require analytical, topological, combinatorial, "nonelementary" and "nonmetric" (whatever it means) arguments. The Theorem also has many equivalent formulations. The attempts (sometimes naive) to find more general versions, equivalents or an elementary proof of Brouwer's Theorem have given rise to interesting questions of "metric type" . We are not going to present here proofs of the Theorem but rather discuss the metric "environment" of it.
To follow let us define the standard simplex in ]Rn as
n
T n - 1 = {x E]Rn: Xi 2: ° for i = 1,2, ... ,n and ~Xi = I}. 1=1
and formulate:
Theorem 1.3 For each n = 1,2, ... , T n - 1 has the t.J.P.p.
(1)
In view of the initial remarks one can immediately formulate the "more general version" .
Theorem 1.4 Any nonempty, convex, closed and bounded subset K of]Rn has the
t.J.P·p··
All three formulations are equivalent. It can be shown using combinations of some of the following facts:
a) All closed balls in]Rn (of positive radius and arbitrary center) are homeomorphic.
b) Tn c ]Rn+l is homeomorphic with Bn.
c) Any nonempty convex and closed subset K of]Rn is a retract of ]Rn. Thus it is a retract of any ball and any set containing it.
d) Any nonempty convex, bounded and closed subset K of]Rn is homeomorphic to a ball Bk for some k ::; n.
There are many other equivalents of Brouwer's Theorem. Let us recall two the most important. First is known as the Nonretraction Theorem.
Theorem 1.5 For n = 1,2, ... the unit sphere sn-l is not the retract of Bn.
The second concerns the contractibility of spheres. Let us recall that a topological space X is said to be contractible to a point Z E X if there exists a continuous function (a homotopy) H : [0,1] x X --> X such that for all x E X, H (0, x) = x and H (l,x) = z. If X is contractible to a point z then it is contractible to any other point of X, so we can simply say X is contractible. In other (topological) terminology X is contractible if the identity mapping I on X is homotopic to a constant map.
The following is the so called No Contractibility Theorem.
Theorem 1.6 For n = 1,2, ... the unit spheres sn-l are not contractible.
Metric environment of the topological fixed point theorems 579
The equivalence of Theorems 1.2, 1.5 and 1.6 (1.2, 1.5, 1.6 for short) can be shown in several ways. We present here arguments which will be useful later in the discussion of infinitely dimensional balls and spheres.
Proo~ of the equivalence. The unit sphere sn-l does not have the t.f.p.p. since the antipodal map -J : x --+ -x is continuous (even isometric) on sn-l without fixed points. Thus, since retractions preserve the t.f.p.p., 1.2implies 1.5.
To prove the contrary let us assume that 1.2 is false. Suppose for certain n there exists a continuous mapping T: B n --+ B n such that Tx oF x for all x E Bn. Extend this mapping to the doubled ball 2Bn by defining
{ Tx for IIxll :s; 1
T1x = (2 - Ilxll) T (fxTI) forI < Ilxll :s; 2 (2)
and the new mapping T2 : B n --+ B n as
Observe that both mappings Tl and T2 are fixed point free and for all x E sn-l we have T2x = O. Now we can contradict 1. 5 by defining a retraction R : B n --+ sn-l as
Rx = X-T2X Ilx -T2x ll'
(3)
Thus 1.5 implies 1.2 and the equivalence of 1.2 and 1.5 is proved.
Suppose now that 1.6 does not hold and let H : [0,1] x sn-l --+ sn-l be a homotopy with H (0, x) = x and H (1, x) = z = canst. Take any radius 0 :s; r < 1 and define
{ z for IIxll :s; r,
Rx = H C~~~II, fxTI) for r < IIxll :s; 1. (4)
The consistency of the above formula for r = 0 comes from the uniform continuity of H. Observe that for each r, R is a retraction of B n onto sn-l. In view of this1.5implies 1.6.
On the other hand, if R is a retraction onto sn-l then sn-l would be contractible to R (0) by the homotopy defined as
H (t,x) = R((I- t)x). (5)
This concludes the proof of equivalence of all formulations 1.2, 1.3, 1.4, 1.5, 1.6 .•
To end this section we turn the readers attention to the fact that all the facts presented here are nonmetric in the sense that they do not depend on the norm II II used in ]Rn.
Geometric properties of balls will be important however, in the infinitely dimensional case we are going to discuss.
2. Schauder's Theorem
From now on let X denote an infinitely dimensional Banach space with norm II II the closed unit ball Bx and the unit sphere Sx (B and S for short) . The crucial difference
580
between infinitely dimensional spaces and ]Rn is caused by the fact that closed, bounded and convex sets do not need to be compact. Recall that X is of finite dimension if and only if Bx is compact.
An important generalization of Brouwer's Theorem was discovered in1930 by J.Schauder and is known as Schauder's Fixed Point Theorem.
Theorem 2.1 Any nonempty, compact, convex subset K of a Banach space has the t.J.p.p ..
Proof. Let K c X be convex and compact. For any c > 0 there exists a finite c-net {aI, a2, ... , ap } C K. Define a collection {ai} of real valued nonnegative functions on K by taking
. ( ) _ { 0 if Ilx - aill ::::: c, . _ 1 2 at x - t - 1 , •• ,p. C - Ilx - aill if Ilx - aill ::; c,
Put
Q;(x) f3;(x)="p .()
L..i=l Q, X
and observe that Ef=l f3i (x) = 1. Now define the function ~ K --+ Ko K n span {aI, a2, ... , ap } by taking
p
~ (x) = Lf3;(x) ai· i=l
Obviously ~ is continuous and represents on K an c-approximation of identity, which means that for x E K we have II~ (x) - xii::; c. Let T : K --+ K be a continuous mapping. Let us modify it, defining To : K --+ K by To = ~ 0 T. Thus To : Ko -; Ko and since Ko is homeomorphic to a compact convex subset of finite dimensional space, To has a fixed point x E Ko. From this we have
Ilx - Txll ::; IIx - Toxll + IITox - Txll = II~ 0 Tx - Txll ::; c.
Therefore ,inf {lix - Txll : x E K} = 0, and since K is compact FixT =10. • It is in general not possible, even in limited ways, to extend Brouwer and Schauder theorems to noncompact settings. Let us recall, in more or less historical order, facts explaining the cause of the failure of Schauder theorem on noncompact sets. The most commonly known are Kakutani's Examples (1943).
Example 1. Consider [2 (N) as a model of a Hilbert space. For the unit ball, the mapping T : B -; B
is clearly continuous with FixT = 0. Moreover it is easy to notice that T is uniformly continuous on Band T (B) C S. Also for any 0 < c < 1 similar mappings
Metric environment of the topological fixed point theorems 581
map B into B (but not into S) are fixed point free and are even lipschitzian. Indeed we have
The next of Kakutani's constructions shows even more. Consider another model on Hilbert space namely 12 (Z) and an the isometric shift operator
A: x = {Xi}iEZ ---> {xi+1hEZ'
Thus the mapping T : B -> B defined by
1 Tx = 2" (1 - IIxll) eo + Ax
where eo = {6i,O}, is a lipschitzian homeomorphism of B onto B with lipschitzian inverse, yet FixT = 0.
All Kakutani's examples have very simple structure. Each one represents a one dimensional perturbation of an isometry. They fail to have fixed points but share a somewhat weaker property. For all the above mappings we have
inf IIx - Txll = O. xEB
(6)
Since it is known that all nonexpansive self-mappings of B in Z2 must have fixed points, Kakutani's constructions show just how fragile is this property.
Balls in less regular spaces than Z2 can accept even nonexpansive mappings without fixed points.
Example 2. Consider the space G [a, b] (e.g. G [-1, 1]) of continuous functions and its unit ball B. Let us recall that there exists a nonexpansive retraction Q : G [a, b] ---> B defined with the use of the function
as
{ -1
a (t) = min {1, max [-1, t]} = ~
for t ::; -1
for - 1 ::; t::; 1
for t 21
Qf (t) = a 0 f (t) = a (f (t)). (7)
Take any function 9 E G[a,b] satisfying g(a)::; -2, g(b) 22 (for example get) = 2t for G [-1, 1]). Define the mapping T : B -> B by the formula
Tf (t) = Q (f (t) + 9 (t)).
For each such mapping we have Tf (a) = -1, Tf (b) = 1 so T: B ---> S. Moreover T is nonexpansive, fixed point free and again satisfies (6).
Example 3. There exist isometries mapping the unit ball in CO into its unit sphere without fixed points. The simplest formula for such a mapping is:
T (Xl. X2, ... ) = (l,Xl, X2, ... ) .
Our next example shows another type of singularity. This one is connected not with the metric regularity of the mapping but with its structure.
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Example 4. One of the properties distinguishing infinitely dimensional Banach spaces from spaces of finite dimension is the fact that if dim X = 00 then X is homeomorphic with the punctured space X"'- {OJ. More, there exists a homeomorphism h : X -+
X"'- {OJ such that hx = x for Ilxll 2: 1. Now the mapping T: B -+ B,
Tx = h- l (-hx)
lacks fixed points but represents an involution which means T2 = I. In other words all point are fixed under T2 (T2x = x).
The first general theorem on the extent of the topological fixed point property has been presented by V. Klee in 1950.
Theorem 2.2 A closed bounded convex subset C of a Banach space has the t.j.p.p. if and only if it is compact.
Recall two commonly known facts. First states that if C is noncom pact then there exist a number l' > 0 and a bounded sequence of points {xn} , n = 0, 1, 2, .. in C such that
dist(xn+!,span(Xl,X2, ... ,xn» 2: l' > 0
The second is the well known Tietze's Extension Theorem.
(8)
Theorem 2.3 Let (M, p) be a metric space and let Zbe a closed subset of M. Then for any continuous function f : Z -+ R there exists a continuous function (extension) 1: M -+ R such that F (x) = f (x) for all x E Z. Moreover if f (Z) c [a, b] then J(M) c [a,b].
With the above we can prove Klee's result.
Proof of Theorem 2.2. Suppose C C X is a closed bounded and convex. Let for {xn},n = 0,1,2, ... be a sequence in C satisfying (8). Without loss of generality we may assume that 0 E C and that IIxoll 2: l' > O. Let us construct a piecewise linear curve joining consecutive points xo, xl, X2, ... by putting
00
r = Urn where r n are segments r n = co (xn,xn+!) = [Xn, Xn+l] (9) n=O
Thus r is a closed subset of C and can be parameterized by a function 'Y : [0, (0) -+ r. For t E [0,(0), let t = n + s where n is the integer part of t and 0 ::; s < 1. Set
'Y (t) = 'Y (n + s) = (1 - s) Xn + SXn+!·
The function 'Y- l : r -+ [0,(0) is continuous and hence (in view of Tietze's Theorem) has a continuous extension 9 : C -+ [0,(0). Now a fixed point free mapping T: C-+ r c C can be defined as
TX='Y(g(x)+I).
Together with Schauder's Theorem this completes the proof.
(10)
• The assumption of boundedness of C is not crucial to our theorem. Unbounded convex sets either contain a halfline and are invariant under translations along it or contain a closed homeomorphic image of the half line and our arguments can be repeated.
Metric environment of the topological fixed point theorems 583
The method of the construction used by Klee has a certain disadvantage. The curve r is obviously the retract of C via the retraction r (x) = 1 (g (x». However r is of infinite length but C is bounded, thus none of continuous mappings satisfying T (C) = r can be uniformly continuous. In view of our initial examples it is natural to ask if there is another method of construction leading to uniformly continuous fixed point free mappings?
Before we pass to the discussion ofthis problem let us notice that formulas (2), (3), (4), (5) used in the proof of equivalence in the introductory section allow us to formulate:
Theorem 2.4 If X is an infinitely dimensional Banach space then:
a the unit ball B does not have the t.j.p.p.,
b the unit sphere S is the retract of B,
c the unit sphere S is contractible.
Since we used explicit formulas all items a, b, c are equivalent in the sense that each one can be derived from another. The search of uniformly continuous extension of Theorem 2.4 has been initiated by B. Nowak (1979) who proved that for some Banach spaces there exists a uniformly continuous (even lipschitzian) retraction of B onto S. The result has been extended to all Banach spaces by Y. Benyamini and Y. Sternfeld in (1983). And finally Y. Sternfeld and P. K. Lin presented the following result.
Theorem 2.5 If C is a closed bounded convex but noncompact subset of a Banach space then for any k > 1 there exists mapping T : C -> C satisfying the Lipschitz condition
IITx - Tyll :S k IIx - yll (11)
and such that
d(T) = inf[llx - Txll : x E C] > o. (12)
The proof is based on an idea similar to those we used showing Theorem 2.2. We only need a modification of the extension theorem.
Theorem 2.6 Let (M,p) be a metric space and let Z be a closed subset of M. Suppose f : Z -> R is a function satisfying the Lipschitz condition
If x - fyl:S kp(x,y) for x,y E z. (13)
Then f has an extension 1 : M -> R (liz = J) which satisfies the same Lipschitz condi!;ion. Moreover, if a :S fx :S b for all x E Z, then 1 can be chosen so that a :S fx :S b for all x E M.
Proof. It suffices to observe that the function defined for x E M by
hx = sup{fy - k(!(x,y) : y E Z},
has the first property. The second holds upon taking
Ix = max { a, min { b, hx} } .
584
• Corollary 2.7 Analogous theorem holds for mappings F = {h, /Z, ... } : Z -+ 100 satisfying
IIFx - Fylll= :::: kp (x, y).
Since 100 consists of bounded sequences with the supremum norm, the function F is represented by a sequence of "coordinate" functions {f;} each of which satisfies (13). Extending each one fi, according to the above theorem we can obtain the extension F of F. Moreover, if the "coordinate functions" f; are bounded all their extensions have the same bounds.
We are now in the position to prove the main result of this section.
Proof of Theorem 2.5. Let e be a bounded, closed and convex but noncompact subset of a Banach space X. As in the proof of Theorem 2.2 assume that 0 E e, Ilxoll 2: r and a sequence {xn }, n = 0,1,2, ... satisfies (8). Construct as before the piecewise linear curve r = r c and define in e a sequence of triangles
41n (e) = co {O, Xn, Xn+1} for n = 0,1,2, ...
and a "fan"
00
41 (e) = U 41n (e) . n=l
Each element x E 41 (e) is of the form x = >"'YC (t) where>.. E [0,1] and 'Yc represents the already used parameterization of r c by t = n + s E [0,(0). We use here e in the index position only to indicate that all objects are in the set e. Let us pass now to the space 100 (No). For each n E No let en denote the standard unit vector {Din} = (0, ... ,0,1,0, ... ) (where 1 is in the nth position). Define the sequence {41n } of standard triangles as
41n = co {O, en, en+il
= {(O, ... ,O,~n,~n+1'O, ... ): ~n 2: 0, ~n+1 2: 0, ~n+~n+1:::: I}
and the "standard fan" as
00
Obviously again each point x of 41 can be represented as x = >"'Y (t) where>.. and t are natural, for 41 and corresponding "frame" curve r generated by unit vectors, parameters. Both "fans" 41 (e) and 41 are locally finite dimensional (two dimensional) except for the neighborhood of zero. We leave to the reader a justification of the fact that both are homeomorphic and that the natural invertible mapping H : 41 -+ 41 (e) defined by
(14)
is onto and moreover both Hand H-1 are lipschitzian.
Metric environment of the topological fixed point theorems 585
The next crucial step is to show that there exists a lipschitzian retraction of 100 (No) onto A. Let e be a vector whose coordinates are all equal to one. For x E 100 (No), set
E(x) = {e ~ 0: (x-ee) VO = {max {Xi -e,On E A},
observe that since IIxll E E (x) ,E (x) i 0 and put
e(x) = infE(x).
Observe that, in the natural coordinatewise order, for x, y E 100 we have
x :::; y + IIx - yll e.
Thus e (y) + IIx - yll E E (x) implying e (x) :::; e (y) + IIx - yll. By symmetry we obtain
Ie (x) - C(y)1 ::; IIx - YII·
Therefore the retraction R : 100 (No) ...... A defined as Rx = (x - e (x)) V 0 fulfills the Lipschitz condition
IIRx - Ryll ::; 211x - yll .
In consequence, A (C) is a lipschitzian retract of C. Indeed, let H-I : A (C) ...... A be the inverse homeomorphism to H defined by (14). By Corollary H-1 has a lipschitzian extension H-l : C ...... 100. The desired lipschitzian retraction Ro : C ...... A (C) can be
now defined as Ro = H 0 R 0 H-l. Only we do not know an exact Lipschitz constant for it.
To complete the proof observe that we can start the construction of a lipschitzian mapping T : C ...... C with positive minimal displacement,
d(T)=inf{lIx-Txll :xEC}>O,
building first, for our standard "fan" A, a lipschitzian mapping TA : A ...... A satisfying dm~.) = inf {lix -TAXIl : x E A} > o. Once we have it, we can define the next mapping T A( 0) : A (C) ...... A (C) , also satisfying d (TA(O») > ° by putting TA(o) = H 0 TA 0 H-I . And we can finish with taking T = TA(O) 0 Ro.
Here is a formula for TA in terms of (.>., t) coordinates. For the initial triangle Ao of A (i.e.,for 0 ::; t ::; 1) set
{
(1- 3.>.t) eo
TAX = TA (.>., (t)) = (1- t) eo + (3'>' - 1) t ( eo1e1 )
,(~+(3'>'-2))
for.>. E [0, ~l ' for .>. E (l, iJ, for.>. E (i,l].
The above mapping maps Ao into Ao U Al without having fixed points. In view of compactness of Ao we have inf {llx - TAXIl : x E Ao} > 0.
Now we can extend TA to all of A by putting for x E U~=l An, X = .>., (t) (t ~ 1) :
{
(1 - 3.>.) eo
TAX = TA (.>., (t)) = (3'>' - 1) 'Y (t - !) ,((t-!) +(3.>.-2))
for.>. E [O,~] ,
for.>. E (~, i] , for.>. E (i,l].
586
We leave to the reader checking that T/1 is lipschitzian and that
d(T/1) = inf {llx - T/1XIl : x E ,6.} = inf {llx - T/1XIl : x E,6.o U,6.d > O.
So we proved that there exists a mapping T : C -+ C satisfying the Lipschitz condition (11) and having positive minimal displacement, which means satisfying (12). We neither evaluated its Lipschitz constant k nor its minimal displacement d (T).
To show that such mappings do exists for any Lipschitz constant k > 1 observe that if T satisfies (11) and (12) then for any a E (0,1) the mapping To< = (1 - a) I +aT : C -+ C is lipschitzian with constant (1 - a) + ak and d (To<) = ad (T). This remark completes the proof of Theorem 2.5. •
Theorem 2.5 has strong consequences concerning balls. Let T : B -+ B be a lipschitzian mapping having d (T) > O. Close analysis of all the reasonings used in the proof of the equivalence of 1.2, 1.5, 1.6 leads to the following result which is stronger than Theorem 2.4
Theorem 2.8 Let X be an infinitely dimensional Banach space with the unit ball B and the unit sphere S. Then:
a For any k > 1 there exists a mapping T : B -+ B satisfying (11) such that d(T) > O.
b There exists a retraction R : B -+ S satisfying (11) with certain (sufficiently large) k > l.
e S is contractible to a point by a lipschitzian homotopy.
The last statement means that there is a homotopy H : [0,1) x S -+ S with H (0, x) =
x, H(l,x) = Xo for which there are two constants A > 0, B > 0 such that for any t,s E [0,1) and x,y E S,
IIH (t,x) - H (s,y)1I ~ Alt - 81 + B Ilx - yll· All three statements are formally equivalent in the sense that each one can be derived from any other.
This theorem completes the qualitative part of our considerations. In the sections to follow we will discuss some quantitative problems connected with concrete estimates for d (T) or a Lipschitz constant for retractions of a ball onto a sphere.
3. Minimal displacement problem
As we already observed, all the fixed point free mappings defined by Kakutani have d (T) = O. This property is natural for certain classes of selfmappings of convex and bounded sets. For example this is true for all nonexpansive mappings, their compact perturbations or the so called k-set contractions. On the other hand for k-lipschitzian mappings we have Theorem 2.5.
The term minimal displacement problem was probably first used by the present author in 1973. It came long before the Sternfeld and Lin result, from the observation that while looking for the fixed points of a mapping T : C -+ C we are often led to evaluate of the quantity
d (T) = inf IIx - Txll xEC
Metric environment of the topological fixed point theorems 587
called the minimal displacement of T.
Sometimes we are able to establish uniform estimates for all T belonging to a certain class T. The minimal displacement for the whole class is defined as
d (T) = sup d (T) = sup inf IIx - Txll . TET TETxEC
The class T under consideration is often divided into subclasses T (k) indexed by some parameter k. In this case we can also investigate the function of this parameter defined by
<p(k) = d(T(k».
Under minimal displacement problems we place all questions related to finding or evaluating the above quantities. We restrict ourselves to investigations by the whole class or subclasses of Lipschitz maps defined on the unit ball B, or more generally on a closed convex and bounded subset C of a Banach space X or to its unit ball B.
We shall denote by L the class of all Lipschitz maps. L is naturally scaled by the increasing system of subclasses L (k),
L=UL(k), (15) k>O
where for k E [0,(0), L(k) consists of all mappings satisfying the Lipschitz condition with constant k
IITx - Tyll ::; k IIx - YII· This notation will be used regardless of the domain of T and other restrictions. Subclasses of interest and their natural scaling by the Lipschitz constant k will be written as LO, L1 ... and LO (k), L1 (k) ....
Let T : C -+ C be of class L (k). If k < 1 then due to the Banach Contraction Principle it has a fixed point. If k = 1 then T is nonexpansive with d (T) = O. So we suppose k > 1. Take any point x E C, any a > 0 and consider the mapping
Tx : y -+ (1 - k ~ a) x + k ~ a Ty. (16)
Since Tx is a contraction there exists exactly one solution of the equation Tx(Y) = y, say Ya, and we have
IIYa-TYall = (1- k~a) IIx-TYall::; (1- k~a)r(x,c), where r (x, C) = sup {lix - zll : z E C} is the radius of C about x. Consequently by choosing appropriate x's and letting a -+ 0 we get
d (T) ::; (1 -~) r (C) ,
where r (C) = inf {r (x, C) : x E C} is the Chebyshev radius of C (with respect to itself).
Since this is the basic estimate let us formulate
588
Theorem 3.1 For any bounded, closed and convex set C and any T : C -+ C, T E £; (k) , we have
(17)
It is known that the estimate (17), is the best possible for certain sets, even balls in some spaces, and certain transformations T .
Example 5. Let S+ be the positive part of the unit sphere in L1 (0, 1),
S+ = {f E L1 (0, 1) : f? 0 and [f = I}.
Then r (S+) = 2. For k > 1, put
tf = sup {t : l f (s) ds = 1 - ~}
and define the mapping T : S+ -+ S+ as
{ 0 for t :::; tf
Tf(t)= kl(t) fort>tf
We leave to the reader to check that T E £; (k) and that for any f E S+
III -Till = 2 (1-D = (I-~) r (S+) = d(T).
Thus T moves all the points of S+ by the same extremal distance. A small modification of the above construction shows that a similar situation occurs for the positive part of the unit sphere in the sequence space ll.
Example 6. In the space of continuous functions C [0, 1] consider the set
K = {x : 0 = x (0) :::; x (t) :::; x (1) = I)}.
Thus K is closed bounded convex and for all x E K we have r (x, C) = r (C) = diam (C) = 1. Obviously the identity function e, (e (t) == t) belongs to K. Take any function a =I e, a E K and define the mapping T", generated by a as
T",x (t) = (a 0 x) (t) = a (x (t)) .
Then Ta inherits the Lipschitz constant of a: if a satisfies la (t) - a (s)1 :::; kit - sl then Ta is also k-lipschitzian. Since each x E K takes all values in [0,1] we have
IIx - Taxll = max Ix (t) - a (x (t))1
=maxls-a(s)1 = lie-ali =cunst.=d1 >0.
The iterations of Ta are of the same type, T~ = Taoa , T~ = Taoaoa, ... . Hence for any n = 1,2, ... , we have
IIx - -r:xll = cunst. = dn > O.
Metric environment of the topological fixed point theorems 589
All the iterations of any Tn have a constant positive displacement. Take the family of special a's given by
ak (t) = min {kt, I} ,k > 1.
Corresponding mappings Tk = Tn. belong to the class .c (k) with d (Tk) = 1 - !. Moreover since for any k > 0, I > 0 ak 0 al = akl, we have Tk 0 Tl = Tkl. Thus the family {Tk : k > I} is a semigroup, extremal with respect to minimal displacements on K.
Example 7. The same behavior of mappings can be observed even on balls. Let B be a unit ball in C [a, b] . Take k > 1 and modify the mapping from Example 2 by putting
Tkf (t) = Q (k (f (t) + 9 (t))) ,
for which we get
for all fEB. Similar examples can be constructed in many spaces with uniform (max) norms such as, balls in CO and the spaces of differentiable functions Cn [a, b] with all kinds of standard norms. An interesting result has been obtained by K. Bolibok. It states that not only C [a, b] but all its subspaces of finite codimension allow a construction with the above property.
Thus, the above examples show that in some extremal situations the inequality (17) is sharp. It is natural to expected that some geometrical properties of the space X or the set C itself can prohibit such extremal behavior. Before giving examples let us return to the general scheme.
For a closed bounded and convex set C let .cc (k) denote the whole family of klipschitzian maps T: C -> C. Define the function 'Pc: [1,00) -> [O,r(C)] by
'Pc (k) = sup {d(T) : T E.cc (k)}
= sup{inf{lIx -Txll: x E C}: T E .cc(k)}.
Similarity arguments make it possible to normalize the size of C and without loss of generality consider only those sets C for which r (C) = 1. From now we will take this to be a standing assumption.
For the whole space X put
'Px (k) = sup {'Pc (k) : C C X, r (C) = I}
and to distinguish the very interesting case, when C is the unit ball B we will write
'l/J (k) = 'l/Jx (k) = 'PB (k).
where it will not lead to any misunderstanding we will omit the subscripts and write 'P, 'l/J instead of 'Pc, 'Px, 'l/Jx. With this notation Theorem 2.5 gives the following,
Corollary 3.2 If C is convex, closed bounded but noncompact then 'Pc (k) > 0 for all k> 1.
Theorem 3.1 (evaluation (17)) implies
(18)
590
for any set C (with r (C) = 1). And so, in particular for the unit ball
1 1/J (k) -::;. 1- k'
There are sets (and balls) for which
let us call them extremal.
1 'Pc (k) = 1 - k'
(19)
The conclusions of our examples can be written in the following form. For the spaces Ll (0, 1) and 11
and for X = C[a,b]
1 'P!s+ (k) = 'PL'(O,I) (k) = 'PI' (k) = 1- k'
1 1/JC[a,bj (k) = 'PC[a,bj (k) = 1 - k'
for all k > 1. Hence the above sets are extremal. However, there are also sets and balls for which the estimation (17) is not sharp. We shall return to these later.
We now discuss some basic properties of the functions 'PC, 'P x, 1/Jx. Let T : C --+ C be of class L (k) with d (T) = d > O. For a E [0,1] the mapping T", = (1- a) I + aT E L (1 - a + ak) and satisfies d (T",) = ad (T) = ad. Consequently,
Property 1. For any a E [0,1] we have
'P (1 - a + ak) :::: a'P (k) .
In other words 'P is concave with respect to k = 1, indeed the function
is nonincreasing for k :::: 1.
'P (k) k-l
The next property comes from the following, previously used, construction. Take A > k and consider the mapping
TAY = (1-~) x+ ~TY. (20)
Since the right hand side of (20) is a contraction with respect to y, it has exactly one fixed point depending on x and A, say Fx = FAX. Thus we have an implicit formula
(21)
from which we get
(A-I) FEL A-k ' (22)
( A-I) TF ELk A-k ' (23)
Metric environment 0/ the topological fixed point theorems
II -F II = IIFx-TFxll > _d_ x x A-1 - A-1'
A Ad IIx -TFxll = A_1I1Fx-TFxll ~ A-1·
Thus for any A > k the observations (23) and (25) imply
Denoting k~ by I we have I > k and (26) takes form
and we have
Property 2. The function
is non decreasing for k ~ 1.
<.p(l)l <.p(k)k -->--1-1-k-1'
<.p (k) k k-l'
The next consequences of the above are
Property 3. The right hand side derivative
always exists and <.p' (1) > o.
Property 4 For all k ~ 1
<.p'(1) = lim <.p(k) k->l k - 1
591
(24)
(25)
(26)
(27)
(28)
(29)
Let us add one more property concerning 7f;x = <.pR. According to Theorem 2.8 there exists a lipschitzian retraction R : B -> S with Lipschitz constant ko say. For 0 < E: < 1, let
Tex ={ -Rm for IIxll::::::E: . -fxrr for IIxll > c
Thus Te E C (~) and Ilx - Txll ~ 1-c for all x E B. Setting k = ~ we obtain
Property 5. For any Banach space X, limk .... oo7f;(k) = 1. More,
ko 7f;x (k) 2: 1 - k'
where ko is the Lipschitz constant of a retraction of B onto s.
However, this does not mean that in generallimk .... oo <.pc (k) = r (e) .
592
Example 8. Let X = l2, let ei = {6'ji} be the sequence of standard unit vectors and let
0= co{ei: i = I,2, ... }
= {x = {xn}: Xn 2: 0 for n = 1,2, ... , and "fxn::; I}. n=1
We have r (0) = 1 but it is a nice exercise to prove that limk-+oo <Pc (k) ::; ,h. Thus the above set is not extremal.
There are spaces which do not contain extremal sets. This follows from some technical evaluations. For example, it is known that for any Banach space X and the function <p = <Px we have
2<p (2) + 6'x (2<p (2» ::; 1
and consequently
<p' (1) + 6'x (<p' (1») ::; 1,
where 6' x is the modulus of convexity of X. This implies that there are no extremal sets in uniformly convex spaces or more generally, in spaces with characteristic of convexity cO < 1 (i.e. for spaces satisfying 6'x (1) > 0 ).
An interesting behavior is shown by II (a very "square" space).
Example 9. Let B be the unit ball in ll. Suppose T : B ~ B, T E £. (k) , is a mapping with d (T) = d > O. Consider equation (20) and for A > k construct the mappings F and TF defined by (21). Suppose, for a moment, that TF(O) is finitely supported, i.e., that TF(O) lies in a subspace of II spanned by n basis vectors. Then if Pn denotes the natural projection of II onto this subspace, the mapping PnT F has finite dimensional range and by Brouwer's Theorem there is a point y E Pn (B) such that y = PnTFy. Thus in view of (22), (23), (24), (25),
(A-I) k A _ k lIyll 2: IITFy - TF(O) II = IITFy - yll + IIY - TF(O) II 2:
2: IITFy - yll + IITF(O)II-llyll 2: 2d (A ~ 1) -Ilyll
implying
On the other hand,
d (A ~ 1) ::; IITFy - yll = IITFy - PnTFyll = IITFYIl-IIYIl ::; I-IIYII·
Combining the above we get
d< (A-I) (A(k+l)-2k) - A A (k + 3) - 4k
(30)
for any A> k.
Metric environment of the topological fixed point theorems 593
IfT F(O) is not "finite dimensional" then the above evaluation can be obtained by taking small c > 0, choosing n so large that liT F(O) - PnT F(O) II < cd A~l' and repeating the above arguments with c-proximity.
Finally, minimizing the right hand side of (30) with respect to A and taking d close to 'I/J (k), we obtain
'l/Jll (k) ::; . { 2+2"'3 (1- l) for 1::; k::; 3+2V3,
~t~ for k > 3 + 2V3. (31)
Thus II space admits extremal sets, for example S+, but the unit ball B is not extremal. We have
and
1 'l/Jll (k) < IPI' (k) = 1 - k
.,,1 (1) < 2+V3. 'I'll - 4
To finish this topic let us mention another observation by K. Bolibok. Let B be the unit ball in C [0, 1] and let a sequence {ti} be dense in [0,1]. If we furnish C[O, 1] with a new (W. Zizler) norm
1
Ilfllz = IIfll + (~(f;:i)) 2) 2, we get a space that is not only strictly convex but uniformly convex in every direction. Such a space has normal structure, but in this case is extremal and has extremal unit ball.
The investigation of minimal displacement can also be restricted to some subclasses of £. Let us conclude with some examples. The Subclasses we are going to present here are selected because they have relevance to topics in the next two sections and to other questions in the metric fixed point theory.
Example 10. Let us consider the subclass £1 of £ consisting of mappings transforming B into S, T : B --- S. Obviously,
where
£1 (k) = {T : B -> S T E £ (k)}.
For this family of mappings let us define,
'l/J1,X (k) = d(£l (k)) = sup {d(T) : T E £1 (k)}.
As before we shall sometimes omit the space; writing 'l/Jl (k) instead of 'l/Jl,X (k) . Obviously, 'l/Jl,X is nondecreasing, and of course
1 'l/Jl,X (k) ::; 'l/Jx (k) ::; 1 - k'
594
The construction justifying Property 5 of 1/J x shows also that
lim 1/Jl (k) = 1, k~oo
while the construction from Example 7 indicates that for some spaces (e.g. C [a, bJ) we may have
1 1/J1,X (k) = 1/Jo,x (k) = 1 - k'
This is not true in general (see the section on Hilbert space).
Example 11. This time let us look at the subclass £2 of £ consisting of all mappings T : B --t X (the image is not necessarily contained in B), sending all the points x E 5 to the origin. This means that Ilxll = 1 implies IITxl1 = 0 or, in other words, T (5) = {O} . This class is also naturally divided into subclasses £2 (k).
Our interest in this class is connected with the optimal retraction problem (see the next section).
Define, analogously to before, the function
1/J2 (k) = sup {d (T) : T E £2 (k)} = d(£2 (k)).
Obviously, 1/J2 (k) ::; 1 for k 2: 1. Let R : B --t 5 be a lipschitzian retraction with Lipschitz constant ko. Then the mapping T = 1- R is of class £2 (ko + 1) and since for all x E B, Ilx - Txll = IIRxII = 1, we have
k 2: ko + 1 implies 1/J2 (k) = 1.
Let us find the first evaluation of 1/J2 (k) for smaller k's. Without loss of generality we can assume that our mappings are defined on the whole space X and take value zero outside of B, (11xll 2: 1 implies Tx = 0). Observe that for all x E B, x i= 0,
IITxl1 = IITX - T (11:11) II ::; k (1 - Ilxll)
and
IITx - T(O)II ::; k Ilxll· Consequently, IIT(O) II ::; k and for all x E X we have
IITX - ~T(O)II ::; ~ IITxl1 + ~ IITx - T(O)II k k k
::; "2 (1 - Ilxll) + "2llxll = "2'
This means that T transforms the ball B(~T(O),~) into itself. We leave to the reader a justification of the simple observation that
k k-l d(T)::; "21/J(k)::; -2-'
Finally,
1/Jz (k) ::; min { 1, ~1/J (k) } ::; min {I, k; 1 }
Metric environment of the topological fixed point theorems 595
and in consequence
t/!~ (1) ~ ~t/!' (1) ~ ~. If we restrict ourselves to the even narrower class £3 C £2 consisting of mappings for which T (B) C B, we obtain the next function t/!3 (k) with
t/!3 (k) ~ min {t/! (k) , 'ljJ2 (k)} .
We leave a proof that £3 =f 0 and that limk ..... oo t/!3 (k) = 1 as an exercise for the reader.
The next three examples are of a slightly different nature and are less connected to the optimal retraction problem (see the next section) but are close to the classical theory of nonexpansive mappings and geometry of Banach spaces.
Recall that a mapping T is said to be nonexpansive if T E £ (1) or, in other words, if
IITx - TYII ~ IIx -YII·
If Tis nonexpansive, so are all of its iterates Tn, n = 0,1,2,3, ... (TO = Id). There is a class of mappings sharing a similar property. A mapping T is said to be uniformly lipschitzian if there exists k ~ 1 such that
for n = 0,1,2, .... The class U£ of uniformly lipschitzian mappings is naturally divided into subclasses U£ (k) . It is known that the class U£ can be viewed as the class of all mappings nonexpansive with respect to equivalent metrics.
Indeed, if T E U£ then d(x, y) = sup;:"=o II rnx - Tny II is a metric, equivalent to the norm metric on B, with respect to which T is nonexpansive.
A basic fact about uniformly lipschitzian mappings is the following,
Theorem 3.3 If the space X is uniformly convex, then there exists a constant 'Y > 1 such that each mapping T : B -> B of class U£ (k) with k < 'Y has a fixed point.
This fact was first observed by W. A. Kirk and the present author in 1973 not only for balls but for all convex, bounded and closed subsets of X. Since then it has been investigated and extended to both more general and special cases by many authors. For example it is known that in the case of Hilbert space, all mappings T E U£ (k) with k < v'2 have fixed points while there exists a mapping T : B -> B, T E U £ (~) with d(T) = 0 but without fixed points (the example has been given by J.-B. Baillon).
Example 12. Let us consider the class £4 divided into natural subclasses £4 (k) consisting of all U £ mappings T : B -> B. For this setting define as before the function t/!4 (k) . There are two natural constants connected with t/!4
and
'YO (X) = sup {k : all mappings T E £4 (k) have fixed point}
'Yl (X) sup {k: all mappings T E £4 (k) or that with d (T) = O}
sUP{k:'ljJ4(k)=O}.
596
Obviously, 1'0 (X) ::; 1'1 (X) and 1(;4 (k) = 0 for k ::; 1'1 (X) . Moreover, 1'1 (X) < =. Again the construction justifying Property 5 actually shows that limk~oo 1(;4 (k) = 1.
The results mentioned above say that for uniformly convex (and some other) spaces, 1'0 (X) > 1. In particular for Hilbert space, H, we have V2 ::; 1'0 (H) ::; ~. Practically nothing is known about 1'1. Here are some questions. Are there spaces with 1'0 < 1'1? For which spaces is 1'1 = I? In other words, does there exist a space with 1(;4 (k) > 0 for k > I? We hope the answer to this last question is affirmative, but an example is unknown. Finally, can one find a formula for 1(;4,H (k), or at least give some good estimations?
Example 13. Our next class £5 consists of all T : B -> B that are 2-periodic (i. e. involutions), which means they satisfy the condition T 2x = x (or, T2 = Id). Obviously £5 c £4 and for each k, £5 (k) C £4 (k) .
Take any x E B and let y = ~ (x+Tx). 1fT E £5 (k), we have
k IITy - Txll ::; k Ilx - yll = "2llx - Txll ,
k IITy - xii = IITy - T2xll ::; k Ily - Txll = "2lJ x - TxlJ·
The above implies
k Ily - TYII ::; "2lJ x - TxlJ·
Now, it is a routine observation that if k < 2 then the sequence of consecutive iterates Xo = x, Xn+1 = ~ (xn + Txn ) converges to a fixed point of T. In other words, all involutions of class £5 (k) with k < 2 have fixed points. For spaces with more regular geometrical structure, uniformly convex spaces or Hilbert spaces, this estimate can be made even larger. For example it is known that for Hilbert space the same holds for k < ,)-1[2 - 3 = 2.62 ....
Define now our last function 1(;5,X (k) = 1(;5 (k) in the usual manner. The above remarks say thatljJ5 (k) = 0 for k < 2 and even larger k in some spaces. Nothing more is known. The question of whether there are lipschitzian involutions of B without fixed points is still unsolved (examples are known of continuous fixed point free involutions, but none of these are uniformly continuous). The question;
Does there exists an involution T of B onto B with d(T) > 0
also remains unanswered.
This last question is connected with the well known problem of geometric nonlinear functional analysis concerning the uniform classification of spheres. It is known that for any infinitely dimensional Banach space X its unit ball B and its unit sphere S are homeomorphic. However the question whether Band S are Lipschitz equivalent is open.
Assume that there exists a homeomorphism h : B -> S such that h (B) = S with hE £ (k) and h-1 E £ (k) for some k. Then
Metric environment of the topological fixed point theorems 597
for all x, YES. Putting
Tx = h-1 (-hx)
we obtain an involution of class I:- (k2) for which
IIx-Txll = Ilx-h-1(-hx)11 = Ilh-1(hx)-h-1(-hx)11 1 2
~ k 112hxll = k'
Thus d (T) ~ f and we would have
Since 1/J5 (k) ::; 1/J (k) ::; 1 - k, we can conclude that if such a homeomorphism exists then its Lipschitz constant k ~ 1 + ..;2. Probably this is a very imprecise evaluation. Thus, finding better estimates for 1/J5 (k) seems to be an interesting problem.
Example 14. Let us recall that a mapping T : C -; C is said to be asymptotically nonexpansive if
for n = 1,2, ... , where the sequence of Lipschitz constants satisfies limn .... oo kn = 1. It is known that the existence of a fixed point for such mapping depends strongly on the geometrical regularity of C. However, the basic question of whether this condition implies d (T) = 0 remains open for general spaces.
To finish this section let us remark that to our knowledge no formula has been found for 1/Jx (k) except for the extremal cases. The natural candidates to work with seem to be, Hilbert space and [I. Other interesting questions are:
Does there exists a Banach space X extremal in the sense that for all Banach spaces Y and k ~ 1
1/Jx (k)::; 1/Jy(k)?
Is Hilbert space or [I such a space?
4. Optimal retraction problem
The optimal retraction problem is the question connected with item b) of Theorem 2.8. As we already noticed, the value of the Lipschitz constant for a retraction R : B -; S appears in some estimations of the various functions 1/J. Close analysis of Theorems 2.6 and 2.8 leads to the conclusion that there exists an universal constant ko such that in any infinite dimensional Banach space there is a retraction R : B -; S belonging to I:- (ko) . Until now there is no known evaluation concerning how large this constant might be. To formalize the question let us assign to each space X the number
ko (X) = inf {k : there exists a retraction R: Bx -; Sx,R E I:- (k)}.
The claim we mentioned above means that for all Banach spaces, the coefficients ko (X) are bounded from above by ko. It is expected that ko (X) may vary depending on the geometry of X.
598
The optimal retraction problem means finding, or estimating, the characteristic ko (X) of X or the upper bound ko.
To date there is no space X for which an exact value of ko (X) is known. We have only some estimates from above and below. The gap is wide.
First let us recall that in any space X the radial projection P = Px : X --+ B defined by
{X for Ilxll ::; 1,
Px = fxlr for Ilxll > 1,
is lipschitzian. This projection is of class £ (2) in arbitrary space X while for some spaces its Lipschitz constant can be smaller (e.g. PH E £ (1) for Hilbert space). In consequence, for any x, y E X with Ilxll ~ r > 0, Ilyll ~ r, we have
1111:11 - II~IIII = Ilp~ - p~11 ::; ~ IIx - YII·
This allows us to extend any retraction R : B --+ S to the whole space by assuming Rx = Px for x E X \ B. Since as we will see, R has (on B) a Lipschitz constant larger then 2 this operation does not increase the constant.
The first simple and rough estimates come from concrete examples.
Example 15. Let X = C [0, 1]. For fEB define the mapping A: B --+ X by
Af (t) = If (t) + 1 - 2 (1 - Ilflll tl- 1 + 2 (1 - Ilfll) t.
An analysis of this mapping leads to the observations that Af = f for all f E Sand inf {IIAfll : fEB} = t (exercise). Now define the retraction
Af Rf = IIAPII = P(7 AI).
Since A E £ (5), P E £ (2) it follows that R E £ (70) and thus ko (C [0,1]) ::; 70.
Some spaces are "resistant" to producing good examples, moreover examples rarely lead to optimal constants. Let us pass to more general reasoning.
Let R : B --+ S be a retraction of class £ (k) (in the terminology of the previous section, R E £l(k)). Then the mapping T = -R also belongs to £1 (k) and moreover T2 = R. Let x E B and let Ilx - Txll = d > 0 (T does not have fixed points). The rectifiable (lipschitzian) curve I : [0,1] --+ S defined by
I (t) = T ((1 - t) x + tTx),
joins two antipodal points Tx = -Rx and T 2x = Rx. Its length exceeds the so called girth of the sphere 9 (X) defined as the infimum of lengths of curves lying on Sand joining antipodal points. For any space g(X) ~ 2, for some "flat" spaces equality holds but for example in Hilbert space 9 (H) = 7r. Let the length of I be l. Thus I ::; kd. Selecting, x we can get d ::; "l/J1 (k) + c: and consequently
(32)
Metric environment of the topological fixed point theorems 599
This implies
ko (X) 2 9 (X) + 1 2 3, (33)
for all spaces. A better estimate can be obtained for spaces for whicb 9 (X) > 2 or if 'l/Jl,X (k) < 1 - i (see the section on Hilbert space).
There are other "tricks" leading to some estimations of ko (X) from below. For example it is known that if 'l/Jx (k) ~ 1 - k!b for large k with certain a > 1, b > 0 then ko (X) 2 a + 2. This, in view of (31, implies koW) 2 4. However, an effective method of searching for such estimates is still to be found.
Let us present here two ways of finding estimations from above.
Example 16. Let T : B --+ B be sucb that \Ix - Tx\l 2 d > ! for all x E B. If TEe (k) then obviously k > 'l/J)/ (!) > 2. Consider the mapping A : B --> X defined by
Ax = x - (1 - Ilxll)Tx. (34)
Observe that A E C (k + 2) and that Ax = x for xES. The mapping A is bounded away from zero. We have
\I Ax II 2 IIxll - (1 - IIxll) IITxl1 2 211xll- 1
and
IIAxll 2 Ilx - Txll - IlxllllTxll 2 d - IIxll· Thus
. 1 IIAxll 2 max{21Ixll- 1,d -lIxll} 2 mm max{2t -I,d - t} = -3 (2d -1).
tE[O,l]
Now putting
and denoting by k(P) and k (R) the Lipschitz constants of P and R respectively, we obtain a retraction R : B --+ S with
k(R) < 3k(P)(k+2). - 2d-l
Since d can be chosen arbitrarily close to 'l/J (k) = 'l/Jx (k) we obtain
ko (X) ~ 3k(P)inf {2'l/J~~~ 1 : k > 'l/J-l G)}· (35)
For extremal spaces with 'l/Jx (k) = 1- i this gives
ko (X) ~ 3k (P) inf k ~k + 2) = 3k (P) (2 + vI2)2 R! (34.97 ... )k (P) < 70. k>2 - 2
This is not the best possible estimate. Modifying (34)
Ax = x - (1 - a (lIxll» Tx, (36)
600
where a : [0,1] --+ [0,1] is an increasing function with a (0) = 0, a (1) = 1 and playing a similar game we get better estimates. For example using a (t) = t 2 leads to ko (X) :::: 44.85 .... Also, some computer experiments performed with functions an (t) = tn for n :::: 20 indicate that in the case k (P) = 2,1/Jx (k) = 1 - t, we get the best result for n = 4, k ~ 4.02 .... The experimental evaluation gives ko (X) :::: 37.74 .... Further use of the formula (34) is limited by the lack of good estimates of 1/Jx for balls which are not extremal.
The next procedure is based on properties of the classes £2 (k) ,£3 (k) introduced in Example 11 and the corresponding functions 1/J2,X (k) ,1/J3,X (k).
Example 17. Let T : B --+ X, T E £2 (k) with d (T) = d > O. Thus since T (8) = 0, the mapping R : B --+ 8,
Rx- -P --x-Tx (X-TX) - Ilx-Txll - d
(37)
is a lipschitzian retraction. It follows that
( k+1) R E £1 k(P) d(T) ,
and consequently
ko (X) :::: k (P) inf 1/J:': (~r (38)
Since we have only that
and we do not know either the strict value of 1/J2 (k) nor its estimates from below, we can not proceed further. However it may be worthwhile to notice that if there is a space X extremal in the sense that 'l/J2,X (k) = min {I, k21} then (37) leads to ko (X) :::: 8. So far such a space has not been found.
Some evaluations can be obtained by means specific to the space under consideration (see also the Hilbert space section).
Example 18. Let X = C [0, 1]. Let Q be the nonexpansive retraction Q : X --+ B defined by (7). For any r > 0 put Qr! = rQ UI) . It is not difficult to check that for any r1 > 0, r2 > 0 and I, gEe [0, 1] ,
As we have already shown there exists a mapping T1 : B --+ B of class £ (k) such that d (T1) = 1 - t. Consider the ball of radius 2 (B2 = 2B) and extend T1 to the mapping T2 : 2B --+ B by putting
{ Td
T21 = TdQJ)
Qk(2-lIflll (T(Q!))
for 11/11:::: 1,
for 1 < 11111 :::: 2 - t, for 2 - t < 11/11 :::: 2.
Metric environment of the topological ,fixed point theorems 601
Again T2 E £ (k) and for all f E S2 (llfll = 2) we have T2f = 0. Moreover, it can be observed that IIf - T2fll 2: 1 - k for all f E B2. Now defining
1 Tf = "272 (21)
we get a mapping T E £3 (k) with d(T) = ~ (1- k). This leads to the observation that
1 ( 1) k - 1 '2 1 - k :s 1JI3,C[O,1] (k) :s 1P2,C[O,I] (k) :s min{1, -2-}' (39)
If we use the left inequality of (37) to generate the retraction
f-Tf (f-Tf) Rf= Ilf-Tfil =p H1-k) ,
we get R E £ (4 k~k~ll)) and therefore
. k(k+1) ( In) 2 ko (C [0, 1]) :s 4mm = 4 1 + v2 = 23.31... .
k>l k-l
Since limk--->oo1P3,X (k) = 1 the left inequality of (39) is not sharp. Thus the above estimate is far from being sharp but to our knowledge it is so far the best known.
As a by-product of the last construction, from (39) we get
1P~,C[O,l] (1) = 1P;,C[O,l] (1) = ~.
The following examples and estimations are connected with an analysis of item c) of Theorem 2.8. Let us begin with
Theorem 4.1 Lei H : [0,1] x S --+ S be a homotopy such that for each xES, H(O, x) = x and H(I,x) = Xo E S. Suppose that
IIH(c,x) - H(d,y)1I :s Alc - dl + 0 Ilx -yll, holds for x, yES and c, d E [0,1] where A, 0 are nonnegative constants. Then there is a retraction R : B --+ S satisfying for x, y E B :
20 IIRx - Ryll :s -lix - yll , r
where r is the solution of the equation
20 A - 201nr r l-r
(40)
Proof. Take any ° < r < 1 and a continuous convex function a : [r,l] --+ [0,1] such that a(r) = O,a(l) = 1. Define the retraction
Rf = { Xo x if Ilxll :s r, H (1- a (lIxl!) , iJXTi) if r < Ilxll :s 1.
602
The Lipschitz constant of R depends strongly on the selection of the radius r and the function a. To obtain an optimal estimate, let x, y E B with IIxll 2: r, Ilyll 2: r. Then
IIRx - Ryll IIH (1 -a (1Ixll), 11=11) - H (1 -a (1Iyll), II~II) II
::; Ala (1Ixll) - a (1Iyll)1 + a 1111:11 - II~IIII ::; Aa' (max {llxll,llyll}) Illxll - Ilylll + max {1~I,IIYII} Ilx - yll
::; max {Aa,(s)+2a}llf_gll. sElr,11 S
In order to make optimal choices for a and r, observe that since
[ (A a' (t) + 2;) dt = A - 2Br
does not depend on the function a, the Lipschitz constant of R is minimal if a is chosen to satisfy
A ' () 2a A - 2a In r ( ) at+-= =cr. t 1- r
Then we have
IIRx - RYII ::; c (r) Ilx - yll· Also, c takes on its minimal value when c (r) = ~ (calculus). • We give see two applications of this theorem.
Example 19. Consider the subspace Co [0, 1] consisting of those functions f for which f (0) = 0. For f E Sand c E [0,1] , set
HI(c, f) (t) = min {I, 1(1 + 2c) f (t) + 11- I}; H2 (c,f) (t) = max{c-l + ct,j (tn;
H3 (c, f) (t) = (1 - c) f (t) + ct.
Notice that since Ilfll = 1, f must assume the value 1 or -1 (or both). Check that HJ joins any function f = HI (0,1) along the path on the sphere with the final function HI (1, f) assuming value 1. Then the second homotopy ends with the function H2 (1, f) (t) 2: t. Finally H3 (1, f) (t) == t.
Now observe that for f, g E Sand c, d E [0, 1] ,
IIHI (c,f) - HI (d,g)11 ::; 21c - dl +311f - gil; IIH2 (c, f) - H2 (d,g)11 ::; 21c - dl + Ilf - gil;
II H3 (c, f) - H3 (d,g)1I ::; Ie - dl + IIf - gil· Let us piece these homotopies together by defining
{
HJ (¥,f)
H(c,f) = H2 C¥ -1,HI (1,1))
H3 (5e - 4, H2 (1, HI(1, 1)))
for e E [O,~] ,
for c E (~, t] , for e E (t, 1] .
Metric environment of the topological.fixed point theorems 603
Thus H joins the functions f E S to the identity function e (t) == t on [0, IJ, and
IIH (c, f) - H (d,g)1I ::; 51c - dl + 311f - gil. Using our theorem and solving (40) numerically for T we obtain T "" 0.345 and c(r) R:O
17.38. In particular we have the estimate
ko (Co [0, 1]) < 17.38.
With the use of Theorem 4.1 and some modifications of it, evaluations for other spaces have been obtained (for references see the Notes section). For example ko (co) ::; 35.18, ko W) ::; 31.64, ko (H) ::; 31.45. For all the subspaces X of C [0, IJ with finite codimension, ko(X) ::; 35.18. It is also known that our last estimate is not sharp, ko (Co [0, 1]) ::; 15.82.
So far, the space with the smallest estimate for ko is L1 (0, 1).
Example 20. Here let us consider the unit sphere Sin L1 (0, 1). For any f E Sand any C E [0, IJ , define
Set
H(Cf)={ If (t)1 ift::;tj(c), , f(t) ift>tj(c),
Obviously, H (0, f) = f and H (1, f) = If I· Thus H contracts S not to a point but to S+. It requires some tricky calculations to show that
IIH(C,f)-H(d,g)11 ::;2Ic-dl+2I1f-gll·
Now let R be the retraction given by
{ ~ (1' - IIfll + If I) if 11111 ::; 1',
Rf = ( j ) H 1 - a (1Ifll) 'm if IIfll > r.
Then for IIfll ::; l' and Ilgll ::; 1',
2 IIRf - Rgil ::; -lif - gil
l'
while as in the previous example for IIfll ::::: 1', Ilgll ::::: l' we get
IIRf - Rgil ::; i Ilf - gil, l'
where l' is chosen to satisfy
2-41nr 4 1- l' T
Thus T "" 0.424, and
ko (L1 (0,1)) < 9.43.
Observe that this example also shows some limitations on the proposed method. The Lipschitz constant 2 for the numerical argument of H can not be improved. Indeed if f < 0 then H joins two antipodal points f and If I along the curve on S of length 2.
604
5. The case of Hilbert space
In spite of geometrical regularity Hilbert space remains very resistant for the investigation of minimal displacement problems and optimal retraction in Hilbert space. Let C be a closed bounded and convex subset of a Hilbert space H. Two facts of a geometrical nature are useful. For any x E H there is a unique point y E C satisfying
IIx - yll = inf {lix - zll : z E C}.
Denoting y = Pcx (or shortly y = Px) we obtain a retraction P = Pc : H --t C. The retraction P is nonexpansive,
IIPx-Pyll:::::: IIx-yll,
for all x, y E H and moreover P and I - P are monotone which means that for all x,yEH
(Px - Py, x - y) 2: 0, ((x - Px) - (y- Py),x -y) = IIx _y1l2 - (Px - Py,x -y) 2: O.
In consequence of the above, for any x E H and any z E C we have
implying
IIx - zll 2: IIx - Pxll and IIx - zll 2: IIPx - zll .
The mapping P = Pc is called the nearest point projection on C. For the unit ball B the nearest point projection PB coincides with the radial projection PH discussed earlier.
The second fact is that if r (C) is the Chebyshev radius of C then there exists a unique point z E C, the Chebyshev center of C, with the property that the ball centered at z of radius r (C) contains C, C c B (z, r (C».
Let r (C) = 1 and assume (without loss of generality) that z = 0 is the Chebyshev center of C. Then C C B. Suppose T : C --t C is a mapping of class C (k) with d = d (T) > O. The mapping T can be extended to the mapping T : H --t C by putting
Tx _ { Tx for x E C, - T Pcx for x ¢ C.
In view of the above facts, TEL (k) , T : B --t Band d (T) = d (T) . Consequently we get
Theorem 5.1 For Hilbert space H The unit ball B is extremal in the sense that for any C with r(C) = 1
CPc (k) :::::: CPB (k) ,
and in consequence
CPH (k) = "pH (k) .
Metric environment of the topological fixed point theorems 605
The next advantage of Hilbert space geometry is connected with the construction, described by equations (20) and (21). Let T : B -+ B belongs to C (k) and for A > k let the mapping F : B -+ B be implicitly defined by
Fx = (1-~) X+~TFX or in another form,
TFx = AFx - (A -1)x.
All the properties (22), (23), (24) and (25) remain valid and in addition we have
k211Fx - FyII2 ;:::: A211Fx - Fyll2 - 2A (A - 1) (Fx - Fy, x - y) + (A - 1)211x _ Yll2
implying
A2 - k2 2 A-I 2 2(Fx-Fy,x-y);:::: A(A-l) IIFx-FYIl +-y-lIx-yll, (41)
or in a weaker form
A-I 2 2(Fx-Fy,x-Y);::::-y-lIx- Y Il . (41')
This shows that F is (strongly) monotone and allows us to find a first evaluation for 'PH (k) .
Theorem 5.2 For Hilbert space H
(42)
and consequently
I I ( 1 'P (1) = 1/J 1)::; -./2' (43)
Proof. Let T,F be as above and let d(T) = d. In view of (41), for each x E B we have
IITF2x - x211 = II (TF2x - Fx) + (Fx _ x)112
= IIA (F2x - Fx) + (Fx - x)112
= A211F2x - Fxl12 + 2A (F2x - Fx,Fx - x) + IIFx - xl1 2
;:::: A211F2x - Fxl12 + (A - 1) IIFx - xll2 + IIFx - xll2
(A2 A) d2 ;:::: + (A - 1)2'
For x = 0,1;:::: IITF2(0) II and we get
A-I ( 1){l; d< - 1-- --- v'A2+A - A A+l'
606
Choosing d close to <p (k) and letting A --4 k we get (42) , then (43) easily follows. •
Actually the above inequality also serves to obtain other evaluations. Suppose T E
.c1 (k) and T: B --4 S. Then, for x = 0 we have IITF2(0)II = /lTF(O) II = A /IF(O)/i = 1 and we obtain
(44)
It follows that "p~ (1) = 0 and that for Hilbert space"pl (k) < "p (k) at least for small k. In both cases we used only the weaker inequality (41') . The stronger, (41) can be used to obtain, estimates from below that one better than (29) ,. Observes that
IIF2x - xl12 = IIF2x - Fxl12 + 2 (F2x - FX,Fx - x) + IIFx - xll2
II 2 112 A2 - k2 II 2 112 A-I 2 2 2': Fx-Fx + A(A-1) Fx-Fx +"""""""}l/lFx-xll +/lFx-xll
d2 [ A2 - k2 A-I] 2': (A-1)2 2+ A(A-1) +"""""""}l .
Since d can be assumed to be close to "p(k) , F2 E .c (((A - 1)/(A - k))2) and x can be arbitrarily selected, standard arguments then lead to
"p ((~)2) > (jJIil)2 [2 + A2 - k2 +~]. A - k - A-I A (A - 1) A
For each fixed I > 1 select A so that I = ((A - l)/(A - k))2 . We leave it to the reader to justify that after making the substitutions and letting k --4 1 we get
(45)
All the above results seem to be imprecise .. The exact formula for "pH (k) as well as the exact value of "pH (1) remain unknown. If we pass to the optimal retraction problem not much is known either. Estimates for ko (H) are far from satisfactory. Let us present some sample results.
Example 21. The girth 9 (H) = 1f. Thus, in view of (44), if k is the Lipschitz constant for a retraction R : B --4 S, the evaluation (32) takes the form
3
k (1- ~r 2': k"pl,H(k) 2': 7r.
This implies ko(H) 2': 4.51... .
The next follows the pattern of Example 18.
Example 22. Let T : B --4 B be of class .c (k) with d(T) = d. Consider the ball Bl+d = (1 + d) B and a natural extension Tl : Bl+d --4 B defined as
T {TX for /lxll ::::: 1, IX = T Px for 1 < IIxll ::::: 1 + d.
Metric environment of the topological fixed point theorems 607
In view of known properties of P, Tl E C (k) and d (T1 ) = d (T) = d. Now consider the sphere 81+d+1/k = (1 + d + i)8 and extend Tl by putting
Observe that Tl E C (k) on B1+d U 81+d+1/k and as such, can be (via the well known Kirzbraun Extension Theorem) be extended to the whole ball B1+d+1/k without increasing the Lipschitz constant. Finally, then, we have a mapping
of class C (k) satisfying Tl(81+d+1/k) = 0 and d (T1) = d. To finish put
T(x) = 11Tl((1+d+-k1)x) l+d+ k
and observe that T : B -> B, T E C(k) , T (8) = 0 and d(T) = d(1+d+i)-l. Construct the retraction R : B -> 8 by
x-Tx (X-TX( 1)) Rx = IIx _ Txll = P -d- 1 + d + k .
Obviously,
and since d can be chosen close to I]! (k) , we get the estimate
. k + 1 ( 1) ko (H) 5: rr~f 1/1 (k) 1 + 1/1 (k) + k .
Using (29) and (42) we can derive from this the simple (but not sharp) estimate
In other words
ko (H) 1/1~ (1) < 2 ( 1 + v'2) 2 = 11.65 ....
The above connects two unknown values. We suggest that the reader try to find an improvement using (45) instead of (29).
Finally let us check how homotopy arguments work in the case of Hilbert space.
Example 23. Let us follow the steps used to estimate ko (L1) . Let L2 (0, 1) serve as a model of Hilbert space. For any function f E 8 and any c E [0,1] define
tf (c) = sup {t : llfl2 dt = c}
608
and
{III if t ::; tJ (c)
H(c,f)= I ift>tJ(c)·
Obviously H (0, f) = I and H (1, f) = III. It is a technicality to prove that H [0,1] x S -+ S is a uniformly continuous homotopy (but not lipschitzian!). Verification of this is left as an exercise. It can be done, for example, by showing that
IIH (c, f) - H (d,g)112 ::; III - gl12 + 4111 - gil + 4 lie - dll·
For any 0 < r < 1, a uniformly continuous retraction R : B -+ S can be defined by
{ P (y'2r-IIJJI+IfI) if 11/11 ::; r
RI= H (1~~~1I, Itrr) if 11/11 > r
Such retractions are not lipschitzian. However their moduli of continuity can be evaluated. This is probably the simplest way to show that in Hilbert space S is uniformly contractible to its positive part and consequently to a point, S is a uniform retract of B, and finally that there are uniformly continuous mappings T: B -+ B with positive minimal displacement (e.g. T = -R).
There have been attempts to use the above (and similar) constructions to obtain some evaluations for ko (H). First T. Komorowski and J. Wosko found the way to modified the above retractions to make them lipschitzian. This let them show that ko (H) ::; 64.25 .... Then K. Bolibok has used other homotopies to get ko (H) ::; 31.45 .... Still all the evaluations known are obtained via individual technical "tricks" and seem to be very imprecise.
6. Notes and remarks
The results, examples, comments and ideas presented above do not form a unified theory. They represent rather a collection of scattered questions, remarks, observations and partial solutions to problems connected with classical fixed point theorems. We believe that some of the questions presented here are intriguing and are waiting for new ideas to obtain final solutions.
As mentioned, there is already some literature devoted to the subject.
The most celebrated Brouwer's Fixed point Theorem was first published in 1912 in [9]. There are many proofs scattered in books on topology or mathematical analysis. The theorem is usually proved by means of either combinatorial arguments, homology theory, differential forms, degree theory or other methods of differential topology. The analytical proof which seems to be the most "metric" is that presented by J. Milnor [33]. There is also an interesting "elementary" proof of the Nonretmction Theorem by Y. Kannai [23]. Schauder's Theorem appeared in 1930 (see [35]). Very soon after that S. Ulam raised the question concerning the existence of a retraction of the unit ball onto its sphere in infinitely dimensional Hilbert space. This problem was placed in the famous collection of mathematical problems known as The Scottish Book (see D. Mauldin [32]). Spread over time, subsequent publications by S. Kakutani 1943 [24], V. Klee 1955 [26], B.Nowak 1979 [34], Y. Benyamini and Y. Sternfeld 1983 [2], P. K.
REFERENCES 609
Lin and Y. Sternfeld 1985 [30] gave the complete answer to the "qualitative" questions concerning the failure of Brouwer and Schauder's theorems in noncompact settings.
The "quantitative" part of the field, namely, the investigations of the minimal displacement problem and the optimal retraction problem was initiated by K. Goebel in 1973 [17]. The first examples of mappings with d (T) > 0 come from this paper. The first rough estimates of ko (X) are due to C. Franchetti, 1985, 1986 [11, 12].
Since then both problems have been intensively discussed and investigated in the mathematical analysis seminar run by K. Goebel at Maria Curie-Sklodowska University, Lublin, Poland. It resulted in a number of M.Sc. and Ph.D. theses containing many partial results, remarks and observations (see T. Komorowski [27], J. Wosko [39], K. Bolibok [3]). Some were published: K. Bolibok [4],[5],[6], K. Bolibok and K. Goebel [7], [8],K. Goebel [18],[19], K. Goebel and W. Kaczor [20], K. Goebel and T. Komorowski [22]'T. Komorowski and J.Wosko [28].
It is worth mentioning that the minimal displacement problem was also discussed for other classes of mappings than subclasses of C.
M. Furi and M. Martelli [13],[14],[15] came up with several results concerning k-setcontractions. These are mappings satisfying the Darbo (a-Lipschitz) condition
a (T (E)) ::; ka (E)
for all sets E on which T acts (a denotes the Kuratowski's measure of noncompactness). S. Reich [36],[37],[38] studied pseudolipschitzian mappings and k-set contractions which are not necessarily self-mappings but satisfy a so called inwardness (or weak inwardness) condition. W. A. Kirk [25] presented some facts concerning the class of Holder maps. J. Wosko in his thesis [39] discussed the minimal displacement problem for the whole class of continuous mappings C. For a given set K, d (C) turned out to be a number characterizing the "degree" of noncompactness of K.
The evaluation of minimal displacement is often an initial step to obtain fixed point results. Let us mention some papers which follow this line: J. Bryszewski, M. Serbinowski [10], J. Garcia-Falset, E. Llorens-Fuster, B. Sims [16], T. Kuczumow, S. Reich, A. Stachura [29], S. Massa, D. Roux [31].
There are not many papers concerning other optimality criterion for the retraction of B onto S, for example (the value ko (X)). An interesting observation has been presented by J. Wosko [39]; it is proved, via a concrete example, that in the space of continuous function C [0,1] , for any c > 0, there exists a retraction R : B --> S that is a (1 + c )-set-contraction.
Finally let us mention that special chapters on the subjects discussed can be found in the book by K. Goebel and W. A. Kirk [21J. Also some facts are discussed in the recent monograph by Y. Benyamini and J. Lindenstrauss [lJ.
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[33J l\~j'nor J. Analytic proof of the "Hairy Ball Theorem" and Brouwer Fixed Point Theorem. Amer. Math. Monthly 85 (1978), 521-524.
[34J Nowak B. On the Lipschitz retraction of the unit ball in infinite dimensional Banach space onto boundary. Bull. Acad. Polon. Sci. 27 (1979),861-864.
[35J Schauder J. Der Fixpuntsatz in Funktionalriiumen. Studia Math 2 (1930), 171-180.
[36J Reich S. Minimal displacement of points under weakly inward pseudo-lipschitzian mappings I. Atti Accad. Naz. Lincei 59 (1975), 40-44.
[37J Reich S. Minimal displacement of points under weakly inward pseudo-lipschitzian mappings II. Atti Accad. Naz. Lincei 60 (1976), 95-96.
[38J Reich S. A minimal displacement problem. Comment. Math. Univ. St. Pauli, 26 (1977), 131-135.
[39J Wosko J. Minimal displacement problem (Polish). Mariae Curie-Sklodowska University, Thesis (1995)
[40J Wosko J. An example related to the retraction problem. Ann. Univ. Mariae Curie-Sklodowska, Sect. A 45 (1991), 127-130.
Chapter 18
ORDER-THEORETIC ASPECTS OF METRIC FIXED POINT THEORY
J acek J achymski
Institute of Mathematics
Technical University of Ladz,
Zwirki 36, 90-924 LadZ, Poland.
jachymfl)ck-sg.p.lodz.pl
1. Introduction
This chapter is intended to present connections between two branches of fixed point theory: The first, using metric methods which is the main subject of this Handbook, and the second, involving partial ordering techniques. We shall concentrate here on the following problem: Given a space with a metric structure (e.g., uniform space, metric space or Banach space) and a mapping satisfying some geometric conditions, define a partial ordering (depending on a structure of a space and/or a mapping) so that one of fundamental ordering principles - the Knaster-Tarski Theorem, Zermelo's Theorem or the Tarski-Kantorovitch Theorem - can be applied to deduce the existence of a fixed point. We emphasize that all the above principles are independent of the Axiom of Choice (abbr., AC) so the above approach to metric fixed point theory is wholly constructive. It seems that such studies were initiated by H. Amann [5J and B. Fuchssteiner [33J in 1977. Subsequently, they were continued among others by S. Hayashi [37J, R. Manka ([59J, [60]), R. Lemmert and P. Volkmann [58J, A. Baranga [8], T. Biiber and W. A. Kirk [20] and J. Jachymski ([41], [42], [43], [44], [46], [47]). On the other hand, some authors have also studied a reciprocal of the above problem: Given a partially ordered set and a mapping on it, define a metric depending on this order so that some theorems of metric fixed point theory could be applied. That was done recently by Y.-Z. Chen [22J, who used Thompson's [81J metric generated by an order. However, in this chapter, we shall not discuss these problems.
The outline of this chapter is as follows. In Section 1 we study consequences of the Knaster-Tarski Theorem. We present here Amann's [5] proof of a fixed point theorem for diametric contractions, Jachymski's [42] proofs of the Angelov [6J theorem for jcontractive mappings on a uniform space, and the Nadler [64J theorem for set-valued contractions. Section 2 is devoted to the famous theorem of Zermelo. We give here a proof of some versions of Caristi's [21J theorem, another proof of Nadler's [64J theorem (via selection methods), and an extension of Zermelo's Theorem with applications. We also establish a common fixed point theorem for two mappings on a partially ordered set. This result yields the fundamental theorem for nonexpansive mappings due to
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W A. Kirk and B. Sims (eds.). Handbook of Metric Fixed Point Theory, 613--641. @ 2001 Kluwer Academic Publishers.
614
W. A. Kirk [49J. Section 3 deals with the Tarski-Kantorovitch Theorem. First we revisit here a proof of Angelov's [6] theorem. Next we give an ordering proof of the Hutchinson-Barnsley ([3S], [9]) theorem - a fundamental result in the theory of iterated function systems.
Finally, we wish to call attention to yet another important ordering principle (not discussed in this chapter), due to Brezis and Browder [14]. In particular, this principle yields Caristi's [21] fixed point theorem (see also Goebel and Kirk [35, pp. 13-14]), some results of the theory of nonlinear semigroups (also cf. 1. Ekeland [30]) and the famous Nash-Moser theorem (cf. W. O. Ray [71]). For further literature on this topic, see, e.g., M. Altman [4], M. Thrinici ([S4], [S5]) and references therein.
2. The Knaster-Tarski Theorem
A partially ordered set is a pair (P, :::S), where P is a nonempty set and j is a relation in P which is reflexive (p :::S P for all pEP), weakly antisymmetric (for p, q E P, p j q and q j pimply p = q) and transitive (for p,q,r E P, p j q and q j r imply p j r). A nonempty subset C of P is said to be a chain if given p, q E C, either p :::S q or q j p. If every chain in (P, j) has a supremum, then (P, j) is called chain-complete (see, e.g., Markowsky [61]). A mapping F : P -t P is said to be isotone or increasing if it preserves ordering, i.e., given p, q E P, p j q implies that Fp j Fq.
We start with recalling the following theorem which is one of fundamental results in fixed point theory. We shall use the abbreviation 'the K-T Theorem' for 'the KnasterTarski Theorem'.
Theorem 2.1 (Knaster-Tarski) Let (P, j) be a partially ordered set in which every chain has a supremum. Assume that F : P -t P is isotone and there is an element Po E P such that Po j Fpo. Then F has a fixed point.
This theorem was proved in 1927 by Knaster [51J for increasing - under set-inclusion -mappings, on and to the family of all subsets of a set. In 1939 Tarski extended Knaster's result to increasing mappings on a complete lattice and he gave its applications in set theory and topology, but his result was unpublished until 1955 (cf. Tarski [SO, footnote no. 2]). The version of the K-T Theorem presented here is due to Abian and Brown [2] and, independently, Pelczar [66], and was established in 1961. This is the reason why, in the literature, this result is also called the Abian-Erown-Pelczar theorem (see, e.g., Schroder [72]). Following Dugundji and Granas [27, p. 14J we incline, however, to the opinion that this theorem should be attributed to Knaster and Tarski, because Knaster initiated the development of fixed point theory on ordered structures, whereas Tarski not only extended Knaster's ideas, but he also found important applications of theorems of this type in other areas of mathematics. Finally, we note that the K-T Theorem is also called the Amann theorem (cf. Zeidler [S9, p. 506]) for which, however, there seems to be no real basis.
By substituting an inverse ordering::: for j in Theorem 2.1, we obtain the following dual version of the K-T Theorem.
Theorem 2.2 Let (P,:::s) be a partially ordered set in which every chain has an infimum. Assume that F : P -t P is isotone and there is an element Po E P such that Fpo :::S Po. Then F has a fixed point.
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In the sequel we shall show how Theorems 2.1 and 2.2 can be used to derive some results of metric fixed point theory. We start with Amann's [5] proof (see also Zeidler [89, p. 512]) of a fixed point theorem for the so-called diametric contractions. In fact, we shall extend his argument by considering a more general dass of mappings: A selfmap I of a bounded metric space is said to be a diametric <p-contraction if there is a nondecreasing function <p : lR+ -+ lR+ (lR+ denotes the set of all nonnegative reals) such that limn -+DO <pn(t) = 0 for all t > 0 (see Matkowski [62] or Dugundji and Granas [27, p. 12]) and
diam/(A) :=; <p(diamA) (2.1)
for all nonempty, dosed and I-invariant subsets A of X.
A more realistic special case here is the following Walter's contraction [86], i.e., a mapping I which satisfies the inequality
d(fx,ly):=;<p(diam(Of(x)UOf(Y))) for all x,yEX,
where Of (x) := Un-1x : n E N} is an orbit of I at a point x and the function <p is nondecreasing, right continuous and <pet) < t for t > 0 (then limn -+DO <pn(t) = 0; cf. Browder [16]). That such an I satisfies (2.1), follows immediately from the fact that x E A implies Of (x) ~ A if A is I-invariant.
Theorem 2.3 Let (X, d) be a complete bounded metric space and I be a diametric <p-contraction on X. Then I has a unique fixed point.
Before proving Theorem 2.3 we establish some new related results in a more general setting. Let (X, T) be a topological space and I be an arbitrary (not necessarily continuous) selfmap of X. Let cl denote the dosure operator. Define
Cf(X) := {A ~ X: A =f 0, A = dA and I(A) ~ A},
and for A,B E Cf(X),
A j. B if A ~ d/(B).
Proposition 2.4 The relation j. is weakly antisymmetric and transitive in Cf(X).
Proof. If A j. B, then A ~ d/(B) ~ dB = B so A ~ B. If, moreover, B j. A, then B ~ A. Consequently, we get that A = B.
Now assume that A j. Band B j. C. Then A ~ Band B ~ d/(C) which yields A ~ d/CC), i.e., A j. C. •
For A, B E C/(X), define
AjB if A=B or A~d/(B).
Then Proposition 2.4 lets us infer that j is a partial ordering in C/(X). This is called Amann's ordering (cf. Zeidler [89, p. 512]). Following Amann we define the mapping Fby
F(A) := cl/(A) for A E C/(X).
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Proposition 2.5 F is an isotone selfmap of Cf(X) with respect to Amann's ordering :S. Moreover, F(A) :S A for all A E Cf(X).
Proof. To show that F(A) E Cf(X) for A E Cf(X), it suffices to prove that F(A) is f-invariant. Since elf(A) S;; elA = A, we get f(F(A)) S;; itA) S;; F(A). To prove the monotonicity of F we may assume, without loss of generality, that A :S. B. Then AS;; elf(B) so f(A) S;; f(elf(B)) = f(F(B)) and hence clf(A) S;; elf(F(B)), i.e., F(A) :S. F(B).
Finally, the trivial inclusion el f(A) S;; el f(A) means that F(A) :S. Aj in particular, F(A):s A. •
Proposition 2.6 Let C be a chain in (Cf(X), :s). Then C has an infimum if and only if the set Ao := nGEC cl f (C) is nonempty. Moreover, if C has no minimum and Ao of 0, then inf C = Ao.
Proof. We start with the 'only if' part. Let Co := infC. If C E C and Co of C, then Co :S. C, i.e., Co S;; clf(C). Hence if Co 'I. C, then Co S;; Ao. If Co E C, then we may infer that Ao = clf(Co) since Co S;; C for C E C and thus clf(Co) S;; elf(C) for such C. In both cases Ao is nonempty.
To prove the 'if' part, assume that Ao of 0. Then Ao E Cf(X) since it is elosed and
f(Ao) S;; n f(elf(C)) S;; n ftC) S;; Ao· GEC GEC
By the definition of Ao, Ao :S. C for all C E C, i.e., Ao is a lower bound of C. Suppose, on the contrary, that Ao is not the greatest lower bound of C. Then there is B E Cf(X) such that B of Ao, Ao :S. Band B :S C for all C E C. Without loss of generality we may assume that C has no minimum. Then B of C so B :S. C for all C E C. Therefore,
Ao S;; elf(B) S;; Band B S;; Ao for all C E C
which yields Ao = B, a contradiction. Thus Ao = infC if C has no minimum. In each case C has an infimum. •
The following example shows that we cannot drop the assumption 'C has no minimum' in the last statement of Proposition 2.6.
Example 2.7 Set X := [0,1]' fx := x/2 for x E X, and endow X with the Euelidean metric. Define C := {X,f(X)}. Then C is a chain in Cf(X) and f(X) = minC. However, Ao = f2(X) of minCo
As an immediate consequence of Propositions 2.4, 2.5 and 2.6, and Theorem 2.2, we obtain the following.
Theorem 2.8 Let (X, T) be a topological space and f be a selfmap of X (not necessarily continuous). If for every chain C in Cf(X) endowed with Amann's ordering,
n elf(C) of 0, (2.2) GEC
then there exists a set A. E Cf(X) such that clf(A.) = A •.
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Proof of Theorem 2.3. We shall apply Theorem 2.2. Let C be a chain in (Cf(X), ::::;). We show that (2.2) holds. If C has a minimum, then by Proposition 2.6, we are done. So assume C has no minimum. Since for A,B E Cf(X), A::::; B implies that A ~ B, C is also a chain with respect to set-inclusion. Therefore, given C1 , • " , Cn E C we have
so the family {clf(C) : C E C} has the finite intersection property. We show that given c > 0 there is a set C E C with diamf(C) < c and then the Kuratowski theorem (see, e.g., Engelking [31, Theorem 4.3.10]), let us infer that (2.2) holds. By hypothesis, given c > 0, there is apE N such that rpP(diamX) < c. Without loss of generality we may assume that p 2: 2. Since C has no minimum, there exist sets C1,'" ,Cp E C such that Cp ::::; •••. ::::;. C1, i.e., Ci ~ clf(Ci-1) for i = 2,··· ,p. Hence and by (2.1) we get
diamCp ::; diamf(Cp_l) ::; cp(diamCp_l) ::; ... ::; cpP-l(diamCl)'
which yields diamf(Cp) ::; rpP(diamX) < c. By Theorem 2.8, there is an A. E Cf(X) for which clf(A.) = A •. By (2.1)
diamf(A.)::; cp(diamA.) = cp(diamf(A.))
which implies that f(A.) is a singleton, since cp(t) < t for t > 0 (cf. [27, p. 12]). Therefore A. = {a.} for some a. E X, and thus a. = fa. since A. is f-invariant. To prove the uniqueness assume that al, a2 are fixed points of f. Then the set A :=
{al,a2} belongs to Cf(X) and f(A) = A. Hence and by (2.1), we may conclude that diam A = 0, i.e., al = a2. •
Remark 2.9 The above proof depends on the countable form of the AC since we used the Kuratowski theorem.
Remark 2.10 We emphasize that in the above proof a contractive condition is used only to show that every chain has an infimum. (Compare with Remarks 2.19 and 3.6.)
The above proof also shows that the K-T theorem implies the Banach Contraction Principle for mappings on a bounded metric space. In the sequel we shall give another proof using a completely different partial ordering introduced by Jachymski [42]. The advantage of this proof is that the boundedness condition can be dropped here and, moreover, it shows the K-T theorem also yields some extensions of the Contraction Principle for mappings on uniform spaces, given by K.-K. Tan [78], Tarafdar [79], D. H. Tan [77] and Angelov [6].
Let (X, 'P) be a complete Hausdorff uniform space with the uniformity generated by a family 'P of pseudometrics, 'P = {p" : a E A}, where A is an index set (see, e.g., Engelking [31]). Let lRt denote the family of all nonnegative real functions on the set A. We define the following relllJion in the Cartesian product X x lRt:
(x,cp)::::;(y,'IjJ) iff p,,(x,y)::;cp(a)-'ljJ(a) for all aEA, (2.3)
where x, y E X and cp, 'IjJ E lRt. Clearly this relation is reflexive. Since the family 'P is separated because of the Hausdorff condition, the above relation is antisymmetric. It is also transitive, which is a simple consequence of the triangle inequality. Thus
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(X x lR~,::s) is a partially ordered set. Let us notice that if A is a singleton {d}, i.e., (X, d) is a metric space, then the above definition takes the form
(x,a) ::S (y, b) iff d(x,y) .:::: a - b, (2.4)
where a, bE lR+. This ordering was introduced by Ekeland [29J in the proof of his variational principle. On the other hand, recently, Frigon [32J slightly modified definition (2.4) and she used such an ordering to prove that the property of having a fixed point is invariant by homotopy for set-valued contractions.
The completeness of (X, P) is crucial in a proof of the following.
Proposition 2.11 Every chain C in (X x lR~,::s) has a supremum.
Proof. We use a similar idea as in Kirk [50J. If u E C, then u = (xO",'PO") for some XO" E X and 'PO" E lR~. Consider the net {(xO"' 'PO" )}O"EC, Obviously this net is increasing, i.e.,
if Ul ::S U2, then p",(XO"pX0"2) .:::: 'P0"1 (a) - 'P0"2(a) for all a E A. (2.5)
Therefore the net {'PO"(a)}O"EC is decreasing in lR+, hence convergent. Denote
'Po(a) := lim 'PO" (a) for all a E A. O"EC
By (2.5), since for each a E A {'PO"(a)}O"Ec is Cauchy, so is the net {XO"}O"EC, By completeness, there is an Xo E X such that Xo = limO"Ec XO"' Taking the limit in (2.5) with respect to U2 yields
p",(xO"'xo) .:::: 'PO"(a) - 'Po(a) for a E A and u E C
which means the pair (xo, 'Po) is an upper bound of C. If a pair (x, 'P) is another upper bound, i.e., for all a E A and U E C, p",(xO"'x) .:::: 'PO"(a) - 'P(a), then, taking the limit with respect to u, we obtain that p",(xo, x) .:::: 'Po(a) - 'P(a) for all a E A, i.e., (xo, 'Po) ::S (x, 'P). That means (xo, 'Po) is the least upper bound of C. •
Now let j be a selfmap of a uniform space (X, P) and j be a selfmap of the index set A. Following D. H. Tan [77J and Angelov [6J j is said to be a j-Lipschitzian mapping if
p",(fx,jy)':::: h",pj(",)(x,y) for all x,y E X and a E A,
where h", 2: O. If, moreover, h", < 1 for all a E A, then we say that j is j-contmctive. The introduction of the mapping j is motivated by applications in the theory of neutral functional differential equations (d. Angelov [6], [7J and Tsachev and Angelov [82]). Define
F(x, 'P) := (fx, h· ('P 0 j)) for all (x, 'P) E X x lR~, (2.6)
where (h· ('P 0 j))( a) := h",'P(j (a)) for a E A. Clearly F is a selfmap of X x lR~.
Proposition 2.12 Let j be a j-Lipschitzian selfmap of (X, P). Let F be an associated map on X x lR~ defined by (2.6). Then F is isotone with respect to the ordering :< defined by (2.3).
Proof. Let (x, 'P) :< (y, 'l/J). Then for all a E A, we have
p",(jx,jy) :s: h",Pj(",)(x,y):s: h",'P(j(a)) - h",'l/J(j(a)),
Order-theoretic aspects 619
which means that F(x, cp) :S F(y, 'l/J). • We show that the K-T Theorem yields the following result obtained independently by K.-K. Tan [78) and Tarafdar [79). In particular, the Banach Contraction Principle is obtainable via the K-T Theorem.
Theorem 2.13 (Tan, Tarafdar) Let I be a j-contractive selfmap 01 (X, P) with j :=id, the identity mapping on A. Then I has a fixed point.
Proof. We consider the ordering :S in X x lR~ and the mapping F, defined by (2.3) and (2.6), respectively. We show that there exists a pair (xo, cpo) such that (xo, cpo) :S F(xo, cpo). Let Xo E X be chosen arbitrarily. It suffices to set
() Pa(xo,lxo) CPo a := h for all a E A.
1- a
By Propositions 2.11 and 2.12, the rest of the assumptions of the K-T Theorem are also satisfied. •
We emphasize that the above proof does not depend on the AC. It does not use any iterative methods either.
Now we consider the following extension of the above theorem, due to Angelov [6) (see also D. H. Tan [77)).
Theorem 2.14 (Angelov) Let I be a j-contractive selfmap of (X, P) such that
sup{hjn-l(a) : n E N} < 1 lor all a E A.
Assume that there is an Xo E X such that
sup {Pjn-l(a)(XO, Ixo) : n E N} :-:; Ca lor all a E A,
where Ca 2 O. Then I has a fixed point.
To derive this result from the K-T Theorem applied to the mapping F on (X x lR~, :S), it is enough to show in view of Propositions 2.11 and 2.12 that there exists a function CPo E lR~ such that (xo, cpo) :S F(xo, cpo). This problem leads to the following functional inequality
cp(a) 2 g(a) + h(a)cp(j(a», (2.7)
where g(a) := Pa(XO, Ixo). This inequality can be considered on any j-invariant subsets of A, in particular, on orbits of the mapping j and we start with just such a case.
Lemma 2.15 Under the assumptions 01 Theorem 2.14, given f3 E A, inequality (2.7) has a solution CPf3 : OJ(f3) -+ IR+.
Proof. Let f3 E A and hf3 := sup{ha : a E OJ (f3)}. Then hf3 < 1. We shall use a similar argument as in Kuczma, Choczewski and Ger [54, p. 149). Let 'Pf3 denote the family of all functions cP : OJ(f3) ..... lR+ which are bounded. Equip 'Pf3 with the sup norm so that it becomes a complete metric space, and define an operator Tf3 by
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Since gIOj(j3) is bounded, Tj3 is a selfmap of <I>j3. Moreover, given 'PI, 'P2 E <I>j3, we easily get that
Now we may apply the Contraction Principle since it is subsumed by the K-T Theorem as shown earlier. Thus Tf3 has a fixed point 'Pf3 and then 'Pf3 is also the desired solution
of (2.7) since hf3 2: ha for 0< E OJ((3). •
To show that (2.7) has a global solution, we shall apply a Hahn-Banach type argument. Instead of functionals on linear subspaces, we shall consider solutions of (2.7) on jinvariant subsets of A. Then orbits of the mapping j (i.e., minimal, with respect to the set-inclusion, j-invariant subsets of A) correspond to one-dimensional subspaces (i.e., minimal, with respect to the set-inclusion, nontrivial linear subspaces). Thus a counterpart of Lemma 2.15 is a result saying that there exists a nonvanishing continuous linear functional on each one-dimensional subspace. Our proof depends on the AC because Zorn's Lemma is used. We slightly modify a formulation of Jachymski [42, Lemma 3] and we simplify the last part of its proof.
Proposition 2.16 Let j be a selfmap of an abstract set A, 9 : A -> lR+ and h : A ->
lR+. Let Ao be a j-invariant subset of A and 'Po : Ao -> lR+ be a solution of (2.7). If for each (3 E A, the functional inequality (2.7) has a solution 'Pf3 : OJ((3) -> lR+, then there exists an extension <Po : A -> lR+ of 'Po, satisfying (2.7) on A.
Proof. Consider the following family of solutions of (2.7):
<I> := {'P : D<p -> lR+ I Ao ~ D<p ~A, j(D<p) ~ D<p, 'PIAn = 'PO
and 'P satisfies (2.7) on D<p}.
We equip <I> with the partial ordering j:
'PI j 'P2 iff D<pJ ~ JJtp2 and 'P2ID"'J = 'Pl·
If C is a chain in (<I>, j), then there is a unique function '1f1 : D", -> lR+ such that D", = U<pEC D<p and '1f1ID", = 'P for all 'P E C. By Zorn's Lemma, there exists a maximal element 'P. : D. -> lR+ in <I>. It suffices to prove that D. = A. Suppose, on the contrary, that there is an 0<0 E A \ D •. We show that OJ (0<0) n D. =1= 0. Suppose that these sets are disjoint. By hypothesis, there exists a solution 'Pan: OJ(O<O) -> lR+ of (2.7). Clearly the set OJ (0<0) U D. is j-invariant and the function 'Pao U 'P. is well-defined, it belongs to <I>, and it extends 'P. which yields a contradiction. Thus we may define an integer
m:= min{n EN: jn(o<o) ED.}.
Then jm-l(o<o) ¢ D •. Set D := D. U Um-l(o<o)}. Then j(D) ~ D. C D. We define the function 'P : D -> lR+ as follows:
It is easily seen that 'P E <I> and 'P. j 'P =1= 'P., a contradiction. Therefore we may infer that 'P. is a global solution of (2.7) and it suffices to set <Po := 'P.. •
Remark 2.17 Proposition 2.16 extends [42, Lemma 3] in which the function h is assumed to take values from [0,1].
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Corollary 2.18 Under the assumptions of Theorem 2.14, the functional inequality (2.7), with g(o) := Pa(XO, fxo) and h(o) := ha, has a solution on A. Hence the K-T Theorem implies Angelov's theorem.
Proof. Let 00 E A. By Lemma 2.15, there is a solution <PaD: OJ(OO) -> IR+ of (2.7). By Lemma 2.15 and Proposition 2.16, <paD can be extended to a global solution <P;"D of (2.7) on A. Then (xo, <P;"D) ~ F(xo, <P;"D) and by Propositions 2.11 and 2.12, all the assumptions of the K-T Theorem are satisfied. Thus there exists a fixed point (x., <P.) of F and then x. = fx.. •
Remark 2.19 We emphasize that in the above proof the contractive condition is used only to show the existence of an element b E X x lR1 such that b ~ Fb (compare with Remarks 2.10 and 3.6).
Finally, we want to discuss relations between theorems of Angelov and Tan-Tarafdar. In particular, Czeslaw Bessaga [11] posed the question whether the latter theorem yields Angelov's result. He suggested that given a j-contractive (with respect to P) mapping f, it could be possible to find another family pI of pseudometrics such that f would satisfy the assumptions of the Tan-Tarafdar theorem on the uniform space (X,PI). However, the following example settles in the negative the above question even if the space (X, P) is metrizable.
Example 2.20 Let X := C(IR'j.), the set of all real continuous functions on the open interval IR'j. := (0,00). Let K(IR'j.) denote the family of all nonempty compact subsets of IR'j.. Endow X with the uniform structure generated by the family of pseudonorms {PA : A E K(IR'j.)}, where PA(<p) := maXtEA 1<p(t)1 for <P EX. Let a,b be positive reals such that 0 < a < 1 < b. Define an operator F by
(F<p)(t) := a<p(bt) for <p E X and t> O.
Then F is a linear selfmap of X and PA(F<p) = apj(A)(<P), where j(A) := bA. Thus F is a j-contractive mapping with hA := a. It can be verified that the condition SUp{pjn(A) (<po - F<po) : n E N} < 00 is satisfied if, e.g., <po is a nonnegative and nonincreasing function. Thus the assumptions of Angelov's theorem are fulfilled. Suppose, on the contrary, that there is a family pi of pseudometrics inducing some Hausdorff uniform structure on X and such that F satisfies the assumptions of the Tan-Tarafdar theorem on (X, PI). Then F has a unique fixed point. On the other hand, it is easily seen that the set of all fixed points of F coincides with the family of all solutions of the Schroder functional equation
<p(t/b) = a<p(t) for t> O.
The latter family, however, is not a singleton according to, e.g., the Kordylewski and Kuczma theorem (cf. [53, Theorem 2.1]). This yields a contradiction.
In the sequel we shall derive Nadler's [64] extension of the Contraction Principle from a set-valued version of the K-T Theorem, due to Smithson [73]. A set-valued mapping F on a partially ordered set (P,~) (that is, F : P -> 2P \ {0}) is said to be order preserving if given PI,P2 E P with PI ~ P2, for each rl E FPI, there is an r2 E FP2 such that rl ~ r2.
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Theorem 2.21 (Smithson) Let (P,~) be a partially ordered set in which every chain has a supremum. Let F be an order preserving set-valued mapping on P such that given a chain C <;;; P and an isotone single-valued mapping w : C --> P satisfying w(c) E Fc for all c E C, there exists d E F(supC) such that w(c) ~ d for all c E C. If there exist elements Po E P and TO E Fpo such that Po ~ ro, then F has a fixed point.
It is easily seen that if F is an isotone single-valued selfmap of a chain-complete (P, ~ ) with Po ~ Fpo for some Po E P, then F satisfies the assumptions of the above theorem. We also emphasize that whereas the K-T Theorem does not depend on the AC, Smithson's [73] proof relies on the AC.
Following Jachymski [42], we shall give a new proof of Nadler's [64] theorem using partial ordering techniques. Let (X, d) be a metric space and Cl(X) denote the family of all nonempty closed subsets of X (not necessarily bounded). For A, B E Cl(X), set
H(A, B) := max { sup{d(a,B) : a E A}, sup{d(b, A) : b E Bn,
where d(a,B) := inf{d(a,b) : b E B}. Then H is said to be a generalized Hausdorff metric since it may have infinite values. We recall a more general form of Nadler's theorem established by Covitz and Nadler [23].
Theorem 2.22 (Nadler) Let (X, d) be a complete metric space and f : X --> Cl(X). Assume there is an h E [0,1) such that
H(jx,fY) :::; hd(x,y) for all X,Y E X.
Then f has a fixed point.
Proof. Let P := X x lR+ and endow P with Ekeland's ordering (2.4). By Proposition 2.11, every chain in P has a supremum. Define a set-valued mapping F on P by
F(x,a) := fx x {hal for (x,a) E P.
We shall show that F satisfies the assumptions of Smithson's theorem. Without loss of generality we may assume, eventually replacing h by (1 + h)j2, that
H(jx, fy) < d(x,y) for all X,y E X with x'" y.
We show that F is order preserving. Let (Xl, all ~ (X2, az) and (Yl, bI) E F(Xl, all, i.e., Yl E fXl and bl = hal. Let bz := ha2. If Xl = Xz, then set Y2 := Yl· If Xl'" xz, then
d(Yl' fxz) :::; H(jXl, fX2) < hd(Xl' X2) :::; heal - az).
Hence there exists Yz E fxz such that d(m, YZ) :::; heal - az). In both cases
Now let C be a chain in P and w : C --> P an isotone mapping such that w(o-) E Fofor (Y E C. Then 0- = (xcr, acr) and the net {( Xcr , acr)} crEC is increasing. Hence following the proof of Proposition 2.11, we may infer by completeness of (X,d) that both nets {Xcr}crEC and {acr}crEc are convergent, say to Xo E X and ao E lR+, respectively. Moreover, (xo, ao) is the supremum of C. Denote (Ycr, bcr) := w(xcr,aa). Then
ba = haa and Ycr E fx".
Order- theoretic aspects 623
Since W is isotone and the net {(x"' au )}"EC is increasing, so is the net {(y", b" )}"EC, Again we may infer that the net {yu }uEC converges to some Yo E X and
(Ya,ba)::5 (YO, hao) , that is, w(xa,aa)::5 (YO,hao).
We only need to show that (YO, hao) E F(xo, ao), i.e., Yo E fxo. For (T E C, we have
d(yo, fxo) :s; d(yo, y,,) + d(y", fxo) :s; d(YO,Ya) + H(fx", fxo)
:s; d(YO,Ya) + hd(xo,x,,).
Hence taking the limit with respect (T, we obtain d(yo'/xo) = O. Since the set fxo is closed, we infer that Yo E fxo.
Finally, we show that there exist (xo,ao) E P and (Yo, bo) E F(xo,ao) such that (xo, ao) ::5 (Yo, bolo Choose arbitrarily Xo E X and Yo E fxo. It suffices to set
d(xo,yo) 0,0:= 1 _ hand bo := hao·
Thus by Smithson's theorem, there exists (x., a.) E P such that (x., a.) E F(x., a.), and then x. is a fixed point of f. •
3. Zermelo's fixed point theorem
The following fixed point theorem is of great importance.
Theorem 3.1 (Zermelo) Let (P,::5) be a partially ordered set in which every chain has a supremum. Assume that F : P -> P is such that
p ::5 Fp for all pEP. (3.1)
Then F has a fixed point.
A mapping F satisfying (3.1) is said to be progressive.
The above theorem is attributed to Zermelo (see, e.g., Dunford and Schwartz [28, p. 5]), although it does not appear explicitly in any of his papers. However, a proof of it can be derived from Zermelo's [90J, [91J proofs of the well-ordering principle. This observation is due to Bourbaki [12], who was the first to formulate the theorem in the above form. (Actually, Bourbaki used well-ordered subsets of P instead of chains so his assumption is formally weaker than that of Theorem 3.1. However, it is more convenient for us to work with chains as will be seen in the sequel; in particular, ef. Remark 3.16.) The proof of Zermelo's Theorem does not depend on the AC. If, however, we allow the use of Zorn's Lemma, then the proof is straightforward; moreover, the assumption on (P,::5) can be weakened then to 'every chain has an upper bound'. This is Kneser's [52J fixed point theorem which turns out to be equivalent to the AC as shown by Abian [1]. In the literature, ZermeJo's Theorem is sometimes called the Bourbaki-Kneser theorem (ef. Zeidler [89, p. 504]).
The fact that Zermelo's Theorem is independent of the AC is important for a constructive approach to metric fixed point theory. It seems that the basic result in this direction is due to Fuchssteiner [33]. He showed that the fundamental fixed point theorem for nonexpansive mappings can be derived from Zermelo's Theorem, though Kirk's [49] original proof of it uses Zorn's Lemma. Fuchssteiner also observed that Zermelo's Theorem yields the Banach Contraction Principle and some extensions of it for mappings on a bounded metric space.
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Our first purpose is to present a result of Jachymski [44] which improves the latter result of Fuchssteiner by omitting the boundedness condition of the space considered. Moreover, it also turns out that a version of Caristi's [21] theorem is subsumed directly by Zermelo's theorem.
Let (X,d) be a metric space and <p be a real function on X. We consider the following relation -:::'<p introduced by Brlimdsted [18]: given X,Y E X,
x -:::'<p y iff d(x,y)::; <p(x) - <p(y).
Proposition 3.2 The relation -:::'<p is a partial ordering in X. If <p is bounded from below and (X,d) is complete, then every chain C is convergent in (X,d); moreover,
(a) lim C is an upper bound of C in (X, -:::'<p) if <p IS lower semicontinuous;
(b) lim C is a supremum of C if <p is continuous.
Proof. That -:::'<p is a partial ordering is straightforward; in particular, the transitivity of -:::'<p follows from the triangle inequality. Let C be a chain. Consider a net {xu hrEC,
where Xu := a. Obviously this net is increasing, i.e.,
(3.2)
Hence <p(XU2 ) ::; <p(XU1 ), i.e., the net {<p(xu )}uEC is nonincreasing and thus it is convergent if <p is bounded below. Hence and by (3.2), we may infer that {XU}UEC satisfies Cauchy's condition so it converges to some Xo EX.
To prove statement (a) it suffices to take the limit with respect to a2 in (3.2). Then we obtain
d(xupxo)::; <p(xuJ - lim <p(XU2 ) U2EC
::; <p(XU1 ) - lim inf <p(x) ::; <p(XU1 ) - <p(xo). X---Xo
Hence Xo is an upper bound of C. Finally, assume that <p is continuous. Let Xo be an upper bound of C, i.e.,
d(xu, xo) ::; <p(xu) - <p(xo) for all a E C.
Taking here the limit with a E C yields that Xo -:::'<p Xo which means Xo is the least upper bound of C. •
One could conjecture that, under the assumptions of Proposition 3.2 (a), if a chain C has a supremum, then it coincides with lim C. The following example shows that this is not the case.
Example 3.3 Let X := [0,1] be endowed with the Euclidean metric. Define a function <p on X by
<p(o) := 0, <p(I):= 1 and <p(x) := X + 2 for X E (0,1).
Then <p is nonnegative and lower semicontinuous. Since <p(x) 2: x for all x E X and <p(o) = 0, ° is the greatest element in (X, -:::'<p). It is easily seen that the sequence (I/(n + 1))~=1 is :5<p-increasing. Hence C := {l/(n + 1) : n E N} is a chain. Clearly
Order-theoretic aspects 625
lim G = O. We show that the only upper bounds of G are 0 and 1. Let Xo E (0,1). Then
cp(I/(n + 1» - cp(xo) = 1/(n + 1) - Xo < 0
for sufficiently large n so Xo is not an upper bound of G. Moreover, 1/(n+ 1) ~'P 1 ~'P 0 for all n E N. Therefore we may infer that supG = 1 # limG.
Theorem 3.4 (Caristi) Let (X,d) be a complete metric space, cp : X ---> lR+ be a lower semicontinuous function and f be a selfmap of X (not necessarily continuous) such that
d(x,fx) :::; cp(x) - cp(fx) for all x EX. (3.3)
Then f has a fixed point.
Caristi's [21] original proof of this result is sophisticated: It is based on an iterative use of a transfinite induction. An elegant proof involving Brondsted's ordering ~'P was given by Kirk [50]. Actually, part (a) of Proposition 3.2 is due to him. Then he applied Zorn's Lemma to get a maximal element x. which is a fixed point of j since, by (3.3), x. ~'P fx •. Another short proof was given by Pasicki [65] based on the Hausdorff maximal chain theorem. A constructive proof of Caristi's theorem was the subject of a paper of Browder [17], but, in fact, he used the countable form of the AC as observed by Bell [10]. Another constructive approach was due to Penot [67], [69], who proved the maximality principle (i.e., a theorem on the existence of maximal elements in (X, ~'P)) with a help of the Cantor intersection theorem. Thus the question arises whether the maximality principle can be proved without using any weaker forms of the AC. This problem has been settled in the negative by Brunner [19], who showed that this principle is equivalent to the Axiom of Dependent Choices (DC). (Incidentally, the maximality principle is equivalent to Ekeland's [29] principle.) Despite this fact, it turns out that Caristi's theorem can be proved wholly without any choice as shown by Manka [59], [60]. On the other hand, we have the following
Proposition 3.5 Zermelo's Theorem implies the restriction of Caristi's theorem to continuous functions cpo More precisely, ij f is a selfmap of a complete metric space (X, d) such that (3.3) holds with a continuous function cp : X ---> lR+, then (X, ~'P) and f satisfy the assumptions of Zermelo'8 Theorem. In particular, Zermelo's Theorem yields the Banach Contraction Principle.
Proof. Clearly (3.3) means that j is progressive on (X, ~'P). By part (b) of Proposition 3.2, every chain in (X, ~'P) has a supremum. To prove the last statement assume that f is a Banach contraction with a constant hE (0,1). Then, given x E X,
d(x, fx) = d(x, fx)/(I- h) - hd(x,jx)/(I- h)
:::; d(x,fx)/(I- h) - d(fx,j2x)/(I- h) = cp(x) - cp(fx),
where cp(x) := d(x, fx)/(1 - h). Clearly cp is continuous and thus j satisfies the assumptions of the above restriction of Caristi's theorem. •
Remark 3.6 We emphasize that in the above proof the contractive condition is used to show that a mapping j is progressive (compare with Remarks 2.10 and 2.19).
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Proposition 3.7 Under the A C, Caristi's theorem is equivalent to its restriction to Lipschitzian functions <po
Proof. The above restriction of Caristi's theorem implies the so-called Drop Theorem of Danes [24]. The proof of this fact is outlined in Dugundji and Granas [27, p. 20] using a function <p of a form <p(x) := Gllxll which is Lipschitzian. Further, the Drop Theorem implies the Ekeland [29] Principle as shown by Danes [25] and, independently, Penot [69]. The latter theorem yields Caristi's result (see, e.g., Kirk [50]). •
Remark 3.8 The above proof indeed uses the AC (at least, the DC is necessary here). This follows from the fact that we have shown the implication
Caristi's theorem for continuous <p =? Ekeland's principle,
whereas the latter theorem implies the DC as shown by Brunner [19], and the former theorem does not depend on any form of the AC.
Question 3.9 Is it possible to prove Proposition 3.7 constructively?
Our next purpose is to present another Jachymski's [44] proof of Nadler's [64] theorem, this time, via Zermelo's Theorem. This proof, however, relies on the AC. It uses a similar idea as Takahashi [76].
Theorem 3.10 Let F be Nadler's set-valued contraction on a complete metric space (X, d) such that Fx E Cl(X) for all x EX. Then there exists a Lipschitzian function <p : X -+ lR+ such that F admits a selection F : X -+ X which is progressive on (X, ~<p). Hence Zermelo's Theorem yields Nadler's theorem.
Proof. Choose a real h' such that h < h' < 1. Then, given x EX, the set
Ax := {y E Fx: h'd(x,y) S d(x,Fx)}
is nonempty. By the AC, there is a mapping F : X -+ X such that Fx E Ax. Clearly F is a selection of F. By a contractive condition, we get
d(Fx,F(Fx)) S H(Fx,F(Fx)) S hd(x,Fx)
which implies that
d( F) = h'd(x,Fx) - hd(x,Fx) < d(x,Fx) - d(Fx,F(Fx)) x, x h' _ h - h' - h .
Therefore F is progressive on (X, ~<p) with <p(x) := d(x, Fx)/(h' - h). Moreover, <p is Lipschitzian since given x, y EX,
1 ( ) _ ()I d(x,y) +H(Fx,Fy) < h+ 1 d( ) <p x <p y S h' _ h - h' _ h x,y.
By Zermelo's Theorem, F has a fixed point x., and thus x. E Fx •. • On the other hand, Nadler's original proof [64] also depends on some form of the AC. In particular, it seems that, in general, the DC is necessary to define an iterative sequence (xn ) satisfying appropriate inequalities and such that Xn+l E Fxn . Thus the following question arises.
Order-theoretic aspects 627
Question 3.11 Is it possible to prove Nadler's theorem constructively?
If the answer is negative, then another problem is worth studying.
Question 3.12 What form of the AC is necessary to prove Nadler's theorem?
Finally, we emphasize that it is not possible to prove Nadler's theorem via the Banach Contraction Principle and a selection method since there exist set-valued contractions having no continuous selections (see, e.g., Jachymski [44, Example 1]).
In the sequel we want to present a constructive proof of Caristi's theorem given by Jachymski [41]. However, as we have already mentioned, Manka was the first to give a wholly constructive proof. He established a generalization of Zermelo's Theorem (cf. [60, Proposition 2]) involving a notion of a sup-function. His argument is a development of an idea of Zermelo's first proof [90]. Our purpose here is to establish the following variant of Zermelo's Theorem which, in fact, is a weaker result than Manka's [60, Proposition 2], but it can be proved in a simpler way and it is strong enough to yield Caristi's theorem. Another extension of Zermelo's Theorem was given recently by Lagler and Volkmann [55].
Given a partially ordered set (P, ~), the family of all chains in P is denoted by C(P).
Theorem 3.13 Let (P,~) be a partially ordered set and f be a progressive selfmap of P. Assume that there exists a function r : C(P) -> P such that for each C E C(P), r( C) is an upper bound of C. Then f has a fixed point.
The idea of considering functions assigning to some subsets of a partially ordered set an upper or lower bound of them goes back to Milgram [63]. We shall derive Theorem 3.13 from the following special form of Zermelo's Theorem. The family of all subsets of a set P is denoted by 2P .
Theorem 3.14 Let P be an abstract set, P be a nonempty subset of 2P , and F be a selfmap ofP. Endow P with the set-inclusion. If for every chain C in (P, ~),
UCEP, (3.4) GEC
and F is progressive on (P, ~), then F has a fixed point.
Condition (3.4) is sufficient for the existence of supC which coincides then with UGEC C. On the other hand, (3.4) is not necessary for the existence of supC as shown in the following.
Example 3.15 Let X := [0,1) be endowed with the Euclidean topology and P .Cl(X). Consider a countable chain C := {[lln,I) : n E N} in (P, ~). Then
U = (0,1] liP, GEC
but supC (= [0,1]) exists.
Proof of Theorem 3.13. We develop here the idea of Manka's proof [59] of Turinici's [83) fixed point theorem. We shall apply Theorem 3.14 with P := C(P) and F defined by
F(C) := C u {J(r(C))} for all C E P. (3.5)
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We show that F is a selfmap of P, i.e., F(C) is a chain in (P, c::). Since by hypothesis, for all x E C, x :::S r(C) and r(C) :::S 1(r(C)), we infer that 1(r(C)) is also an upper bound of C. Thus F(C) is a chain. Clearly F is progressive on (P, c::). Moreover, if C is a chain in (C(P), c::), then it is easily seen that the union UCEC C is a chain in (P, :::S). By Theorem 3.14, F has a fixed point Co. Then it follows from (3.5) that 1(r(Co)) E Co so, by the property of r, 1(r(Co)) :::S r(Co). Simultaneously r(Co) :::S l(r(Co)) since I is progressive. Thus r(Co) is a fixed point of f. •
Remark 3.16 A counterpart of Theorem 3.13 for W(P), the family of all nonempty well-ordered subsets of P, substituted for C(P), cannot be proved in the above way, since condition UCECC E W(P) need not hold for every chain C in (W(P),c::) (cf. Example 3.17). This is the reason we used chains in Theorem 3.13 instead of wellordered subsets. Incidentally, such a counterpart of Theorem 3.13 is also true and it corresponds to Manka's [60, Proposition 2].
Example 3.17 Let (P,:::s):= ([0,1],::0::) and An:= {11k: k = 1,··· ,n} for n E N. Then each An is well-ordered and the family {An : n E N} is a countable chain in (W(P), C::). However, the union UnEN An is not well-ordered.
Now we are ready to give a constructive proof of Caristi's theorem.
Proof of Caristi's theorem. By part (a) of Proposition 3.2, not only every chain has an upper bound, but we are also able to define, without the AC, a mapping on the family of all chains in (X, :::S'I') assigning to every chain an upper bound of it. Namely, the limit operator turns out to be such a mapping. By (3.3), I is progressive and it suffices to apply Theorem 3.13 with (P,:::s) := (X, :::S'I') to complete the proof. •
Finally, note that Theorem 3.13 can also be applied to give a constructive proof of a recent result due to Khamsi and Kreinovich [48], who established, with the help of Zorn's Lemma, a fixed point theorem for the so-called dissipative mappings on probabilistic metric spaces. The details of the constructive proof of their theorem are given in lachymski [41].
In the sequel we want to present another proof of the Browder-G6hde-Kirk [15], [36], [49] theorem, using an ordering argument. Fuchssteiner [33] was the first, who gave a constructive proof of this result with a help of his iteration theorem derived from Zermelo's Theorem. A minor variant of Fuchssteiner's proof is given in Goebel and Kirk [35, pp. 42-43]. (Also, there is another constructive proof, in a uniformly convex setting, due to Goebel [34] which relies on the Cantor intersection theorem; an order is not used here.) We shall derive here the Browder-G6hde-Kirk theorem, in a version of Kirk [49], from the following common fixed point theorem for two progressive mappings on a partially ordered set (d. lachymski [47]).
Theorem 3.18 Let (P,:::s) be a partially ordered set and Po c:: P be nonempty and such that every chain in Po has a supremum in Po. Let F : Po -; Po and G : P -; P be progressive mappings. II G(FixF) c:: Po, then F and G have a common fixed point.
We preface the proof of Theorem 3.18 with the following extended formulation of Zermelo's Theorem, containing a constructive formula for a fixed point which, exceptionally, is indispensable here. This formula was given by Bourbaki [12].
Order-theoretic aspects 629
Theorem 3.19 Under the assumptions of Zermelo's theorem, givenp E P, denote
Hp:= supn{A ~ P: pEA, : f(A) ~ A and for every chain C ~ A, supC E A}.
Then H is well-defined, Hp is a jixetJ-point of F and P::5 Hp.
Proof of Theorem 3.18. By Theorem 3.19, there exists - independently of the AC - a progressive mapping H : Po --+ FixF. Then the mapping Go H is progressive and Go H is a selfmap of Po since G(FixF) ~ Po. By Zermelo's Theorem, Go H has a fixed point p.. Then
p. ::5 Hp. ::5 G(Hp.) = P.
which gives p. = Hp*. Hence Gp* = G(Hp.) = p. so P. is fixed under G. (Actually, Fix( Go H) = Fix G n Fix H for any progressive mappings G and H.) Since F 0 H = H, p. is a fixed point of F 0 H. Repeating the above argument yields P. = Fp., i.e., p. is a common fixed point of F and G. •
We recall Kirk's [49J version of the fundamental fixed point theorem for nonexpansive mappings. We refer the reader to Goebel and Kirk [35, Chapter 4J for the terminology used below. Throughout this discussion X denotes a Banach space.
Theorem 3.20 (Kirk) Let K be a nonempty, weakly compact, convex subset of X and assume K has normal structure. Then every nonexpansive mapping f : K --+ K has a fixed point.
Given a nonempty, weakly compact, convex subset K of X and an arbitrary mapping f : K --+ K, set
M := {A ~ K : A:f- 0 and A = cl(conv A)}
and
Mf:= {A EM: f(A) ~ A}.
The mapping A --+ C(A) (A EM), where C(A) stands for the Chebyshev center of A, is said to be the Chebyshev operator and will be denoted by C. It is well-known that C is a selfmap of M (see, e.g., Goebel and Kirk [35, p. 38]). Moreover, given A EMf, if A = cl(conv f(A)) and f is nonexpansive, then C(A) is f-invariant (see, e.g., Goebel and Kirk [35, proof of Theorem 4.1]).
The idea of the proof of Kirk's theorem via Theorem 3.18 is as follows. We consider the operator F defined by
F(A) := cl(conv f(A)) for A EMf. (3.6)
Then it straightforward that f has a fixed point if and only if there exists a singleton in the family FixF. Zermelo's Theorem ensures that FixF:f- 0. To show FixF contains a singleton, it suffices to prove that F has a common fixed point with an operator on M which has the property that every fixed point of it is a singleton. In particular, the Chebyshev operator has such a property if the set K has normal structure.
Proof of Kirk's Theorem 3.20. We apply Theorem 3.18 setting P := M, Po := M f, ::5:=:2, G := C and F defined by (3.6).
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Step 1. It follows from the definition of the Chebyshev operator that C(A) <::; A for A E M which means C is progressive on (M, :2). Let A E Mj. Since f(A) <::; A, we may conclude that cl(conv f(A)) <::; cl(conv A) = A, i.e., F(A) <::; A. Hence
f(F(A)) <::; f(A) <::; F(A).
Thus we have shown that F is a selfmap of Mj and F is progressive on (Mj,2). Consider a chain C in (M j, :2). Then the family C has the finite intersection property so, by weak compactness, the set nC is nonempty. Moreover, it is closed, convex and f-invariant so it belongs to Mj. It is easily seen that nC = supC in (Mj, :2).
Now Zermelo's Theorem implies the existence of a set Ao E Mj such that
Ao = cl(conv f(Ao)).
In fact, such a set exists for an arbitrary mapping f: We used neither the nonexpansive condition, nor continuity of f in Step 1.
Step 2. Since f is nonexpansive, we know that if A E Mf and A = cl(conv f(A)), then C(A) EMf. That means condition C(FixF) <::; Mj is satisfied. By Theorem 3.18, C and F have a common fixed point A. EMf, i.e., A. = C(A.) = F(A.).
Step 3. Since A. = C(A.) and K has normal structure, we obtain A. = {a.} for some a. E K. Then F(A.) = {J(a.)} and since A. = F(A.), we conclude that a. is a fixed point of f. •
Finally, note that Zermelo's Theorem 3.19 can also be applied to prove more abstract formulations of Kirk's theorem dealing with nonexpansive mappings on a metric space which possesses the so-called convexity structure, a notion introduced by Penot [68J. In this setting Biiber and Kirk [20J gave a constructive proof of Soardi's [74J fixed point theorem. We emphasize that the success of their approach is due to the fact that, under the assumptions of Zermelo's Theorem, there exists - independently of the AC (!) - a retraction from a set P onto the set of all fixed points of the mapping considered.
4. The Tarski-Kantorovitch Theorem
A selfmap F of a partially ordered set (P,::S) is said to be ::s-continuous if for every countable chain C having a supremum, the image F(C) has a supremum and
supF(C) = F(supC).
It is easily seen that a ::5-continuous mapping is isotone. (P,::5) is said to be ::5-complete if every countable chain has a supremum. (in the literature (P,::5) is also called then chain w-complete; d. Markowsky [61J.) Following Dugundji and Granas [27, p. 15], we recall the Tarski-Kantorovitch Theorem (abbr., 'the T-K Theorem').
Theorem 4.1 (Tarski-Kantorovitch) Let (P,::5) be a ::5-complete partially ordered set and a mapping F : P --t P be ::5-continuous. If there exists Po E P such that Po ::5 Fpo, then F has a fixed point; moreover, p. := sup{Fnpo : n E N} is fixed under F.
A similar fixed point theorem is known in the literature as Kleene's fixed point theorem (see, e.g., Baranga [8]) in which increasing sequences are substituted for chains in the definitions of ::5-continuity and ::5-completeness. It turns out that the assumptions of both theorems are equivalent according to the following.
Order- theoretic aspects 631
Proposition 4.2 Let (P,:5) be a partially ordered set and F : P --> P. Then
(a) (P,:5) is :5-complete if and only if every strictly increasing sequence in P has a supremum;
(b) F is :5-continuous if and only if F is isotone and for every strictly increasing sequence (Pn) having a supremum, there exists sup{FPn : n E N} and it coincides with F(suP{Pn: n EN}).
Proof. The 'only if' parts of both statements are trivial. We prove the if' part of (a). Let C be a countable chain, C = {cn : n EN}. If maxC exists, then we are done. Otherwise, we may define by induction a subsequence (CkJ setting
kl := 1 and kn+1 := min{m > kn : Ckn :5 em and Ckn # cm}.
Then (Ckn ) is strictly increasing so by hypothesis, there exists Co := SUp{Ckn : n EN}. Since for each n E Nand kn ::; m < kn +1, Cm :5 Ckn , we may conclude that Co = supC. A similar argument can be used to prove the 'if' part of (b). •
Our first purpose is to show, following Jachymski [43J, that the T-K Theorem yields Angelov's theorem in a little more general setting: It suffices to assume that a uniform space X is only sequentially complete. Moreover, thanks to the constructive formula for a fixed point in the T-K Theorem, it will be possible to prove the convergence of successive approximations of a mapping f, the fact non-obtainable via the KnasterTarski Theorem. We also emphasize that this time all our proofs are independent of the AC. We consider the same partially ordered set (X x lR~,:5) as in Section 1. We start with two results on completeness (cf. Jachymski [43]).
Proposition 4.3 Let (X, P) be a Hausdorff uniform space with a family of pseudometrics P := {Pa : n E A}. The following statements are equivalent.
(i) Every sequence (xn) in X such that
00
LPa(Xn,Xn+1) < 00 for all n E A (4.1) n=l
is convergent.
(ii) Every increasing sequence ((Xn, ipn))~=l in (X x lRt,:5) is such that (Xn) converges to some Xo EX, (ipn) converges pointwise to some ipo E lR~ and
(Xo, ipo) = sup{(xn, ipn) : n EN}.
(iii) (X x lR~,:5) is :5-complete.
In particular, if (X, P) is sequentially complete, then (X x lR~,:5) is :5-complete.
Proof. (i) => (ii): Assume that ((xn,ipn))~=l is increasing, i.e.,
p,,(xn,xn+1)::; ipn(n) - ipn+1(n) for n E A and n E N.
Summing up with respect to n yields
m-l
LP,,(Xn,Xn+1)::; ipl(n) - ipm(n)::; ipl(n) for all m:O:: 2 n=l
(4.2)
632
which implies (4.1). By (i), (xn) converges to some Xo E X. Moreover, by (4.2), 'PnH(a) ::; 'Pn(a) and since the functions 'Pn are nonnegative, the sequence ('Pn) converges pointwise to some 'Po E lR.~. Let m, n E Nand m > n. By (4.2), we get
m-l
p",(xn,xm) ::; L p",(Xk,Xk+1) ::; 'Pn(a) - 'Pm(a) for a E A. k=n
Hence letting m -+ 00 we obtain p",(xn,xo) ::; 'Pn(a) - 'Po(a) which means (xo,'Po) is an upper bound of the set {(xn, 'Pn) : n EN}. Let (x, 'P) be another upper bound of this set, i.e., p",(xn,x) ::; 'Pn(a) - 'P(a) for a E A and n E N. Letting n -+ 00 we get (xo, 'Po) ::5 (x, 'P). Thus (xo, 'Po) is the least upper bound of the sequence «xn, 'Pn»~l.
(ii) ~ (iii): Since (ii) implies that every increasing sequence in (X x lR.~,::5) is convergent, we may apply part (a) of Proposition 4.2 to deduce (iii).
(iii) ~ (i): Assume that (4.1) holds. Define
00
'Pn(a) := LP",(Xk,XkH). k=n
Since p",(xn,Xn+1) = 'Pn(a) - 'Pn+1(a), the sequence «xn,'Pn»~=l is increasing. By (iii), this sequence has a supremum (xo, 'Po). Then
p",(xn, xo) ::; 'Pn(a) - 'Po(a)
which implies that 0 ::; 'Po (a) ::; 'Pn(a). Since 'Pn(a) -+ 0, we get 'Po(a) = O. Thus p",(xn,xo) ::; 'Pn(a) which yields that Xo = limxn .
Finally, if (X, P) is sequentially complete, then condition (i) holds since every sequence satisfying (4.1) is Cauchy, hence convergent. The ::5-completeness follows from (i) ~ (ii). •
In a metric setting a more pellucid result can be established. We use here Ekeland's ordering defined by (2.4). Part 'only if' of the following theorem was proved by Baranga [8, Corollary 3].
Proposition 4.4 Let (X,d) be a metric space. Then (X,d) is complete if and only if (X x lR.+,::5) is ::5-complete.
Proof. The 'only if' part follows from Proposition 4.3. To prove the 'if' part observe that every Cauchy sequence (xn) has a subsequence (Xkn ) such that
00
Ld(Xkn,Xkn+l) < 00.
n=l
By implication (iii) ~ (i) of Proposition 4.3, this subsequence converges to some Xo E X. Then Xo is also the limit of (xn). •
Lemma 4.5 Let (X, P) be a Hausdorff uniform sequentially complete space with P := {p", : a E A}. Let f : X -+ X be a j-Lipschitzian mapping and F be an associated mapping on X x lR.~ defined by (2.6). Then F is ::5-continuous.
Proof. We apply part (b) of Proposition 4.2. Let «xn, 'Pn»~l be increasing in (X x lR~, ::;). By Proposition 4.3, (Xn) converges to some Xo E X and (CPn) converges
Order-theoretic aspects 633
pointwise to some rpo E lR~, and (xo, rpo) = sup{(xn, rpn) : n EN}. By Proposition 2.12, F is isotone and hence (F(xn, rpn))~l is increasing. Again by Proposition 4.3, there exists (YO,,,po) := sup{F(xn,rpn) : n E N}j moreover, Yo = limfxn = fxo since f is continuous, and "po(o:) = limh",rpn(j(o:)) = h",rpo(j(o:)). Thus
(Yo, "po) = (fxo, h· (rpo 0 j)) = F(xo, rpo)
which means F is j-continuous. • We are ready to prove the convergence of successive approximations in the Tan-Tarafdar theorem and Angelov's theorem via the T-K Theorem. We assume in Theorems 4.6 and 4.7 given below that the uniform space (X, P) is sequentially complete (not necessarily complete).
Theorem 4.6 Under the assumptions of Theorem 2.13, f has a fixed point x. and fnx -t x. for all x EX.
Proof. By Proposition 4.3 and Lemma 4.5, (X x lRt, j) is j-complete and the mapping F is j-continuous. Fix an x EX. If we set
rp(o:) := p",(x,Jx)/(l- h",) for all 0: E A,
then (x,rp) j F(x,rp). By the T-K Theorem, (x.,rp.) := sup{~(x,rp) : n E N} is a fixed point of F. Then x. = fx •. By monotonicity of (Fn(x,rp))~=l and Proposition 4.3, (x.,rp.) = (limrx,limhnrp)j in particular, x. = limrx. •
Theorem 4.7 Under the assumptions of Theorem 2.14, f has a fixed point. Moreover, limrxo is a fixed point of f.
Proof. We show there exists a function rpo E lR~ such that (xo, rpo) j F(xo, rpo). Then rpo should be a solution of the functional inequality (2.7). First we use an argument similar to the one used in the proof of Lemma 2.15. In our present notation we slightly change the definition of Tf3 by setting
Tf3 := 9 + h· (rp 0 j) for all rp E iPf3.
(Actually, we consider here the restrictions of g, hand j to an orbit OJ((3).) Then Tf3(iPf3) ~ iPf3 and Tf3 is a Banach contraction with a contractive constant hf3. Hence it has a fixed point rpf3. This time we can use the convergence of successive approximations of Tf3' a fact covered by the T-K Theorem in view of Theorem 4.6, and consequently we shall be able to circumvent the AC. In particular, consider the sequence (T!10). Then rpf3 = limn ..... "" T!10 and hence we obtain
rpf3(O:) = g(o:) + ~ (ll h(jP(O:))) g(r(o:)) for 0: E OJ((3).
Set rpo«(3) := rpf3«(3) for (3 E A. It can easily be verified that rpo is a solution of (2.7) on A. By the K-T Theorem, the pair (x.,rp.) := sup{Fn(xo,rpo): n E N} is a fixed point of F. Since ~(xo, rpo) = (rxo, rpn), where rpn(O:) := h",rpn-l (j(o:)) for n E N, and (Fn(xo,rpO))~=l is increasing, we may conclude by Proposition 4.3 that x. = limrxo .
•
634
A number of authors studied a class of mappings which satisfy a nonlinear contractive condition. Namely, given a function cp : 1R+ --t 1R+ such that cp(t) < t for t > 0 and cp(O) = 0, we say that a selfmap f of a metric space (X, d) is cp-contractive if
d(fx,fy)s:cp(d(x,y» for all x,yEX.
It seems that the theory of such mappings was initiated by Rakotch [70J and subsequently it was developed, e.g., by Browder [16], Zitarosa [92], Boyd and Wong [13], Matkowski [62], Dugundji and Granas [26]' Jachymski ([39], [40], [45]) and Alber, Guerre-Delabriere and Zelenko [3J. In [43J we examined possibilities of deriving fixed point theorems for cp-contractive mappings from the T-K Theorem restricted to Ekeland's ordering -:!::E in X x 1R+ and a mapping F of the following form:
F(x,a) := (fx,Ta) for x E X and a E 1R+,
where T : 1R+ --t iR.+. Then it is natural to assume that the set {a E iR.+ : Ta < a} is nonemptYi otherwise, the above restriction of the T-K Theorem would be trivial since the condition (xo, cpo) -:!:: F(xo, cpo) would then imply that Xo = fxo. It is rather surprising that our previous methods turn out to be ineffective in the setting of cpcontractive mappings. In particular, Rakotch's [70J fixed point theorem involving a function cp such that t,..... cp(t)jt (t > 0) is non-increasing (this forces subadditivity of cp, i.e., cp(s + t) S: cp(s) + cp(t) for s,t E iR.+) cannot be derived from the T-K Theorem in the above way, according to the following result (for a proof, cf. Jachymski [43]).
Theorem 4.8 Let a function cp : 1R+ --t 1R+ be nondecreasing, subadditive and such that cp(t) < t for t > O. The following statements are equivalent.
(i) There is an h E (0,1) such that cp(t) S: ht for all t E iR.+.
(ii) Given a complete metric space (X,d) and a cp-contractive mapping f : X --t X, there exists a function T : iR.+ --t iR.+ such that T(ao) < ao for some ao E iR.+, and the mapping (f, T) is -:!::E-continuous on X x 1R+.
Briefly, if a function cp has the property that for every cp-contractive mapping the T-K Theorem can be used in the above way, then every such mapping is a Banach contraction. Thus, in some sense, the T-K Theorem can be applied only to a class of mappings satisfying a linear contractive condition.
Finally, we want to show how some results on the theory of iterated function systems (IFS) can be derived from the T-K Theorem. The theory of IFS deals with fixed points of the following Hutchinson-Bamsley operator (cf. Hutchinson [38J and Barnsley [9, p. 80]).
N
F(A) := U fi(A) for A ~ X, (4.3) i=l
where X is a set and iI,'" ,fN are selfmaps of X. The fundamental HutchinsonBarnsley theorem says that if (X, d) is a complete metric space and all the maps Ii are Banach contractions, then F has a unique fixed point A. in K(X), the family of all nonempty, compact subsets of X. Such a set A. is called then a fractal in the sense of Bamsley. There are at least two different proofs of the above result based on the Banach Contraction Principle. The first, due to Hutchinson [38J, uses a fact that every finite composition
Ii, ... ip := fil 0'" 0 fip for pEN and i1,'" ,ip E {I,·" ,N}
Order-theoretic aspects 635
is also a Banach contraction so it has a fixed point. Then it can be shown that A* coincides with the closure of the set of fixed points of all finite compositions hl ... i p ' The second proof uses a well-known theorem, a proof of which is rather technical, saying that completeness of (X,d) implies completeness of K(X) endowed with the Hausdorff metric H (see, e.g., Barnsley [9, Theorem 11.7.1)). Next it can be shown that the operator F is a Banach contraction on (K(X),H) and the Contraction Principle then yields the theorem (see, e.g., Barnsley [9, Theorem 111.7.1)). Our purpose here is to give yet another proof via partial ordering techniques. Such an idea also appears in papers of Soto-Andrade and Varela [75J and Hayashi [37J; however, they considered other versions of Tarski's fixed point theorem. We start with the following theorem on ::5-continuity (cf. Jachymski, Gajek and Pokarowski [46, Lemma 1]).
Proposition 4.9 Let (P,::5) be a partially ordered set which is ::5-complete and such that for any p, q E P, there exists an infimum inf{p, q}. Assume that for any increasing sequences (Pn) and (qn) in P,
inf{sup{Pn: n E N},sup{qn: n E N}} = sup{inf{Pn,qn}: n EN}. (4.4)
Let Fl, ... , FN be ::5-continuous selfmaps of P and define a mapping F by
Fp:= inf{F1P,'" ,FNP} for pEP.
Then F is ::5-continuous.
Proof. For the sake of simplicity, assume that N = 2; then an easy induction shows that our argument can be extended to the case of an arbitrary N E N. By Proposition 4.2, it suffices to show that given an increasing sequence (Pn) in P, Fp = sUPnEN Fpn, where P := sUPnEN Pn. Since Fl and F2 are isotone, so is F. Hence the sequence (FPn) is increasing and by hypothesis, it has a supremum. Then by (4.4) and ::5-continuity of Fl and F2,
sup{FPn: n E N} = sup{inf{F1Pn,F2Pn}: n E N} = inf{sup{FIPn: n EN},
SUp{F2Pn: n E N}} = inf{FIP,F2p} = Fp,
which proves the ::5-continuity of F. • Corollary 4.10 Let X be a Hausdorff topological space, It,··· , fN be continuous selfmaps of X. Endow K(X), the family of all nonempty, compact subsets of X, with the set-inclusion 2. Then the operator F defined by (4.3) is 2-continuous on K(X).
Proof. Given an increasing sequence (An) in (K(X),2) (i.e., (An) is decreasing in the usual sense), the set n~l An is nonempty by compactness, and it is a supremum of (An). (Actually, every chain in (K(X),2) has a supremum.) Thus (K(X),2) is 2-complete. Set
Fi(A) := /i(A) for i = 1,··· , N and A E K(X).
By topological continuity, all the Fi are selfmaps of K(X). We show each Fi is 2-continuous. By Proposition 4.2, it suffices to show that given a decreasing - in the usual sense - sequence (An) in K(X),
636
This result can be found, e.g., in the Polish edition of Engelking [31] published in 1989 - cf. Exercise 3.1O.A (c) of this book. For the sake of completeness, we give a proof of it. The inclusion ~ is obvious. Let y E n~=l fi(An), i.e., y = fi(Xn) for some Xn E An. Set En := f-1({y}) n An. Then Xn E En and En are compact, i.e., En E K(X). Moreover, En+! ~ En so, by compactness, n~=l En # 0. If X E n~=l En, then fix = y and x E n~=l An which means y E fi(n~=l An).
Observe that condition (4.4) in (K(X), 2) means given decreasing - in the usual sense - sequences (An) and (En) in K(X),
00 00 00
n=l n=l n=l
This equality holds indeed even if the sets considered are not compact. By Proposition 4.9, F is 2-continuous. •
Now Corollary 4.10 and the T-K Theorem yield the following.
Theorem 4.11 Let X be a topological space, fr,··· ,f N be continuous selfmaps of X and F be the Hutchinson-Barnsley operator (4.3). The following statements are equivalent.
(i) There exists a nonempty, compact set A. ~ X such that
F(A.) = A •.
(ii) There exists a nonempty, compact set Ao ~ X such that
F(Ao) ~ Ao·
Proof. It suffices to show (ii)=;.(i). Apply the T-K Theorem setting (P,:::5) .(K(Ao), 2). The 2-continuity of F on K(Ao) follows from Corollary 4.10 applied to the restrictions filAo of the mappings fi to the set Ao. Moreover, n~=l Fn(Ao) is a fixed point of F. •
We are ready to give another proof of the Hutchinson-Barnsley theorem as done in Jachymski, Gajek and Pokarowski [46] (also cf. Hayashi [37]). According to Theorem 4.11, it suffices to show the existence of a nonempty, compact set Ao such that F(Ao) ~ Ao. That can be done in a simple way if, e.g., all the mappings fi are Banach contractions
and a metric space (X, d) has the Heine-Borel property (see, e.g., Williamson and Janos [88]), i.e., every closed and bounded subset of X is compact. It is worth noting that many results of the theory of IFS were obtained in the above class of spaces (cf. Lasota and Myjak [56] and Lasota and Yorke [57]).
The following lemma is essentially due to Williams [87]. We give it in a slightly modified form.
Lemma 4.12 Let (X,d) be a metric space and f : X -> X be a Banach contraction. Then given Xo E X, there exists ro > 0 such that for all r ~ ro, the closed ball E(xo, r) is f -invariant. Hence given a finite family fr,···, f N of Banach contractions and Xo E X, there exists R > 0 such that F(E(xo,R)) ~ E(xo,R), where F denotes the Hutchinson-Barnsley operator (4.3).
REFERENCES 637
Proof. Let h denote the Lipschitz constant of f. Let Xo E X, r > 0 and x E B(xo, r). Then
d(fx, xo) :s: d(fx, fxo) + d(fxo,xo) :s: h: r + d(fxo, xo).
Hence d(fx,xo):S: r if h: r+d(xo,fxo):S: r, i.e., r;C:: d(xo,fxo)j(l- h). Thus it is enough to set ro := d(xo,!xo)j(l - h).
Now it is easily seen that if the mappings Ii are Banach contractions with Lipschitz constants hI,'" , hN, then it suffices to set
R:= max{d(xo'!iXO)j(l- hi) : i = 1"" ,N}.
• Now Theorem 4.11 and Lemma 4.12 yield the Hutchinson-Barnsley theorem for an appropriate IFS on a Heine-Borel metric space; in particular, on the Euclidean space ~n. However, the uniqueness of a fixed point of F is not obtainable via the T-K Theorem.
Finally, note that the T-K Theorem can be applied in the theory of IFS in somewhat more general settings because of using conditions involving a suitable behaviour of countable chains only. This is the reason why countable compactness can be substituted for compactness. In particular, if X is a Hausdorff topological space and C(X) denotes the family of all nonempty, closed subsets of X, then 2-completeness of C(X) is equivalent to countable compactness of X (cf. Engelking [31, Theorem 3.10.2]). If, moreover, X is sequential (i.e., every sequentially closed subset of X is closed), then, in case N = 1, the Hutchinson-Barnsley operator F is 2-continuous and it maps C(X) into itself if and only if the mapping f is continuous in a topological sense (cf. Jachymski, Gajek and Pokarowski [46, Proposition 5]). (It is interesting that the last equivalence need not hold if we drop the assumption that X is sequential; cf. [46, Example 3].) Therefore for a sequential space X and a mapping f : X -> X, the assumptions of the T-K Theorem with (P,~) := (C(X), 2) and F(A) := f(A) for A E C(X) are satisfied if and only if X is countably compact and f is continuous with respect to the topology.
ACKNOWLEDGEMENTS
I wish to thank Art Kirk for the invitation to write this chapter and his patience as I was writing it. Also, I am grateful to Czeslaw Bessaga, Andrzej Granas, Roman Manka, Janusz Matkowski and Jim Stein, Jr. for fruitful discussions and useful remarks on this stuff. Further, I thank Yakov Alber, Vasil Angelov, Yong-Zhuo Chen, Susumu Hayashi, Ludvik Janos, Wataru Takahashi, Mihai Thrinici and Peter Volkmann for providing me with reprints of their papers. However, most of all I owe gratitude to my wife Zofia and children for putting up with my absence at home as I was working on this chapter and my other papers.
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Chapter 19
FIXED POINT AND RELATED THEOREMS FOR SET-VALUED MAPPINGS
George Xian-Zhi Yuan
Department of Mathematics
The University of Queensland
Brisbane 4072 QLD Australia
In this chapter, we focus in the discussion of fixed point theory for set-valued mappings by using Knaster-Kuratowski-Mazurkiewicz (KKM) principle in both topological vector spaces and hyperconvex metric spaces. In particular, the fixed point theory of set-valued mappings of Browder-Fan and Fan-Glicksberg type has been extensively studied in the setting of locally convex spaces, H-spaces, G-convex spaces and metric hyperconvex spaces. By using its own feature of hyperconvex metric spaces being a special class of H-spaces, we also establish its general KKM theory and then its various applications. In section 2, we first discuss some recent developments of KKM theory itself and the general Ky Fan minimax principle is given in section 3. In sections 4 and 5, two types of Ky Fan minimax inequalities and their equivalent fixed point forms for set-valued mappings are given. In section 6, the general Fan-Glicksberg type fixed point theorem is discussed in G-Convex spaces. These spaces include locally convex H-spaces, locally convex topological vector spaces and metric hyperconvex metric spaces as special cases. Finally, the general KKM theory and its various applications in metric hyperconvex spaces and the generic stability of fixed points are discussed in section 7.
1. Introduction
The classical Knaster-Kuratowski-Mazurkiewicz principle (often called the KKM theorem, or KKM lemma [100]) has numerous applications in various fields of pure and applied mathematics. These studies and applications are called the KKM Theory today.
In 1961, Ky Fan proved a generalization of the classical KKM theorem in infinite dimensional Hausdorff topological vector spaces and established an elementary but very basic geometric lemma for set-valued mappings. In 1968, Browder gave a fixed point form of Fan's geometric lemma and this is now called the Browder-Fan fixed point theorem. Since then there have been numerous generalizations of the BrowderFan fixed point theorem and their applications in coincidence and fixed point theory, minimax inequalities, variational inequalities, nonlinear analysis, convex analysis, game theory, mathematical economics, and so on. By applying his geometric lemma, Ky Fan obtained a minimax inequality in 1972 which plays a fundamental role in nonlinear analysis and mathematical economics and has been applied to potential theory, partial
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W.A. Kirk and B. Sims (eds.), Handbook o/Metric Fixed Point Theory, 643-690. © 2001 Kluwer Academic Publishers.
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differential equations, monotone operators, variational inequalities, optimization, game theory, linear and nonlinear programming, operator theory, topological group and linear algebra. In particular, by using Ky Fan's minimax inequality, a more general form of the Fan-Glicksberg fixed point theorem is derived for set-valued mappings which are inward (or outward) as defined by Fan in 1969 which are more general than Halpern's definitions for inward (or outward) mappings in 1965.
In 1983 and 1987 Horvath obtained some generalizations of Fan's geometric lemma and his minimax inequality by replacing convexity assumptions with topological properties: pseudo-convexity and contractibility. By extending Horvath's concepts, Bardaro and Ceppitelli [13] in 1988 obtained generalizations of Ky Fan minimax inequalities to topological spaces which have a so called H-structure (also called H-spaces). Following this line, a number of generalizations of Ky Fan's minimax inequalities are given by Horvath [84], Baradaro and Ceppitelli [14], Ding and Tan [47], Ding et al. [48], Chang and Ma [31], Park [127], Tarafdar [154], and Tan et al. [150] in topological spaces which need not have a linear structure but with an H-structure.
Throughout this Chapter all spaces are assumed to be Hausdorffunless otherwise specified. Let X and Y be non-empty sets. We shall denote by 2Y the family of all non-empty subsets of Y, F(X) the family of all non-empty finite subsets of X. Let X be a topological space. For each non-empty subset A of X, we denote the closure of A in X by clx A (in short, cl A) or A if there is no confusion. A subset A of X is said to be compactly closed (resp., open) if An C is closed (resp., open) in each non-empty compact subset C of X.
2. Knaster-Kuratowski-Mazurkiewicz Principle
In this section, we will discuss some recent developments of KKM theory in H-spaces.
A topological space X is said to be contmctible if the identity mapping Ix of X is homotopic to a constant function. Let j\j and ]R denote the set of all natural numbers and the set of all real numbers, respectively. Let vo, VI, ••• ,Vn be n + 1 points in a Euclidean space which are not contained in any linear manifold (i.e., a translation of a vector subspace) of dimension less than n. The convex hull of these n+ 1 points is called an n-dimensional simplex and is denoted by VOVl ••• Vn . The points vo, vb· .. ,vn are also called vertices of the simplex. For 0 ::; k ::; nand 0 ::; io < i l < ... < ik ::; n, the ksimplex VioVil •.• Vik is a subset of the n-simplex VOVl ••. Vn . It is called a k-dimensional face (or say, simply k-face) of the simplex VOVl •.• Vn .
For the convenience of our discussion, for n E j\j and N = {O, 1,··· ,n}, we also denote by AN = co{ eo, ... ,en} the standard simplex of dimension n, where {eo,··· en} is the canonical basis of ]Rn+1 and for J E F(N), AJ = co{ej : j E J}. The points eo, el, ... ,en are called vertices of the simplex AN. Suppose that T = {TO, Tb ... ,Tm}
is finite collection of n-simplexes in VOVl ..• Vn such that
(i) U:l Ti = VOVI ••• Vn ; and
(ii) T; n Tj is either empty or a common face of any dimension of Ti and Tj, where i,j E {1,2,··· ,m}.
Then T is called a triangulation of VOVI ..• V n . Let T be a triangulation. A vertex of a simplex in T is called a vertex of T. An (n - I)-face of a simplex from T is called a boundary (n - 1) -simplex of T if it is a face of exactly one n-simplex of T.
Fixed point theorems for set-valued mappings 645
We next present an extremely useful result which was published by Knaster, Kuratowski, and Mazurkiewicz [100J in 1929. It is regarded as one of the most important principles in nonlinear analysis, and it provides the foundation for the other results of this section. Its proof is based on the celebrated Spemer's combinatorial lemma [146J published in 1928. There are several versions of Spemer's Lemma and its extensions, for more details of its proofs, see Border [25, p.23-24J, Cohen [36], Dugundji and Granas [54J, Fan [69], Shashkin [131, p.9-14]' Shih and Lee [132]' Wong [164], Yuan [174], Zeidler [175J and many other books for its numerous and different proofs.
KKM Principle. Let CO,," , Cn be closed subsets of the standard n-dimensional simplex D.N and let {eo,'" ,en} be the set of its vertices. If for each J E F(N), D.J C UjEJCj . Then ni=OCi 'I- 0.
On the other hand, Shih and Tan [134J (see also Kim [93J or Lassonde [104]) provided the following dual form of KKM theorem by replacing the word closed with the word open.
The Dual Form of KKM Principle. Let Co, .. , , Cn be open subsets of the standard n-dimensional simplex D.N and let {eo,'" , en} be the set of its vertices. If for each J E F(N), D.J C UjEJCj. Then ni=oCi 'I- 0.
By following Shih and Tan [134]' a set-valued mapping B : F --> 2.6N is said to be a Shapley mapping if for each T E F, we have
T C UOPE:FB(p).
Let CO,C1,'" ,Cn be subsets of a standard n-simplex D.N. If we define a mapping B : F --> 2.6N by B(i) = Ci for each i E N = {0,1,'" ,n} and B(p) = 0 for all non O-dimensional faces of D.N' Then it is easy to see that B is a Shapley mapping if and only if the following condition is satisfied: D.J C UjEJCj = UjEjB(j) for each J E F(N).
Now we state the following selection theorem due to Shih and Tan [134J.
Theorem 2.1 If B : F -> 2.6N is a Shapley mapping such that each B(p) is an open subset of D.N for each p E F. Then B admits a Shapley-selection A : F --> 2.6N such that each A(p) is a closed subset of D.N and A(p) C B(p) for all p E F.
As an application of Shih and Tan's selection theorem for a Shapley mapping and the KKM principle, we can now prove the dual form of the KKM principle above.
Proof of the dual form of the KKM principle. We define a mapping B : F --> 2.6N
by B(i) = Ci for each i E N = {0,1,·.· ,n} and B(p) = 0 for all non-O-dimensional faces of D.N. As we note above, B is a Shapley mapping if and only if
D.J C UjEJCj = UjEjB(j)
for each J E F(N). By Theorem 2.1, there exists another Shapley mapping A: F ->
2.6N such that A(p) is closed and A(p) C B(p) for each p E F. Let Di = A(i) for each i E N = {a, 1"" ,n}. Then we have D; is closed and D; C C; for all i EN. Since A is also a Shapley mapping, it follows that
D.J C UjEJDj = UjEjA(j)
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for each J E F(N). By the KKM principle, we have n?=oA(i) # 0. Thus it implies that n?=oB(i) # 0. •
The following notions introduced by Bardaro and Ceppitelli in [13] were motivated by the earlier work of Horvath [82].
Definition 2.2 A pair (X,{rA}) (also called an H-structure) is said to be an Hspace (also called C -space according to Horvath [84]) if X is a topological space and {r A} AEF(X) a given family of non-empty contractible subsets r A of X, indexed by A E F(X) such that rA C FB whenever A C B. Let (X,{rA}) be an H-space. A non-empty subset D of X is said to be:
(i) H -convex (also called an F -set by Horvath [84]) if rAe D for each A E F(D)j
(ii) weakly H-convex if rAn D is contractible for each A E F(D) (or equivalently, (D, {rAn D}) is an H-space), andj
(iii) H -compact in X if for each A E F(X), there exists a compact, weakly H-convex subset DA of X such that D u A C DA.
It is clear that the product space of a family of H-spaces is also an H-space.
The following example from Horvath [84, p.345] shows that an H-space need not be a convex subset of a topological vector space.
Example 2.3 Let X be a convex set in a topological vector space E, and let Y be any topological space. Suppose that f : X -+ Y is a continuous bijection. For given A E F(Y), let DA:= co{x EX: f(x) E A}. Then DA is convex, so that DA is contractible. Since DA is also compact f : DA -t f(DA) is a homeomorphism. Let r A = f(DA). Then r A is contractible and rAe r A' whenever A c A' E F(Y). Therefore (Y, {r A}) is an H-space. Note that the space Y itself may be a torus, the Mobius band, or the Klein bottle. This example shows that an H-space does not have to be contractible.
Definition 2.4 Let D be a non-empty subset of an H-space (X, {r A}). Following Tarafdar [154], the H-convex hull of D, denoted by Hco(D), is defined by
H co(D) = n{B eX: B is H-convex and DeB}.
Clearly, H co(D) is the smallest H-convex subset of X containing D, and the intersection of any family of H-convex set is also H-convex. Let X be a topological space. Following Horvath [84], an n-dimensional singular face structure on X is a mapping F : F(N) -+ 2x such that (a) for each J E F(N), F(J) is not-empty and contractible and (b) for any J, J' E F(N), J c J' implies F(J) C F(J').
Let N = {a, 1,··· ,n}. We denote by 8f:j.N and f:j.1V the boundary and kth-skeleton of the n-dimensional simplex f:j.N, respectively. Because if X is a contractible topological space, each (single-valued) continuous function f : 8f:j.N -+ X can be continuously extended to f:j.N, i.e., there exists a continuous function 9 : f:j.N -+ X such that g(x) = f(x) for all x E 8f:j.N, we have the following continuous selection result for H-structures.
Fixed point theorems for set-valued mappings 647
Lemma 2.5 let X be a topological space. For each non-empty subset J of N = {O, 1,,,, ,n}, let FJ be a non-empty contractible subset of X with FJ c FJ' whenever 0 # J c J' c {O, 1,'" ,n}. Then there exists a continuous junction f : f:j.N --t X such that f(f:j.J) C FJ for each non-empty subset J of {O, 1, ... ,n}.
Proof. For each singleton {i}, take any fixed point Xi E F{i}' Then fO : f:j.N --t X defined by fO(ei) := Xi is a continuous function on the Oth-skeleton f:j.fjy of f:j.N' Now assume that a continuous function fk : f:j.~ --+ X on the kth-skeleton of f:j.N has been constructed in a such way such that fk(f:j.J) C FJ if cardJ S k + 1. Let f:j.J be a face of dimension k + 1 of f:j.N and for each i E J, let Ji := J \ {i}. Then the boundary set UiEJf:j.J, of f:j.J is contained in the kth-skeleton of f:j.N and
Because FJ is contractible, fk can be extended to a continuous function fy+l : f:j.J --t
FJ. If f:j.J and f:j.J' are two different k + 1 dimensional faces of f:j.N with non-empty intersection, then fy+l and fy;rl coincide with the continuous function fk on f:j.Jnf:j.J"
By gluing the functions fy+l together, we obtain a continuous function fk : f:j. t+l --> X, which has the desired property. •
Proposition 2.6 Let X be a topological space. Let F : F(N) --> 2x be a singular face structure on X and {Mi : i = 0"" , n} be a family of closed (resp., open) subsets of X such that for any J E F(N), F(J) c UiEJMi . Then nf=oMi # 0.
Proof. By Lemma 2.5, there is a continuous function f : f:j.N --> X such that for each J E F(N), f(f:j.J) C F(J). Let Ci = f-l(Mi ) for each i = 0"" ,n. Then {Ci}f=o is a family of closed (resp., open) subsets of f:j. such that for any J E F(N), f:j.J C UiEJCi. By the KKM principle (resp., its dual form), nf=oci # 0. Take any Xo E nf=oCi, then f(xo) E nf=oMi # 0. •
As an application of Proposition 2.6, we have the following result.
Theorem 2.7 Let X be a contractible topological space, {Mi : i = 0, ... ,n} be a closed (resp., open) covering of X and {Fi : i = 0"" ,n} a family of contractible subsets of X such that
(i) for any i E {O,'" ,n}, Fi n Mi = 0, and
(ii) for any J E F(N) with JoiN, niEJ Fi is non-empty and contractible.
Proof. Define F : F(N) --> 2x by F(N) = X and F(J) = nifpFi if J E F(N) with JoiN. Then F is a singular face structure on X. Note that for any J E F(N), F(J) C UiEJMi since F(J) C UiEJM; U Ui¢JMi and F(J) n Mi = 0 whenever i rJ. J. Then Proposition 2.6 implies that n~oMi # 0. •
Theorem 2.8 Let X be a contractible topological space and Y be a topological space, {Mi : i = 0,··· ,n} be an open (resp., closed) covering ofY and {F; : i = 0"" ,n} be a family of contractible subsets of X. Let S : X --> Y be continuous such that
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(a) for each i E {O, 1" .. ,n}, F; C S-l(Mi)i and
(b) for each J E F(N) with J =f N, niuF; is non-empty and contmctible.
Proof. Suppose the contrary, so that uY=oMF = Y, where MF = Y \ Mi denotes the complement of Mi in Y for each i = 0,1,'" ,no Thus {MF: i = 0,'" ,n} is an open (resp., closed) covering of Y. So that {S-l(MF) : i = 0,,,, ,n} is an open (resp., closed) covering of X and by condition of (a), for each i = 0, 1"" ,n, F;nS- 1(Mf) = 0. Therefore Fi and S-l(MF) for i = 0",' ,n satisfy all hypotheses of Theorem 2.7. By Theorem 2. 7, n~os-l (MF) =f 0 which contradicts the fact that {Mi : i = 0, ... ,n} is a covering of Y. Thus nY=oMi =f 0. •
Since Chang and Yang [32] gave a generalization of the KKM theorem in which the domain need not be a subset of its range, there have been several generalizations in this direction. For example, Chang and Ma [31] extended this definition into H-spaces and later Zhou [178] gave a more generalized definition and obtained a characterization of the generalized HKKM mapping which is also a generalization of the corresponding result given by Chang and Zhang [33].
Definition 2.9 Let X be a non-empty set and Y a topological space. A mapping C: X -+ 2Y U {0} is said to be tmnsfer closed valued (e.g., see Zhou and Tian [181]) if for each x E X and y 1. C(x), there exist x, E X and an open neighborhood N(y) ofy in Y such that y' 1. C(X') for each y' E N(y).
It is obvious that if a mapping C : X -+ 2Y is transfer closed valued, then for each x E X and y E Y with y 1. C(x), there exists some x, E X such that y 1. cl(C(X'».
The following lemma was first proved by Zhou and Tian [181] for the case when the domain X is a topological space and in the present form by Zhou [178]; for completeness, we include its proof here.
Lemma 2.10 Let X be a non-empty set, Y a topological space and C : X -+ 2Y . Then nXEXC(X) = nxEx cl(C(x» if and only if the mapping Cis tmnsfer closed valued.
Proof. Sufficiency: It is clear that nXEXC(X) c nxEx cly(C(x)). It is sufficient to show that nxEx cl(C(x» C nXEXC(X). Suppose that y 1. nXEXC(X). Then there exists some x E X such that y 1. C(x). Since C is transfer closed valued on X, there exists some x' E X such that y 1. cly(C(X'», so that y 1. n,,,Excl(C(x». Therefore nxEXcl(C(x» = nXEXC(X).
Necessity: Suppose (x,y) E X x Y such that y 1. C(x); then y 1. nzEXC(Z) = nzEx cl(C(z» so that there exists x' E X such that y 1. cl(C(x'». But then there exists an open neighborhood N(y) of y in Y such that N(y) n C(X') 0 so that y' 1. C(X') for all y' E N(y). Thus C is transfer closed valued. •
Definition 2.11 Let D be a non-empty subset of an H-space (X, {r A})' A mapping F : D -+ 2x is called HKKM if rAe UXEAF(x) for each A E F(X). When X is a non-empty convex subset of a topological vector space and r A = co A, the convex hull of A for each AE F(X), then (X, {fA}) becomes an H-space. In this case, the notion
Fixed point theorems for set-valued mappings 649
of an HKKM mapping F : D --> 2x coincides with the notion of a KKM mapping F, i.e., coA C UxEAF(x) for each A E F(D).
Definition 2.12 Let X be a non-empty set and Y a topological space. A mapping G: X --> 2Y is said to be a generalized HKKM (in 'short, GHKKM) if for each finite subset A = {Xl,'" ,xn} of X, there exists a corresponding finite subset B = {Y1,Y2'" ,Yn} (Yi'S need not be distinct here) in Y and a family {r a }aE.r(B) of non-empty contractible subsets of Y such that ra C ral whenever G C G' E F(Y) such that
r{Yij:jEJ} c Uj=lG(Xi)
for 0 # J C {O,l,··· ,n}.
It is clear that each HKKM mapping is GHKK and an example in Chang and Yang [32] shows that the converse does not hold.
Example 2.13 Let E = (-00,+00) and X = [-2,+2]. Define a set-valued mapping G: X --> 2E by
G(x) = [- (1+ ~2), ~1 for each X EX. Since UXEXH(x) = [-9/5,9/5] and X rf- G(x) for each x E [-2, -9/5) U (9/5,1]. This shows that G is not a KKM mapping. Next we prove that G is a generalized KKM mapping. In fact, for any finite subset {Xl,'" ,xn } C X, take {Yl,'" ,Yn} C [-1,1], then for any finite subset {Yill'" ,Yik} C {Y1,'" ,Yn}, we have
k
CO{Yill'" ,Yik} C [-1,1] = n G(x) C UG(Xi) xEX j=l
Thus G is GHKKM.
Theorem 2.14 Let X be a non-empty set and both Y and Z two topological spaces. Suppose S : Y --> Z is continuous and G : X --> 2z is such that;
(1) the composition mapping S-l 0 G : X --> 2Y defined by
(S-l 0 G) (x) = UzEG(x){y E Y : z = sty)}
for each x EX, is a generalized HKKM mapping; and
(2) for each x EX, G(x) is closed (resp., open) in Y.
Then the family {G(x) : x E X} has the finite intersection property, i.e., for each A E F(X), nXEAG(X) # 0.
Proof. For any finite subset {XO,X1,'" ,xn } of X, since S-1 0 G : X --> 2Y is a generalized HKKM mapping, there exists a finite subset B = {Yo, Yl, ... ,Yn} of Y and a family {ra}cE.r(B) of non-empty contractible subsets of Y such that ra c ral whenever G C G' such that
s
r{YiO'Yil .... ,Yi.} c U(S-l o G)(Xi;) j=O
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for each finite subset {Yio,Yil"" ,Yi,} of {YO,Yl,Y2,'" ,Yn}, where (O:S: s:S: n). Let Mi = S-l(G(Yi)) for each i = 0,1"" ,n; and define a mapping F : F(N) -+ Y by F(J) = r{Yk:kEJ} for each J E F(N). Since S is continuous, Mi is closed (resp., open) in Y for i = 0,1,'" ,n by the assumption (2). Moreover the mapping F is a singular face structure on Y. Therefore all hypotheses of Proposition 2.6 are satisfied. By Proposition 2.6, n?=oMi i=- 0. Take any yo E n?=oMi, then S(yo) E n?=OG(Xi) i=- 0. •
As an application of Theorem 2.14, we have the following result due to Zhou [178].
Theorem 2.15 Let X be a non-empty set and Y a topological space. Let G : X -+ 2Y
be such that
(a) G is transfer closed valued on X; and
(b) there exists a non-empty finite subset Xo of X such that the set Yo = nxExo cl G(x) is non-empty and compact in Y.
Then the intersection nxExG(x) is non-empty and compact if and only if the mapping cl G is a generalized HKKM mapping.
Proof. Necessity: Suppose nXExG(x) is non-empty and compact. Take any
Yo E nxEXG(x).
Note that the singleton set {yo} is contractible. For each A = {Xl,'" ,xn } E F(X), take B = {Yl," . ,Yn} with Yi = Yo for all i = 1,2"" ,n and let rB' = {yo} for all B' E F( B). Since Yo E cl G (x) for all x EX, it is clear that the mapping cl G is generalized HKKM.
Sufficiency: Since the mapping clG is a generalized HKKM by Theorem 2.14 with Y = Z and S being the identity mapping on Y, the family {clG(x) : x E X} has the finite intersection property. Now define a mapping G'(x) = cly G(x) n Yo for each x E X. Then the family of non-empty compact subsets {G'(x) : x E X} has the finite intersection property, so that nxEx cly G(x) = nxExG'(x) i=- 0. Since G is transfer closed, nXEXG(X) = nxExclG(x) by Lemma 2.10. Therefore nxExG(x) i=- 0 •
As an immediate consequence of of Theorem 2.15, we have the following characterization of non-empty intersections for generalized HKKM mappings.
Theorem 2.16 Let X be a non-empty set and Y a compact topological space. Let G : X -+ 2Y be transfer closed valued on X. Then the intersect nxExG(x) is nonempty if and only if the mapping cl G is a generalized HKKM mapping.
Theorem 2.16 is a generalization of the corresponding results given by Chang and Yang [32] and Chang and Ma [31].
As a special case of Theorem 2.16, we have the following famous extension of the classical KKM principle (KKM Lemma) in topological vector spaces due to Fan [59] which plays a very important role in the study of nonlinear analysis today.
Theorem 2.17 Let X be a non-empty subset of a topological vector space E and F : X -+ 2E be a set-valued KKM mapping. Then the family {F(x) : x E X} has the finite intersection property, i.e., nXEAF(x) i=- (1 for each A E F(X).
Fixed point theorems for set-valued mappings 651
As a finite dimensional version of Theorem 2.17, we have the following classical FKKM principle.
Corollary 2.18 (FKKM Principle) Let X be a non-empty subset of a finite dimensional space lR,n, where n EN. Suppose F : X -+ 2IRn is a set-valued KKM mapping. Then the family {F(x) : x E X} has the finite intersection property.
3. Ky Fan Minimax Principle
In this section, as an application of FKKM lemma, we will prove the Ky Fan minimax principle in topological vector spaces. For other results in more general setting such as H-spaces and G-convex spaces, please see Yuan [174] and references wherein for more details.
The minimax inequality of Fan [66] is fundamental in proving many existence theorems in nonlinear analysis. There have been numerous generalizations of Fan's minimax inequality by weakening the compactness assumption or the convexity assumption. In [13], using Horvath's approach [82], Bardaro and Ceppitelli obtained some minimax inequalities in topological spaces which have H -space structure. Following this line, there are many generalizations given by Horvath [82], Tarafdar [154], Ding and Tan [43], Ding et al. [49], Chang and Ma [31], Park [127], Tan et al. [150]. These results generalize most of the corresponding results given by Fan [59] and [69], Dugundji and Granas [53], Lassonde [103], and Zhou and Chen [180] to topological spaces which have the so-called H-structure.
Theorem 3.1 Let X be a non-empty set and Y a compact topological space and </J : X x Y ---> lR, U { -00, +oo} be such that:
(a) the mapping x 1--+ {y E Y: </J(x,y) SO} is tmnsfer closed valuedi
(b) the mapping x -+ cly{y E Y: </J(x,y) S O} is genemlized HKKM on X.
Then there exists y' E Y such that </J( x, y') S 0 for all x EX.
Proof. Define a mapping G : X -+ 2Y by G(x) = {y EX: </J(x,y) SO} for each x E X. Then we have: (1) the mapping G is transfer closed valued and (2) the mapping clG is generalized HKKM. By Theorem 2.16, nXExG(x) =1-0. Take any y' E nxExG(x), then SUPXEX </J(x,y*) sO for all x E X. •
Remark 3.2 It is clear that condition (a) of Theorem 3.1 is equivalent to the following condition which first appeared in Tan et al. [150]:
The fact (a)': for each y E Y with {x EX: </J(x, y) > O} =I- 0, there exists x' E X such that y E inty{y' E Y: </J(x',y') > O}.
Theorem 3.3 Let X be a non-empty subset of a compact H -space (Y, {r A}) and </J : X x Y -+ lR, U { -00, +oo} be such that:
(a) the mapping x 1--+ {y E Y: </J(x,y) SO} is tmnsfer closed valued on Xi and
(b) the mapping x 1--+ cly{y E Y: </J(x,y) SO} is HKKM on X.
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Then there exists y' E Y such that ¢(x, yO) :s; 0 for all x EX.
Proof. Since (Y, {r A}) is an H -space, each HKKM mapping is automatically a generalized HKKM mapping. Therefore all hypotheses of Theorem 3.1 are satisfied. By Theorem 3.1, there exists y' E Y such that ¢(x, yO) :s; 0 for all x E X. •
Corollary 3.4 Let X be a non-empty subset of a compact H -space (Y, {r A}) and ¢ : X x Y --> lR U { -00, +oo} be such that:
(a) for each x E X, Y f-t ¢(x,y) is lower semicontinuous on Y; and
(b) the mapping x f-t cly{y E Y: ¢(x,y):S; O} is HKKM on X.
Then there exists y* E Y such that ¢(x,y*):S; 0 for all x E X.
Proof. Suppose y E Y is such that {x EX: ¢(x,y) > O} =I- 0. Fix any x' EX with ¢(x',y) > O. By (a), there exists an open neighborhood N(y) of y such that ¢(x',y') > 0 for each y' E N(y). Hence y E inty{y' EX: ¢(x',y') > OJ. Now the conclusion follows from Theorem 3.3 and the fact (a)' preceding it. •
Corollary 3.5 Let X be a non-empty subset of a non-empty compact convex set Y in a topological vector space and ¢ : X x Y --> lR U { -00, +oo} be such that:
(a) the mapping x f-t {y E Y : ¢(x,y) :s; O} is transfer closed valued on X; and
(b) the mapping x f-t cly{y E Y: ¢(x,y):S; O} is KKM on X.
Then there exists y* E Y such that ¢( x, y') :s; 0 for all x EX.
Proof. For each A E F(Y), let r A = coCA). Then (Y, {r A}) is an H-space and the mapping x f-t cly{y E Y : if>(x,y) :s; O} is HKKM on X. Thus the conclusion follows from Theorem 3.3. •
Corollary 3.6 Let X be a non-empty subset of a non-empty compact convex set Y in a topological vector space and ¢ : X X Y --> lR U { -00, +oo} be such that:
(a) the mapping x f-t {y E Y : ¢(x, y) :s; O} is transfer closed valued on X;
(b) for each A E F(X) and each y E co(A), minxEA ¢(x, y) :s; O.
Then there exists y' E Y such that ¢(x, yO) :s; 0 for all x E X.
Proof. By Corollary 3.5, we only need to prove that the map x --> cly{y E Y : ¢(x,y) :s; O} is KKM on X. Suppose not, then there exist A E F(X) and y E coCA) such that y ~ UxEA clx{y EX: ¢(x, y) :s; OJ. It follows that ¢(x, y) > 0 for each x E A, so that minxEA .p(x, y) > 0 which is a contradiction. •
As seen from the proof of Corollary 3.4, the condition 'for each x EX, Y --> .p(x, y) is lower semicontinuous' implies the condition 'for each y E X with {x EX: .p(x,y) > O} =I- 0, there exists x' E X such that y E intx{Y' EX: ¢(x',y') > OJ'. Thus Corollary 3.5 and hence Theorem 3.1 and Theorem 3.3 generalize Theorem 1 of Yen [170J (see
Fixed point theorems for set-valued mappings 653
also Theorem 2.2 of Simons [139)) and Theorem 2.11 of Zhou and Chen [180]. The following is an example for which Theorem 3.3 is applicable while Theorem 1 of Yen [170] and Theorem 2.11 of Zhou and Chen [180] are not.
Example 3.7 Let Y = [0,1] and X be the set of all rational numbers in [0,1]. Define tP:XxY-+lRby
tP(x,y) = { x - y, 2,
if y is rational,
if y is irrational
for each (x, y) E X x Y. Suppose (x, y) E X x Y and tP(x, y) > O. If y is irrational, then clearly y < 1. If y is rational, then since tP(x, y) = x - y > 0, we also have y < x ~ 1. In either case, take x' = 1 and note that {y' E Y : tP(x', y') > O} = [0,1), so that y E [0,1) = inty{y' E Y : tP(x',y') > O}. Thus condition (a)' and hence condition (a) of Theorem 3.3 is satisfied. Moreover, for each x E X, clx{y E Y : tP(x, y) ~ O} = clx{y E Y : y is rational and y ~ x} = [x, 1]. It follows that the map x ....... cly{y E Y : tP(x,y) ~ O} is KKM on X. Thus condition (b) of Theorem 3.3 is also satisfied. Therefore Theorem 3.3 is applicable. However, for each x E X, the map y ...... tP(x,y) is not lower semicontinuous and hence Theorem 1 of Yen [170] and Theorem 2.11 of Zhou and Chen [180] are not applicable.
Example 3.7 shows that for each x EX, the lower semicontinuity of the mapping y ....... tP(x, y) is not essential for the existence of solutions for minimax inequalities.
Corollary 3.8 {Ky Fan Minimax Inequality) Let X be a non-empty subset of a non-empty compact convex set Y in a topological vector space and tP : X x Y -+
lR U { -00, +oo} be such that;
(a) the mapping y ...... f(x,y) is lower semicontinuous for each given x E Xi
(b) for each A E F(X) and eachy E co(A), min"'EAtP(x,y) ~ O. Then there exists y' E Y such that tP(x,y*) ~ 0 for all x E X.
4. Ky Fan Minimax Inequality-I
Ky Fan's minimax inequality [66] has become a versatile tool in nonlinear and convex analysis. In this part, we shall first give a minimax theorem which includes those generalizations of Ky Fan's minimax inequality due to Aubin [4], Aubin and Ekeland [9], Ben-El-Mechaiekh et al [21], Chang [34], Deguire [38], Ding and Tan [44], Ding and Tarafdar [50], Dugundji and Granas [53], Fan [66], Granas [77], Lassonde [104], Park [125], Shih and Tan [136], Tan and Yuan [147], Tarafdar [153], and Yen [170]. Then several equivalent fixed point forms are then formulated.
For convenience of our discussion, we first state the following FKKM Lemma again which is a special case of Theorem 2.17.
FKKM Lemma. Let X and Y be non-empty sets in a topological vector space E and F : X -+ 2Y be such that
(i) for each x E X, F(x) is closed in Yi
(ii) for each A E F(X), co(A) c n"'EA F(X)i
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(iii) there exists an Xo E X such that F(xo) is compact.
Then nxEX F(x) =1= 0.
Theorem 4.1 Let X be a nonempty convex subset of a topological vector space E and f, 9 : X x X --; JR U {-oo, +oo} be such that
(1) f(x,y)::; g(x,y) for each (x,y) E X x X;
(2) for each A E F(X) and y E coA, minxEA g(x, y) ::; 0;
(3) there exist a nonempty compact convex subset Xo of X and a non empty compact subset K of X such that for each y E X \ K, there is an x E co(Xo U {y}) with y rt clx{y EX: g(x,y)::; O}.
Then nXEX clx{y EX: f(x, y) ::; O} n K =1= 0.
Proof. Define F, G : X --> 2x by
F(x) := clx{y EX: f(x,y) ::; O} and G(x) := clx{y EX: g(x,y)::; O}
for each x EX. Then G(x) C F(x) for each x E X by (1). In order to derive the conclusion, it suffices to prove that the family {F(x) n K : x E X} has the finite intersection property. For each A E F(X), let D := co(Xo U A). Then D is non-empty compact and convex. We first note that by (2), g(x,x) ::; 0 for each x E X. Define H : D --; 2D by H(x) = {y ED: g(x,x) ::; O} for each xED. Then we have (a): D(x) =1= 0 for each xED and (b): coA C UxEAD(x) by (2) again. Otherwise there exists x E co A such that g(x, x) > 0, which is impossible. By the FKKM Lemma, nxED clD H(x) =1= 0. Taking any Yo E clD H(x). Then the condition (3) implies that Yo E K, so that nxEAH(x) n K =1= 0. As clD H(x) C F(x) for each xED, we have nxEAF(x) n K =1= 0. That is, the family {F(x) n K : x E X} has the finite intersection property. Therefore nxEx F( x) =1= 0. •
In order to study Ky Fan type minimax inequalities for functions which may not be lower semicontinuous, we introduce the following definition:
Definition 4.2 Let X be a non-empty set, Y be a topological space and A E JR. A function f : X x Y --; JR U {-oo, +oo} is said to be A-transfer lower (resp., upper) semi continuous on Y if for each (x, y) E X x Y with f(x, y) > A (resp., f(x, y) < A), there exists Xl E X and a non-empty open neighborhood N(y) of y in Y such that f(x l , w) > A (resp., f(x l , w) < A) for all W E N(y). Moreover f is said to be A-transfer compactly lower (resp., upper) semi continuous on Y if f is A-transfer lower (resp., upper) semicontinuous on each non-empty compact subset of Y.
It is clear that if f is lower (resp., upper) semicontinuous on Y, then f is A-transfer lower (resp., upper) semicontinuous on Y for each A E JR.
We also recall the following definition which is in fact equivalent to the definition 8 of Zhou and Tian [181J.
Definition 4.3 Let X be a non-empty set and Y a topological space. A set-valued mapping F : X --> 2Y is then (1): said to be transfer (resp., compactly) closed valued
Fixed point theorems for set-valued mappings 655
on X if for each (x,y) E X x Y with y 1. F(x), there exist an x' E X and a nonempty (resp., compactly) open neighborhood N(y) of y in Y such that w 1. F(x') for all W E N(y); (2): F is said to be transfer (resp., compactly) open valued if the mapping x ...... Y \ F(x) is transfer (resp., compactly) closed valued and (3): F is said to be transfer (resp., compactly) open valued if the mapping FI : X ---> 2Y defined by FI(X) := Y \ F(x) for each x E X is transfer (resp., compactly) closed valued.
Of course, each set-valued mapping with closed (resp., open) values is compactly transfer (resp., open) closed valued. It will be convenient to have the following dual version of Definition 4.3:
Definition 4.4 Let X be a non-empty set and Y be a topological space. A setvalued mapping G : Y -> 2x is said to be transfer (resp., compactly) open inverse valued provided for each (x,y) E X x Y with y E G-1 (x), there exist x' E X and a non-empty (resp., compactly) open neighborhood N(y) of y such that N(y) C G-1(x'). Moreover F is said to be transfer (resp., compactly) closed inverse valued if the mapping FI : Y ...... 2x defined by FI(y) := X\F(y) for each y E Y is transfer (resp., compactly) open inverse valued.
Let X be a non-empty set (without any topology) and Y be a topological space. From the definitions, it is clear that a set-valued mapping G : Y ...... 2x is transfer (resp., compactly) open inverse valued if and only if the mapping F : X ...... 2Y defined by F(x):= Y\ G-1 (x) for each x E X is transfer (resp., compactly) closed valued on X.
Before we prove some general minimax inequalities which are extensions of the Ky Fan minimax inequality, we need the following lemma:
Lemma 4.5 Let X be a non-empty set, Y a topological space and f : X x Y ->
JRU {-oo, +oo} be an extended real-valued function. For each given A E JR, define a setvalued mapping F). : X ...... 2Y by F).(x):== {y E Y: f(x,y) :s: A} for each x EX. Then nxEx F).(x) = cly F). (x) if and only if the mapping f is A-transfer lower semi continuous on Y. In particular, if nxEx cly F).(x) is compact, then nxExF).(x) = cly F>.(x) if and only if f is A-transfer compactly lower semicontinuous on Y.
Proof. Sufficiency: It is clear that nXEXF).(X) C nxEX cly F).(x). It suffices to prove that nxExclF).(x) C nXExF).(x). Suppose that y rj. nXExF).(x). Then there exists some x E X such that y 1. F).(x), i.e., f(x,y) > A. As f is A-transfer lower semicontinuous on Y, there exist x' E X and a non-empty open neighborhood N(y) of y such that f(x',w) > A for all w E N(y). Then w rj. F).(x') for all wE N(y) and we have y 1. cly F).(x'). Note that N(y) is a non-empty open neighborhood of y, so that y 1. nxEx cly F).(x). Therefore nxEXF).(X) = nxEx cly F).(x).
Necessity: Suppose there exists (x,y) E X x Y such that f(x,y) > A, i.e., y 1. F).(x). Then y 1. nXExF>.(x) = nxEx cly F).(x) , so that there exists Xl E X such that y 1. cly F(x'). But then there exists an open neighborhood N(y) of y in Y such that N(y)nF).(x') = 0. Therefore w rj. F>.(XI) for all wE N(y). This means that f(x',w) > A for all w E N(y), so that f is A-transfer lower semicontinuous on Y.
In the case that nxEx cly F>.(x) is compact, it is easy to check that the preceding arguments are still applicable, so the function f is A-transfer compactly lower semi continuous. •
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A result similar to Lemma 4.5 was also given by Zhou [178J. Now as an immediate consequence of Theorem 4.1, we have the following result. Here and for the remainder of this section X denotes a non-empty convex subset of a topological vector space.
Theorem 4.6 Let ¢, 1/J : X x X -> R U {-oo, oo} be such that
(a) ¢(x,y)::; 1/J(x,y) for each (x,y) E X x X;
(b) ¢ is O-transfer compactly lower semicontinuous in its second variable;
(c) for each A E F(X) and for each y E corA), minxEA 1/J(x,y) ::; 0;
(d) there exist a non-empty closed and compact subset K of X and Xo E X such that 1/J(XO,Y) > 0 for all y EX \ K.
Then there exists y E K such that ¢(x,y)::; 0 for all x E X.
Proof. From the condition (d), fj rt- {y EX: ¢(xo, y) ::; O} for each fj E X \ K. Hence {y EX: ¢(XO,y) ::; O} c K. As K is closed, we must have that
clx{y EX: ¢(XO,y)::; O} C K.
Therefore for each fj E X \ K, fj rt- clx{y EX: ¢(XO,y) ::; O}. Now let Xo := {xo} in Theorem 4.1, it follows that nXExclx{y EX: ¢(x,y)::; O} =J 0. As ¢ is O-transfer lower semicontinuous on its second variable, Lemma 1.2 implies that
n {y EX: ¢(x,y) ::; O} = n clx{y EX: ¢(x,y) =J O} =J 0. xEX XEX
Thus there exists fj E K such that ¢( x, YO) ::; 0 for all x EX. • Theorem 4.6 has a number of equivalent formulations; for example.
Theorem 4.6' (First Geometric Form). Let B,D C X x X be such that
(a) BCD;
(b) the mapping Bl : X -> 2x defined by Bl(X) := {y EX: (x,y) E B} for each x E X is transfer compactly open valued;
(c) for each A E F(X) and for each y E corA), there exists x E A such that (x,y) rfD;
(d) there exist a non-empty closed and compact subset K of X and Xo E X such that (xo, y) E D for all y E X \ K.
Then there exists fj E K such that {x EX: (x, fj) E B} = ¢.
Sketch of proofs.
(1) Theorem 4.6 ==? Theorem 4.6': Let ¢,1/J : X x X -> lR be the characteristic function of B, D respectively.
(2) Theorem 4.6' ==? Theorem 4.6: Define B := {(x,y) E X x X : ¢(x,y) > O} and D:= {(x,y) E X x X: 1/J(x,y) > O}.
Fixed point theorems for set-valued mappings 657
Lemma 4.7 Let ¢,'I/J: X x X ---> Ru {-oo,+oo} be such that
(i) 'I/J(x,x) ~ 0 for each x EX;
(ii) for each y E X, the set {x EX: 'I/J(x,y) > O} is convex.
Then for each A E F(X) and for each y E coCA), min"'EA 'I/J(x,y) ~ O.
Proof. Suppose the conclusion were false, then there exist A E F(X) and y E coCA) such that min"'EA 'I/J(x,y) > O. It follows that A C {x EX: 'I/J(x,y) > O} so that y E coCA) c {x EX: 'I/J(x,y) > O} by (ii) and 'I/J(y,y) > 0, which contradicts (i). Therefore the conclusion must hold. •
In view of Lemma 4.7, Theorem 4.6 implies the following.
Theorem 4.8 Let ¢, 'I/J : X x X ---> R U {-oo, +oo} be such that
(a) ¢(x,y) ~ 'I/J(x,y) for each (x,y) E X x X and 'I/J(x,x) ~ 0 for each x EX;
(b) ¢ is O-transfer compactly lower semicontinuous on its second variable;
(c) for each fixed y E X, the set {x EX: 'I/J(x,y) > O} is convex;
(d) there exist a non-empty closed and compact subset K of X and a point Xo EX such that 'I/J(xo, y) > 0 for all y E X \ K.
Then there exists ii E K such that ¢(x, ii) ~ 0 for all x EX.
The following is a fixed point version of Theorem 4.8.
Theorem 4.8'. Let P, Q : X ...... 2x be such that
(a) for each x E X, P(x) c Q(x)j
(b) P is transfer compactly open inverse valuedj
(c) for each y EX, Q(y) is convex;
(d) there exists a non-empty closed and compact subset K of X and Xo E X such that X \ K c Q-l(xO);
(e) for each y E K, P(y) i- 0.
Then there exists a point x E X such that x E Q(x).
Theorem 4.8". Let P, Q : X ---> 2x be such that
(a) for each x E X, P(x) c Q(x)j
(b) P is transfer compactly open inverse valued;
(c) there exist a non-empty closed and compact subset K of X and Xo E X such that X \ K C (coQ)-l(xO);
(d) for each y E K, P(y) i- 0.
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Then there exists x E X such that x E coQ(x).
Theorem 4.8111• Let P, Q : X -. 2x be such that
(a) for each x E X, P(x) c Q(x);
(b) P is transfer compactly open inverse valued;
(e) there exist a non-empty closed and compact subset K of X and Xo E X such that X \ K C Q-I(xO);
(d) for each y E K, Ply) f. 0.
Then there exists x E X such that x E coQ(x).
Theorem 4.8''''. Let Q : X -. 2x be such that
(a) for each y E X, Q-I(y) contains an subset Oy which is open in each non-empty compact subset C of X and UyEXOy = X;
(b) there exist a non-empty closed and compact subset K of X and a point Xo E X such that Xo E co(Q(y)) for all y E X \ K.
Then there exists a point x E X such that x E co(Q(x)).
Theorem 4.8111". Let X be a non-empty convex subset of a topological vector space
and Q : X -. 2x be such that
(a) for each x E X, Q(x) is convex;
(b) for each y EX, Q-I(y) contains a subset Oy (which may be empty) of X which is compactly open in X;
(c) there exist a non-empty closed and compact subset K of X and a point Xo E X such that Xo E Q(y) for all y E X \ K and K C UyEXOy.
Then there exists a point x E X such that x E Q(i).
Indication of proofs.
(1) Theorem 4.8 ===> Theorem 4.8': Define ¢,1/1 : X x X -. R by
¢(X,y):={ 1, ifxEP(y), 0, if x ¢ Ply);
and ( ) {I, if x E Q(y),
1/1 x,y := 0, ifx¢Q(y).
(2) Theorem 4.8' ===> Theorem 4.8: Define P, Q : X -. 2x by
PlY) := {x EX: ¢(x,y) > O} and Q(y):= {x EX: 1/1(x,y) > O}
for each y E X respectively.
(3) Theorem 4.8' <==> Theorem 4.8" <==> Theorem 2.4.5'" is obvious.
(4) Theorem 4.8" ===> Theorem 4.8"": Define P : X -. 2x by
P(x):={YEX:XEOy} for each xEX.
(5) Theorem 4.8"" ===> Theorem 4.8": Let Oy := P-l(y) for each y E X.
(6) Theorem 4.8"" <==> Theorem 4.8""': Obvious.
Fixed point theorems for set-valued mappings 659
5. Ky Fan Minimax Inequality-II
As an application of Theorem 4.1 and Lemma 4.5, we have the following Ky Fan type minimax inequality.
Theorem 5.1 Let X be a nonempty convex subset of a topological vector space E and f, g : X x X -> lR U {-oo, +oo} be such that
(1) f(x,y) :::; g(x,y) for each (x,y) E X x X;
(2) f is O-transfer compactly lower semicontinuous on its second variable;
(3) for each A E F(X) and y E coA, minxEAg(x,y) :::; 0;
(4) there exist a nonempty compact convex subset Xo of X and a nonempty compact subset K of X such that for each fj E X \ K, there is an x E co(Xo U {fj}) with fj rt. clx{y EX: g(x,y) :::; O}.
Then there exists a fj E K such that f (x, fj) :::; 0 for all x EX.
Proof. By Theorem 4.1 and our condition (4), nxExclx{y EX: f(x,y):::; O}nK oj 0. As f is O-transfer compactly lower semicontinuous on its second variable, Lemma 4.5 implies that nxEx clx{y EX: f(x, y) :::; O} n K = nxEX{y EX: f(x, y) :::; O} n K. Take any fixed fj E nxEX{y EX: f(x,y) :::; O}. Then sUPxEX f(x,fj):::; O. •
Theorem 5.2 Let X be a non-empty convex subset of a topological vector space and ¢,1jJ : X x X -> R U { -00, +oo} be such that
(a) ¢(x,y):::; 1jJ(x,y) for each all x,y E X, and 1jJ(x,x) :::; 0 for all x E X;
(b) ¢ is O-transfer compactly lower semicontinuous on its second variable;
(c) for each y E X, the set {x EX: 1jJ(x,y) > O} is convex;
(d) there exist a non-empty compact convex subset Xo of X and a non-empty compact subset K of X such that for each y E X \ K, there exists an x E co(Xo U {y} ) with y rt. clx{w EX: 1jJ(x,w):::; O}.
Then there exists a fj E K such that ¢(x, fj) :::; 0 for all x EX.
Proof. By (a) and (c), Lemma 4.7 implies that for each A E F(X) and for each y E co(A), minxEA¢(x,y) :::; O. Then ¢ and 1jJ satisfy all hypotheses of Theorem 2.l. By Theorem 2.1, there exists a fj E K such that ¢(x,fj) :::; 0 for all x E X. •
As an immediately consequence of Theorem 5.2, we have the following minimax inequality.
Corollary 5.3 Let X be a non-empty convex subset of a topological vector space and ¢ : X x X -> R U { -00, +oo} be such that
(a) ¢(x,x) :::; 0 for all x E X;
(b) for each fixed x EX, Y I-> ¢(x, y) is compactly lower semicontin1LDus;
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(c) for each y E X, the set {x EX: ¢(x,y) > O} is convex;
(d) there exist a non-empty compact convex subset Xo of X and a non-empty compact subset K of X such that for each y EX \ K, there exists an x E co(Xo U {y}) with y ¢ {w EX: ¢(x,w)::; O}.
Then there exists ayE K such that ¢(x,fj) ::; 0 for all x E X.
Proof. Let 1jJ(x,y) := 1jJ(x,y) in Theorem 5.2. Then the conclusion follows from Theorem 5.2. •
Example 5.4 Let X = [0,00) and ¢,1jJ :--+ R be defined by
¢(x,y) = ' {I if(x,Y)EXXX:x<y<x+2~xforO::;x<l, 0, otherwise;
and
1jJ(x,y) = { 1, if (X,Y).E X x X and x> y, 0, otherwIse.
Then all the conditions (a), (b), (c) and (d) in Theorem 4.6 are satisfied with K =
{O} = {xo} so that there exists y = 0 E X such that ¢(x, fj) ::; 0 for all x EX. However, there does not exist a non-empty compact convex subset Xo of X and a non-empty compact subset K of X such that for each y E X \ K, there exists an x E co(Xo U {y}) with ¢(x,y) > o. Thus Corollary 5.3 is not applicable.
Example 5.5 Let X := [0,00) and ¢,1jJ : X x X --+ R be defined by
¢(x,y) =1jJ(x,y) = { 1, ify>x.ory>x+3~X' 0, otherWIse.
Then all the conditions (a), (b), (c) and (d) of Corollary 5.3 are satisfied with Xo = K = {O} and 1jJ(x,y) = ¢(x,y) for each (x,y) E X x X. Therefore there exists y EX such that ¢(x, fj) ::; 0 for all x EX. However, there does not exist a non-empty subset K of X and Xo E X such that 1jJ(XO,Y) > 0 for all y E X \ K. Therefore Theorem 4.6 is not applicable and hence the above two examples show that Theorem 4.6 and Corollary 5.3 are not comparable.
Theorem 5.1 has a number of equivalent formulations; for example.
Theorem 5.1' (First Geometric Form). Let X be a non-empty convex subset of a topological vector space and B, D c X x X be such that
(a) BcD;
(b) the mapping Bl : X --+ 2x defined by BI(x) := {y EX: (x,y) E B} for each x E X is transfer compactly open valued;
(c) for each A E F(X) and for each y E coCA), there exists x E A such that (x, y) II D:
Fixed point theorems for set-valued mappings 661
(d) there exist a non-empty compact and convex subset Xo of X and a non-empty compact subset K of X such that for each y E X \ K, there exists x E co[Xo U {y}] with y rt- clx{w EX: (x,w) rt- D}.
Then there exists fj E K such that {x EX: (x, fj) E B} = cp.
There are various ways to recast Theorem 5.2 as a fixed point theorem; for example.
Theorem 5.2'. Let X be a non-empty convex subset of a topological vector space and P, Q : X ----+ 2x be such that
(a) for each x E X, P(x) c Q(x);
(b) P is transfer compactly open inverse va.lued;
(c) for each y E X, Q(y) is convex;
(d) there exist a non-empty compact and convex subset Xo of X and a non-empty compact subset K of X such that for eachy E X\K, there exists x E co[XoU{y}] with y rt- clx[X \ Q-l(x)];
(e) for each y E K, P(y) # 0.
Then there exists a point x E X such that x E Q(x).
Theorem 5.2". Let X be a non-empty convex subset of a topological vector space and Q : X -+ 2x be such that
(a) for each y EX, Q-l(y) contains an subset Oy wh'ich is open in each non-empty compact subset C of X and UyExOy = X;
(b) there exist a non-empty compact and convex subset Xo of X and a non-empty compact subset K of X such that for each y E X \ K, there exists x E co[Xo U {y}] with y rt- clx[X \ (coQ)-l(x)],
Then there exists a point x E X such that x E co(Q(x)).
Theorem 5.2"'. Let X be a non-empty convex subset of a topological vector space and Q : X ----+ 2x be such that
(a) for each x E X, Q(x) is convex;
(b) for each y E X, Q-l(y) contains a subset Oy (which may be empty) of X which is compactly open in X;
(c) there exist a non-empty compact and convex subset Xo of X and a non-empty compact subset K of X such that for each y E X\K, there exists x E co[XoU{y}] with y rt- clx[X \ (coQ)-l(x)].
Then there exists a point x E X such that x E Q( x).
The following example shows that a transfer open inverse mapping may not have open inverse values.
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Example 5.6 Let X := [0,1] and F : X ~ 2x is defined by
F(x) := { [x,l] , [0, 1] ,
if x is rational;
if x is irrational.
Then it is clear that F is not open inverse valued but F is transfer open inverse valued.
6. Fan-Glicksberg Fixed Points in G-Convex Spaces
In this section we establish Fan-Glicksberg type fixed point theorems for upper semicontinuous set-valued mappings with closed acyclic values in the setting of another abstract convexity structure - called a locally G-convex space. This notion of convexity generalizes the usual convexity found in locally H-convex spaces, hyperconvex metric spaces, and locally convex topological vector spaces.
A topological space X is said to be contractible if the identity mapping Ix of X is homotopic to a constant function. We recall that a non-empty space is said to be acyclic if all of its reduced Cech homology groups over the rationals vanish. In particular, each contractible space is acyclic and thus any non-empty convex or star-shaped set is acyclic.
In the last few years, many papers have been devoted to prove the existence of fixed points for set-valued mappings in nonlinear analysis without traditional linear structures. As a result, several generalizations of convexity notions, such as hyperconvex metric spaces, H-spaces, G-convex spaces have been introduced and studied by Aronszajn and Panitchpakdi [3], Bardaro and Ceppitelli [13], Bielawaski [24], Horvath [81][85], Kirk et al. [96], Park [119]-[123], Park and Kim [130], Tarafadr [156], Tarafdar and Watson [157], Wu [165], Yuan [174] and many others. Recently, in order to cover the general economic models without linear convex structures, Park and Kim [130] introduces another abstract convexity notion-called G-convex spaces which includes many abstract convexity notions such as H-convex spaces as a special case. The formal definition is due to Park [123] (see also Park and Kim [130]).
Definition 6.1 A generalized convex space, or say, a G-convex space (X, D; r) consists of a topological space X, a non-empty subset D of X and a function r : F(X) ~ X\ {0} such that
(1) for each A, BE F(X), r(A) c r(B) if A c B; and
(2) for each A E F(X) with IAI = n + 1, there exists a continuous function cPA Lln ~ r(A) such that cPA(LlJ ) c r(J) for each 0 =J J c {0,1,'" ,n}, where A = {XO,Xl,'" ,Xn}, and LlJ denotes the face of Lln corresponding to the subindex of J in {O, 1, 2,'" ,n}.
A subset C of the G-convex space (X, D; r) is said to be G-convex iffor each A E F(D), rAe C for all Ace.
Throughout this section, for A = {XQ,Xl,X2,'" ,Xn} E F(X), we use r A or rN (where N = {O, 1, 2"" ,n}) to denote r(A). A space X is said to have a G-convex structure if and only of X is a G-convex space. By Lemma 2.5, it is clear that the notion of G-convex spaces includes the corresponding H-spaces as a special class, which, in turn, includes topological vector spaces as special cases. We now introduce the notion of a locally G-convex space.
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Definition 6.2 A G-convex X is said to be a locally G-convex space if X is a uniform topological space with uniformity U which has an open base fJ := {Vi : i E J} of symmetric neighbourhoods such that for each V E fJ, the set Vex) := {y EX: (y,x) E V} is a G-convex set for each x EX.
Let X and Y be two topological spaces. We recall that a set-valued mapping F : X ---> 2Y is said to have transfer open inversed values, if X = UyEY int(F-1(y)), where int(F-l(y)) denotes the relative interior of the set F-1(y) := {x EX: y E F(x)} for each y E Y in X; and F is said to be upper semicontinuous if the set {x EX: F(x) c U} is open for each open neighborhood U in Y. When Y is a G-convex space, the mapping T: X ---> 2Y is said to have G-convex values ifT(x) is a G-convex set for each XEX.
We first have the following selection result for a set-valued mapping with transfer open inversed values.
Lemma 6.3 Let Y be a compact topological space and X be a G-convex space. Suppose T : Y ---> 2x \ {0} is an open inverse valued mapping with G-convex values. Then there exist some positive integer n E N and two single-valued continuous mappings <jJ : An --->
X and"p : Y ---> An such that the mapping f : Y ---> X defined by fey) := <jJ("p(y)) for each x E Y is a continuous selection ofT, i.e., fey) E T(y) for all y E Y, where An is the standard n-dimensional simplex with vertices eo, el, ... , en.
Proof. Since Y is compact and T has open inversed values, it follows that Y has a finite open cover {Oi: i = 0,1,··· ,n}, where Oi C int(F-1(Xi)) for some Xi EX. Let A = {XO,Xl,··· ,xn }. Suppose {"pi}f=o is a partition of unity the corresponding to the finite covering {Oi : i = 0,1,··· ,n}, that is, "pi(y) ~ 0 and L,i"pi(y) = 1 and if "pi(Y) > 0, then y E Oi. Set J(y) := {i E J : "pi(y) > O} for each y E Y. Then for each i E J(y), it follows that yEO; C T-1(Xi), so that Xi E T(y). This means An T(x) =1= 0 for all x E X. Secondly, the mapping "p : Y ---> An defined by "p(y) := L,~=o "pi (y)ei for each y E Y is continuous. Since X is a G-convex space, there exists a continuous mapping <jJ : An ---> r A such that <jJ(AJ) c r J for each nonempty finite subset J in {O, 1, ... , n}. Then the mapping f : Y ---> X defined by fey) := rfJ(1fJ(y)) for each y E Y, is continuous. Since T(y) is G-convex, it follows that fey) := <jJ("p(y)) C <jJ(AJ(y)) c r J(y) c T(y) for each y E Y (as r J(y) C T(y)), i.e., fey) E T(y) for each y E Y. •
By employing the arguments of Eilenberg and Montgomery [55, p.106-107], and using Theorem 6.3 of Gorniewicz [75, p.111] in his paper published in 1975, instead of the coincidence theorem used in [55, p.106-107], the following lemma was obtained by Shioji [138, p.188]. As it involves a long proof parallel to the one used by Eilenberg and Montgomery in their paper published in 1946, we omit the details here (see also Lemma 2.1 of Park et al. [129]).
Lemma 6.4 (Eilenberg and Montgomery, Gomiewicz and Shioji and Park). Let AN be an n-dimensional simplex with the Euclidean topology and Y be a compact topological space. Let"p: Y ---> AN be a single-valued continuous mapping and T : AN ---> 2Y
be a set-valued upper semicontinuous mapping with non-empty compact acyclic (e.g., contractible) values. Then there exists xo E AN such that Xo E 1fJ 0 T(xo)·
We now have the following fixed point theorem in locally G-convex spaces given by Yuan [173], which is a generalization of the Fan-Glicksberg type fixed point theorems
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for upper semicontinuous set-valued mappings with non-empty closed acyclic values found in several papers (e.g., see Kirk and Shin [95], Park [119]-[123], Tarafdar [154], Wu [165] and others in locally convex spaces).
Theorem 6.5 Let X be a compact locally G-convex space and F : X -+ 2x be an upper semicontinuous set-valued mappings with non-empty closed acyclic values. Then F has a fixed point, i.e., there exists Xo E X such that Xo E F(xo).
Proof. Since (X, r A) is a locally Hausdorff G-convex space, X has a uniform structure U and without loss of generality, suppose V := {V; : i E I} is a symmetric and open base family for the uniform structure U. Then for each base V; and x EX, the set V;(x) := {y EX: (y,x) E V;} = {y EX: (x,y) E V;} is an open G-convex set, and the niEIV; = .6. = {(x,x) : x E X}. For each V;, we define a set-valued mapping T; : X -+ 2x by T;(x) := {y EX: (y, x) E V;) = V;(x) for each x EX. Then it is clear Ti is a set-valued mapping from X to X with non-empty G-convex values and UyEX int(Ti- 1(y)) c X because for each y E X, the set
Ji-l(y) = {x EX: (x,y) E V;}
is open in X, so that int(T;-l(y)) = Ti- 1(y). By Lemma 6.3, there exist some positive integer n EN and two (single-valued) continuous mappings
gi : .6.n -+ X and <Pi: X -+ .6.n
such that fi(X) := gi(<Pi(X)) E Ti(X) for each x E X, where .6.n is a standard ndimensional simplex. Now we define another set-valued mapping P; : .6.n -+ 2x by P;(x) := F(gi(X)) for each x E .6.n . Then P; is an upper semicontinuous set-valued mapping with non-empty closed and acyclic values. By Lemma 6.4, the mapping
<Pi 0 P; : .6.n -+ 2lin
has a fixed point, i.e., there exists uv, E .6.n such that U v, E ¢>i (P; (uv.)). Then there exists xv, E Pi(uv,) = F(gi(Uv.)) c X and uv, = <Pi(Xv.). Let
Yv, = gi(Uv.) = gi(<Pi(Xv.)) = fi(Xv.) E Ti(xv.).
Thus xv, E F(yv.), i.e., (xv"yv.) E GraphF and (xv"Yv.) E V;, where GraphF denotes the graph of the mapping F. Since X x X is compact, we may assume that {xv" Yv, }iEI converges to (xo, yo). Note that F is an upper semicontinuous mapping with non-empty closed values, it follows that the graph of F is closed and thus (xo, yo) E GraphF, i.e., Yo E F(xo). In order to finish the proof, it suffices to show that xo = yo. Since {V;};EI is an open symmetric base family for the uniform structure U and
lim(xv"yv.) = (xo,Yo), iEI
without loss of generality, there exists Va E {V; : i E I} such that (XVj , xo) E Vj and (YVj' Yo) E Vj for all Vj E {V; : i E I} with Vj c Va. Now for any U E {V; : i E I} with U c Vo, let V" E {V; : i E I} with V" c Va be such that V" 0 Vu 0 V" c U. Since (xvu , YvJ E V", it follows that (xo, YvJ E V" 0 Vu and thus (xo, Yo) c V" 0 V" 0 V" c U. Therefore we have show that (xo, Yo) E U for all U E {V; : i E I} with U c Va. Note that n{U : U E V, and U C Va} = n{v; : i E I} =.6.. This implies that xo = yo and thus Xo is a fixed point of F. •
It is clear each locally convex H-space is a locally G-convex space. Note that if X is a compact convex subset of a locally convex Hausdorff topological vector space E and
Fixed point theorems for set-valued mappings 665
if r A := co(A) for each A E F(X), then (X, r A) is a locally convex H-space. Thus Theorem 6.5 is really a generalization of celebrated Fan-Glicksberg fixed point theorem for upper semicontinuous set-valued mappings with non-empty closed and acyclic (e.g., contractible) values in both locally convex H-spaces, I.c.spaces (locally H-convex spaces) and locally convex topological vector spaces. Theorem 6.5 shows that Theorem 2.1 of Tarafdar [156] is true when a mapping F is only upper semicontinuous. Theorem 6.5 also generalizes Theorem 3.2 of Tarafdar and Watson [157] and the corresponding Fan-Glicksberg type fixed point theorems given by Wu [165] to locally G-convex spaces.
Definition 6.6 An H-space (X, r A) is said to be a locally convex H -space if X is a uniform topological space with the uniform structure U and there exists a base family {V; : i E I} for U such that for each V; E {V; : i E I}, the set
V;(x) := {y EX: (y,x) E V;}
is H -convex for each x EX.
Definition 6.6 includes the l.e.-spaces of Horvath [84] (see also Tarafdar [156] and [157]). Each non-empty convex subset S in locally convex spaces has a uniform structure U induced by the family of seminorms; on the other hand, each convex S in topological vector spaces has a natural H-structure r A given by r A := co(A) for each non-empty finite subset A E F(S). Thus locally convex H-spaces include locally convex topological vector spaces as a special case.
The following definition of locally convex H -spaces for metric spaces corresponds to Definition 6.6.
Definition 6.7 Let the metric space (Y, r) be an H -space with the metric d. Then Y is said to be a locally metrically convex H -space if for each r > 0, the set
{y E Y: d(y,E) < r}
is H -convex for each H -convex set E in Y and also all open balls are H -convex.
It is clear each locally convex H-space is a locally G-convex space. Note that if X is a compact convex subset of a locally convex Hausdorff topological vector space E and r A := co (A) for each A E F(X), then (X, r A) is a locally convex H-space. Thus Theorem 6.5 is really a generalization of celebrated Fan-Glicksberg fixed point theorem for upper semicontinuous set-valued mappings with non-empty closed and acyclic (e.g., contractible) values in both locally convex H-spaces, I.c.spaces (locally H-convex spaces) and locally convex topological vector spaces. Theorem 6.5 shows that Theorem 2.1 of Tarafdar [156] is true when the mapping F is only upper semicontinuous. Theorem 6.5 also generalizes Theorem 3.2 of Tarafdar and Watson [157] and the corresponding Fan-Glicksberg type fixed point theorems given by Wu [165] to locally G-convex spaces.
Thus we have the following Fan-Glickberg fixed point theorem which is first given by Wu [165]
Theorem 6.8 Let (X, r A} be a compact Hausdorff locally convex H -space. Suppose F : X ~ 2x is an upper semieontinuous set-valued mapping with non-empty closed acyclic values. Then F has at least one fixed point.
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Remark 6.9 Although we have established several fixed point theorems for upper semicontinuous (resp.,hemicontinuous) set-valued mappings in topological vector spaces which have sufficient enough continuous linear functionals, it is clear that a Hausdorff topological vector space may not have enough continuous linear functionals (e.g, the Hardy space HP for p E (0,1)), thus Schauder's famous conjecture which asks 'if a continuous (even single-valued) mapping defined on a non-empty compact and convex subset of a Hausdorff topological vector space has a fixed point' remains open since 1934. For more historic details on Schauder's conjecture and its development, see Dugundji and Granas [54], Granas [77], Idzik [87], Park [117] and references therein.
The celebrated Ky Fan minimax inequality has numerous applications in various and diverse fields of mathematics and applied science. It also has numerous equivalent formulations and generalizations. For example, this chapter gives us a picture and relationships between the Brouwer fixed point theorem [26), Sperner's lemma [146], KKM lemma, Fan's geometric lemma [59], Browder-Fan fixed point theorem, the FanGlicksberg fixed point theorem, Ky Fan type minimax theorems and the equivalent forms of the Ky Fan minimax inequality, one of the most important principles in nonlinear analysis. In this line, more details have been illustrated by Aubin [7], Aubin and Ekeland [9], Border [25], Park [122), [174] and references therein. Nonlinear analysis began early this century with the birth of the Brouwer fixed point. However, the Ky Fan minimax inequality is a more convenient form of this fact as it can be used to imply a whole series of existence theorems for solutions of nonlinear equations or inclusions (see Aubin [7, p.135]) in a very general form. Such theorems are very useful in many applications, particularly in optimization theory, mathematical economics, and game theory as we can see from Yuan [174]. The book of Singh et a1. [143] also gives an extensively study of fixed point theory and best approximations using the KKM-map principle.
7. Nonlinear Analysis of Hyperconvex Metric Spaces
The aim of this section is to discuss general KKM theory and its various applications in metric hyperconvex spaces.
7.1. Fan-Glicksberg Fixed Points in Hyperconvex Metric Spaces
Sine [140] and Soardi [145] proved independently that the fixed point property for nonexpansive mappings holds in bounded hyperconvex spaces. Since then many interesting results for nonexpansive mappings have been established in the framework of hyperconvex structures, e.g., see Baillon [11), Goebel and Kirk [74], Khamsi [92), Sine [140][141] and others. In this section, we first show that each hyperconvex metric space is indeed a complete locally metrically H-space (thus, it is a locally convex H-space) and then using fixed point theorem established in the previous section, we give a hyperconvex version of fixed point theorem for upper semicontinuous set-valued mappings with closed acyclic values in hyperconvex metric spaces, which, in turn, unifies the corresponding fixed point theory of set-valued mappings in the existing literature.
For the definition and basic properties of hyperconvex metric spaces, see the article by R. Espinola and M. A. Khamsi elsewhere in this Handbook. We also adopt the notation of that article. In particular cov(A) will denote the admissible cover of A ~ M; thus
cov(A) = n{B eM: B is a closed ball such that A c B},
Fixed point theorems for set-valued mappings 667
and A := {A eM: A = cov(A)} will denote the family of all admissible subsets of M.
Lemma 7.1 Each hyperconvex metric space (Y, d) is a complete locally metrically convex H -space (of course, (Y, d) is a locally convex H -spaces).
Proof. Following Horvath in [85], for each y E Y, F((y)) = {y} which, of course, is an F-set. We show that for any H- convex set E and any r > 0, the set
{y E Y: d(y, e) < r}
is an H-set. Let Yo,' " ,Yn E Y such that d(Yi, E) < r and Yo E F((Yl,' .. ,Yn)), i.e., Yo belongs to any closed ball containing {Yl,' .. ,Yn}' We want to show that d(yo, E) < r. Take any fixed points fh,'" ,Yn E E such that d(Yi,Yi) < r for i = I",' ,no Since F({Vl,'" ,Vn}) C E, it suffices to show that d(yo,F({Y1,''' ,Yn})) < r. By the definition of F({Yl,'" ,Vn}), F({Yl,'" ,Yn}) = niEIB(ui,ri), where (ui,ri) E Yx R+. Choose r' E (0, r) such that d(Yi' Vi) < r' for i = 1,' .. ,n. As {ZIl,' .. ,Yn} C Bi(Ui, ri) for i E I, it follows that {Yl,' .. ,Yn} C B(Ui, ri +r) for i E I and Yo E niElB(Ui, ri +r'). By Sine's Lemma in [141, p.864], there exists a retraction
R: niEIB(ui,ri + r') -4 niEIB(ui,r;)
such that d(y, R(y)) ::; r' for any Y E niEIB(Ui, ri + r'). Then R(yo) E F( {Yl ... ,Vn}) and d(Yo,R(yo)) ::; r' < r. •
Because each hyperconvex metric space is a locally convex H-space (thus, an H-space), we have the following hyperconvex version of the Fan-Glicksberg Theorem.
Theorem 7.2 Let X E A(M) be a compact subset of a hyperconvex metric space M. Suppose F : X -> 2x is an upper semi continuous set-valued mappings with non-empty closed admissible values. Then F has at least one fixed point.
Proof. M is complete, and by Theorem 8.3.5 of Engelking [57J M has a uniform structure. Thus by Lemma 7.1 X is a locally convex H-space, and the conclusion follows by Theorem 6.8. •
We would like to note that Theorem 7.2 improves the corresponding fixed point theorems given by Horvath [85], Kirk and Shin [95], and Khamsi [92J as special cases.
7.2. KKM Theory in Hyperconvex Metric Spaces
In this section, we establish the general KKM theory in hyperconvex metric spaces. As applications, we give hyperconvex versions of Fan's celebrated minimax principle and Fan's best approximation theorem for set-valued mappings.
Let (M,d) be a metric space. Following Khamsi [92J, a subset SCM is said to be finitely metrically closed if for each F E F(M), the set cov(F) n S is closed. Note that cov(F) is always defined and belongs to A(M). Thus if S is closed in M, it is obviously finitely metrically closed. We also recall that a family {A,}aED of M has the finite intersection property if the intersection of each non-empty finite sub-family is not empty.
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Definition 1.3 Let X be any non-empty set and let M be a metric space. A set-valued mapping G : X -+ 2M \ {0} is said to be a genemlized metric KKM mapping (in short, GMKKM) if for each non-empty finite set {x!,··· ,xn} C X, there exists a sequence of points of M, Yl.··· ,Yn, not necessarily all different, such that for each subsequence {Yil'··· ,Yik}' we have
In particular, the mapping G : X -+ 2M is said to be a metric KKM (in short, MKKM) mapping iffor each finite subset FE F(X), we have cov(F) C UXEFG(X).
As seen in Lemma 7.1 each hyperconvex metric space is an H-spacej thus each GMKKM mapping is a GHKKM mapping. As a direct application of Theorem 2.15 we have the following characterization of GMKKM mappings in hyperconvex metric spaces. This theorem is found in Kirk et al [96].
Theorem 7.4 Let X be a non-empty set and let M be a hyperconvex metric space. Suppose G : X -+ 2M \ {0} is a set-valued mapping with finitely metrically closed values. Then the family {G(x) : x E X} has the finite intersection property if and only if the mapping G is a genemlized metric KKM mapping.
Theorem 7.4 now provides the following characterization.
Theorem 7.5 Let X be a non-empty set and M be a hyperconvex metric space. Suppose G : X -+ 2M \ {0} is a set-valued mapping with non-empty closed values and there exists xo E X such that G(xo) is compact. Then nxExG(x) =f 0 if and only if the mapping G is a genemlized metric KKM mapping.
Proof. Necessity: Since nxExG(x) =f 0, it follows that the family {G(x) : x E X} has the finite intersection property. Since G(x) is closed for each x E X, it is finitely metrically closed. By Theorem 7.4, G is a generalized metric KKM mapping.
Sufficiency: Since G is a generalized metric KKM mapping, it follows by Theorem 7.4 that the family {G(x) : x E X} has the finite intersection property. Rewriting this to {G(x) n G(xo) : x E X} and noting G(xo) is compact, we have
o =f nxEXG(x) n G(xo) = G(xo) nxEx G(x) = nxEXG(x).
• As a special case of Theorem 7.4, we also have the following result which includes Theorem 3 of Khamsi [92, p.303].
Corollary 7.6 Let X be a non-empty subset of a hyperconvex metric space M. Suppose G : X -+ 2M\ {0} is a metric KKM mapping with finitely metrically closed values. Then the family {G(x) : x E X} has the finite intersection property.
By Theorem 7.4 and Lemma 2.10, we have the following characterization of the finite intersection property of set-valued mappings.
Fixed point theorems for set-valued mappings 669
Theorem 7.7 Let X be a non-empty set and let M be a hyperconvex metric space. Suppose G : X -+ 2M\ {0} is a set-valued mapping such that Gis tmnsfer closed valued and there exists a finite subset Xo of X such that the set nxExoclG(x) is non-empty and compact. Then nxExG(x) is non-empty if and only if the mapping clG is a genemlized metric KKM mapping.
Theorems 7.4, 7.5 and 7.7 tell us that in hyperconvex metric spaces the finite intersection property for a family of subsets is equivalent to the existence of a non-empty intersection for a set-valued mapping which is a generalized metric KKM mapping. In order to study in which situations such generalized metric KKM mappings can be derived, we now introduce the following definitions.
Definition 7.8 Let X E F(M) be a non-empty subset of a hyperconvex space M. Then:
(1) the function f : X -+ [-00, +00] is said to be hyper quasi-convex (resp., concave) if the set {x EX: f(x) ::; A} (resp., the set {x EX: f(x) ~ A}) for each A E IR is an admissible set in M.
(2) the function 'l/J(x, y) : X x X -+ [-00, +00] is said to be hyper diagonal quasiconvex (resp., concave) in y if for each A E F(X) and any Yo E cov(A), we have 'l/J(YO, Yo) ::; maxy,EA 'l/J(YO, Yi) (resp., 'l/J(YO, yo) ~ minY,EA 'l/J(YO, Yi)).
(3) 'l/J(x, y) is said to be hyper I-diagonal quasi-convex (resp., concave) in Y for some I E [-00, +00] if for any A E F(X) and Yo E cov(A) , we have that I ::; maxy,EA 'l/J (Yo , Yi) (resp., I ~ minY,EA 'l/J(yo, Yi)).
Remark 7.9 We note that the inequality' ::;' (resp., ' ~') could be replaced equivalently by the strict inequality' <' (reps., ' >') in the Definition 7.8(1). It is clear that if 'l/J(x, y) is hyper diagonal q\lasi-convex (resp., concave) in y, then 'l/J(x, y) must be hyper I-diagonally quasi-convex (resp., concave) in y, where I = infxEx'l/J(x,x) (resp, 1= sUPxEX'l/J(x,x)).
Definition 7.10 Let X E F(M) be a non-empty set in a hyperconvex metric space M and I E (-00, +00]. Suppose 'l/J : M x X -> (-00, +00] is a function. Then 'l/J is said to be hyper I-genemlized quasi-convex (resp., concave) in M if for each nonempty finite subset {Xl,'" ,xn } C X, there exists a sequence Yl,'" ,Yn in M such that for each subsequent Yill'" ,Yik and any xo E COV{Xill'" ,Xik}' we have I ::; maxl~j9'l/J(xO'Yij) (resp., I ~ minl~j~k'l/J(XO,Yij))'
The relevant definitions imply the following.
Lemma 7.11 Let X be a non-empty set, M a hyperconvex metric space and let IE IR be a given real number. Suppose 'l/J : M x X -> (-00, +00] is a mapping. Then the following two statements are equivalent:
(1) the mapping G: X -+ 2M \ {0} defined by G(x) := {y EM: 'l/J(y,x)::; I} (resp., G(x) := {y EM: 'l/J(Y,x) ~ I}) for each x E X is a genemlized metric KKM mapping.
(2) the function 'l/J is hyper I-genemlized quasi-concave (resp., convex) in M.
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We now give the following minimax inequality which is a hyperconvex version of Fan's celebrated minimax inequality principle.
Theorem 7.12 Let X E A(M) be a compact subset of a hyperconvex space M. Suppose 'IjJ : X X X -+ (-00, +ooJ is a mapping such that
(1) for· each fixed x E X, y>--t 'IjJ(x, y) is hyper O-generalized quasi-concave; and
(2) for each fixed y E X, X >--t 'IjJ(x, y) is lower semicontinuous.
Then there exists Xo E X such that sUPYEX 'IjJ(xo, y) ~ O.
Proof. We define a mapping G : X -+ 2x by
G(x) := {y EX: 'IjJ(y,x) ~ O}
for each x E X. Then it is clear that G is a GMKKM mapping by Lemma 7.11. Now by Theorem 7.5, it follows that nxExG(x) i' 0. Take any point Xo E nxExG(x). Then we have sUPYEX'IjJ(xO,y) ~ O. •
As an application of the generalized metric MKKM principle established above, we give the following hyperconvex version of Fan's best approximation in hyperconvex spaces for set-valued mappings.
Theorem 7.13 Let M be a hyperconvex metric space and X E A(M) be a non-empty compact subset of M. Suppose F : X -t A(M) is a set-valued continuous mapping. Then there exists Xo E X such that
d(xo,F(xo)) = inf d(x,F(xo)). xEX
In particular if F(xo) is compact and Xo ric F(xo), Xo must be a boundary point of X (i.e., xo E Bd(X)).
Proof. Define a mapping G : X -t 2M \ {0} by
G(x) := {y EX: d(y, F(y)) ~ d(x, F(y))}
for each x E X. As F is continuous, G(x) is closed and non-empty for each x E X. Now we claim that G is a metric KKM mapping. Suppose it were not. Then there exists a non-empty and finite subset {Xl,· .. , xn } and y E cov( {Xi : i = 1, ... , n}) such that
d(xi,F(y)) < d(y,F(y))
for i = 1,·· . , n. Let E > 0 such that d(Xi, F(y)) ~ d(y, F(y)) - E for i = 1,··· , n. Let r = dist(y, F(y)) - E. Then for i = 1,··· , n, Xi E F(y) + r, where
F(y) + r := U{B(a; r) : a E F(y)}
and B(a,r) is a closed ball centered at a with the radius r. Note that F(y) E A(M), it follows that F(y) + r E A(M) (e.g., see Sine [141, p.864]). Thus
COV{Xl,X2,··· ,xn } C F(y) + r.
This in turn implies y E F(y) + r, that is, dist(y,F(y)) ~ r = dist(y,F(y)) - E, which is impossible. Therefore G must be a metric KKM-mapping.
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Note that since X is compact, we have nxExG(x) i= 0. Take any point Xo E nXEXG(x), then it is clear that d(xo,F(xo)) ::; d(x,F(xo)) for all x E X, which implies that d(xo, F(xo)) = infxEX d(x, F(xo)).
If F(xo) is compact and Xo 1. F(xo), then there exists Uo E F(xo) such that d(xo,uo) = d(xo, F(xo)) > O. We now claim that Xo is a boundary point of X. Suppose it were not, i.e., Xo E intX, where intX is the set of all relatively interior points of X in M. Then there exists a positive number r > 0 small enough such that B(xo, r) c intX c X and
0< r < d(xo,F(xo)) ::; d(z,F(xo))
for all z E B(xo, r). Note that d(xo, uo) = d(xo, F(xo)) = r + d(xo, F(xo)) - rand d(xo, F(xo)) - r > 0, it follows by the hyperconvexity of M that
B(xo,r) nB(uo,d(xo,F(xo)) - r) i= 0.
Since Uo E F(xo) and thus B(uo, d(xo, F(xo)) - r) C B(F(xo),d(xo,F(xo)) - r), there exists
Yo E B(xo, r) n B(F(xo), d(xo, F(xo)) - r)
such that d(Yo,F(xo)) ::; d(xo,F(xo)) - r. However, we have
d(Yo,F(xo)) 2': d(xo,F(xo)) > 0,
which is a contraction. Therefore Xo must be a boundary point of X, i.e., Xo E Bd(X) .
• As a special case of our best approximation theorem, we have the following best approximation result for single-valued mappings, first given by Khamsi in [92J.
Corollary 7.14 Let M be a hyper-convex metric space and X E A(M) be a non-empty compact subset of M. Suppose F : X -; M is a single-valued continuous mapping. Then there exists Xo E X such that
d(xo, F(xo)) = inf d(x, F(xo)). xEX
7.3. Intersection Theorems
In what follows, we shall give a dual form of above KKM principle in hyperconvex space in the sense that the family {G(x) : x E X} of Theorem 7.4 still has the finite intersection property if G(x) takes finitely metrically open values instead of taking finitely metrically closed values for each x EX.
Theorem 7.15 (The dual form of FKKM principle in hyperconvex space.) Let X be a non-empty subset of a hyperconvex metric space M. Suppose
G : X -; 2M \ {0}
is a MKKM mapping with finitely metrically open values. Then the family {G(X)}xEX has the finite intersection property.
Proof. As each hyperconvex metric space is an H-space by Lemma 7.1, the conclusion follows by Proposition 2.6. •
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As applications of both Theorems 7.4 and 7.15, we have the following Ky Fan matching theorem for both closed and open covers in hyperconvex spaces.
Theorem 7.16 Let X E A(.M) be a non-empty subset of a hyperconvex metric space M. Suppose {Ai}f=l is a family of closed subsets of X such that U~IAi = X and Xl,'" , Xn are (not necessarily distinct) n points of X. Then there exist k indices il < i2 < '" < ik between 1 and n such that cOV{XiuXi2"" ,Xi.} n (nj=IAij ) f. 0.
Proof. Let Xo = {XI,X2,'" ,xn} and define a set-valued mapping F: Xo -> 2x by F(Xi) := X \ Ai for each i = 1,2, . " , n. Suppose the conclusion is false. Then F(x) is open for each X E Xo and for each non-empty finite subset B C Xo, we have
cov{x : X E B} C U{F(x) : x E B}.
By Theorem 7.15, it follows that nf=IF(x;) f. 0. But this means that Uf=IAi f. X, which contradicts our hypothesis. •
As an application of Theorem 7.4, we also have the following Ky Fan matching theorem for open covers in hyperconvex spaces.
Theorem 7.17 Let X E A(M) be a non-empty subset of a hyperconvex metric space M. Suppose {Ai}f=l is a family of open subsets of X such that Uf=lAi = X and Xl,'" ,Xn are (not necessarily distinct) n points of X. Then there exist k indices i l < i2 < ... < ik between 1 and n such that cov{ Xiu Xi2' ... , Xi.} n (nj=l Aij) f. 0.
Proof. Let Xo = {Xl, X2,' .. , xn} and define F : Xo -> 2x by F(Xi) := X \ Ai for each i = 1,2,··· , n. Suppose the conclusion is false. Then F(x) is closed for each X E Xo and for each non-empty finite subset B C Xo, we have cov{x : X E B} C U{F(x) : x E B}. By Theorem 7.4, it follows that nf=IF(Xi) f. 0. But this means that Uf=IAi f. X, which contradicts our hypothesis. •
We note that in Theorem 7.17 X is not assumed compact as required by Theorem 7.16.
As applications of the Ky Fan matching theorems for closed covers, we establish some fixed point theorems for set-valued mappings and some existence of intersection theorems which are hyperconvex versions of the corresponding results given by AlexandroffPasynkoff and others.
First we have the following fixed point theorem for set-valued mappings in hyperconvex spaces.
Theorem 7.18 Let X E A(M) be a non-empty subset of a hyperconvex metric space M. Suppose A : X -> 2x is a set-valued mapping with closed values (maybe taking empty values for some point x EX) and there exists n points Xl, ... , Xn (not necessarily distinct) of X such that Uf=lA(Xi) = X. Suppose also that the set A-I(y) := {x EX: y E A( x)} for each y E X is admissible. Then A has a fixed point.
Proof. By Theorem 7.16, there exist k indices il < i2 < ... < ik between 1 and n such that COV{Xi"Xi2"" ,Xi.} n (nj=lA(Xij» f. 0. Take any
wE COV{Xiu Xi2"" ,Xi.} n nj=lA(Xij)'
Fixed point theorems for set-valued mappings 673
Then Xi; E A-I(w) for j = 1,'" ,k. Because A-I(w) is admissible, it follows that w E cov{ Xi" ... ,Xik } c A( w). Therefore w is a fixed point of A. •
As an application of Theorem 7.17, we also have the following Browder-Fan type fixed point theorem for set-valued mappings in hyperconvex spaces.
Theorem 7.19 Let X E A(M) be a non-empty subset of a hyperconvex metric space M. Suppose A : X -+ 2x is a set-valued mapping with open values (maybe tak'ing empty values for some point X EX) and there exist n points Xl," . ,Xn (not necessarily distinct) of X such that U?=IA(Xi) = X and the set A-I(y) := {x EX: y E A(x)} for each y E X is admissible, Then A has a fixed point.
Proof. By Theorem 7.17, there exist k indices il < i2 < ... < ik between 1 and n such that COV{Xil' Xi2'" . ,Xik } n (nj=IA(Xij)) 1= 0. Take any
wE COV{Xi"Xi2"" ,Xik} n (n;=IA(Xij))'
Then Xij E A-I(w) for j = 1,'" ,k. Because A-I(w) is admissible, it follows that wE COV{Xi,,··· ,Xik} C A(w). Therefore w is a fixed point of A. •
As equivalent forms of Theorems 7.1S and 7.19, we have the following fixed point theorems for set-valued mappings in hyperconvex spaces.
Theorem 7.20 Let X E A(M) be a subset of a hyperconvex space M. Suppose that F : X -+ 2x is a set-valued mapping with non-empty admissible values. Suppose there exists a non-empty finite subset C of X such that F(x) n C 1= 0 for each X E X and F-I (y) is closed for each y EX. Then F has a fixed point.
Proof. Define A: X -+ 2x by A(x) := F-I(x) for each X EX. Then it is easy to verify that A satisfies all the hypotheses of Theorem 7.1S. Thus the conclusion follows .
• Theorem 7.21 Let X E A(M) be a subset of a hyperconvex space M. Suppose that F : X -+ 2x is a set-valued mapping with non-empty admissible values. There exists a non-empty finite subset C of X such that F(x) n C 1= 0 for each X E X and F-I(y) is open for each y EX. Then F has a fixed point.
Proof. Define A : X -+ 2x by A(x) := F-I(x) for each x E X. Then it is easy to verify that A satisfies the hypotheses of Theorem 7.19 and the conclusion follows. •
The following are consequences of Theorems 7.20 and 7.21.
Theorem 7.22 Let X E A(M) be a non-empty subset of a hyperconvex space M. Suppose G : X -+ 2x is a upper semicontinuous set-valued mapping with non-empty closed admissible values and there exists a non-empty finite subset C of X such that G(x) n C 1= 0 for each x EX. Then G has a fixed point.
Proof. As G is upper semicontinuous with non-empty closed values, it follows that the graph of G is closed and thus the set G-I(y) is closed for each y E X. Therefore the hypotheses of Theorem 7.20 are satisfied. Thus the conclusion follows. •
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Theorem 7.23 Let X E A(M) be a compact subset of a hyperconvex space M. Suppose P : X -+ 2x is a set-valued mapping with non-empty admissible values and P-1(y) zs open for each y EX. Then P has a fixed point.
Proof. Note that p-l(y) is open for each y E X and P(x) of. 0 for all x E X, it follows that UyEXp-1(y) = X. As X is compact, there exists a finite subset C of X such that X = UyECp-1(y). Therefore for each x E X, we have P(x) n C of. 0. Thus P satisfies the hypotheses of Theorem 7.21 and the conclusion follows. •
Remark 7.24 Theorem 7.23 is a hyperconvex version of the famous Browder-Fan fixed point theorem. In next section, we will establish a more general form for non-compact hyperconvex spaces. We also note that both Theorems 7.18 and 7.19 are hyperconvex versions of the corresponding fixed points given by Kim [93] and Lassonde [104].
In the remainder of this section, as applications of both Theorems 7.15 and 7.17, we will establish some intersection theorems which are hyperconvex versions of the corresponding results due to Alexandroff-Pasynkoff [1] and others.
Theorem 7.25 Let X E A(M) be a non-empty subset of a hyperconvex metric space M. Suppose {Ai: i = 1,2"" ,n} is a closed cover of X (i.e., Ai is a closed subset of X for i = 1"" ,n and U?=lAi = X) and there exist n points {Xi, i E 1= (1,,, . ,n)} (not necessarily distinct) of X such that
cov{Xj : i E I and j of. i} C Ai
for each i E I. Then n?=lAi of. 0.
Proof. By Theorem 7.16, it follows that there exists a non-empty subset J C I such that cov{Xj : j E J} n (njEJAj) of. 0. On the other hand, our assumption implies that cov{Xj : j E J} c n{Ai : i E 1\ J}. Therefore, it follows that there exists Xo E cov{ Xj : j E J} such that Xo E niEI Ai, i.e., n?=l A; of. 0. •
Theorem 7.26 Let X E A(M) be a non-empty subset of a hyperconvex metric space M. Suppose {Ai: i = 1,2,'" ,n} is an open cover of X (i.e., Ai is an open subset of X for i = 1,'" ,n and U?= 1 Ai = X) and there exist n points {Xi, i E I = (1"" ,n)} (not necessarily distinct) of X such that
cov{Xj: i E I andj of. i} C Ai
for each i E I. Then n?=lAi of. 0.
Proof. By Theorem 7.17, it follows that there exists a non-empty subset J C I such that cov{Xj : j E J} n njEJAj of. 0. On the other hand, our assumption implies that cov{Xj : j E J} c n{Ai : i E 1\ J}. Therefore, it follows that there exists Xo E cov{Xj : j E J} such that Xo E niElAi , i.e., n~lAi of. 0. •
Theorem 7.27 Let X E A(M) be a non-empty subset of a hyperconvex metric space M. Suppose {Ai: i = 1,2", . ,n} is a closed cover of X such that njEJAj of. 0 for each J C I with J of. I, where I = {I, 2,'" ,n}. Then n?=lAi of. 0.
Proof. For each i E I = {I,,,, ,n}. Take any Xi E n{Aj : j E I andj of. i}. Then by Theorem 7.16, it follows that there exists a non-empty subset J C I such
Fixed point theorems for set-valued mappings 675
that cov{Xj : j E J} n (njEJAj) =J 0. On the other hand, our assumption implies that {Xj : j E J} c n{Ai : i E I\J}. Therefore, it follows that there exists Xo E {Xj : j E J} such that Xo E niEIAi , i.e., ni=lAi =J 0. •
Theorem 7.28 Let X E A(M) be a non-empty subset of a hyperconvex metric space M. Suppose {Ai: i = 1,2"" ,n} is an open cover of X such that njEJAj =J 0 for each J c I with J =J I, where I = {I, 2"" ,n}. Then ni=lA; =J 0.
Proof. For each i E I = {I"" ,n}. Take any Xi E n{Aj : j E I andj =J i}. Then by Theorem 7.17, it follows that there exists a non-empty subset J C I such that cov{Xj : j E J} n njEJAj =J 0. On the other hand, our assumption implies that {Xj : j E J} c n{Ai : i E I\J}. Therefore, it follows that there exists Xo E {Xj : j E J} such that Xo E niE1A;, i.e., ni=lAi =J 0. •
7.4. Fixed Point Theorems
In this part, we shall use the generalized metric KKM principle and Fan's best approximation theorem in hyperconvex metric spaces established above to derive hyperconvex versions of the Browder-Fan fixed point theorem and Schauder-Tychonoff fixed point theorem for set-valued mappings.
Theorem 7.29 Let X E A(M) be a compact subset of a hyperconvex metric space M. Suppose that F : X -> 2x is a set-valued mapping with non-empty values such that:
(1) F is transfer open inversed valued; and
(2) for each x E X, F(x) is admissible.
Then there exists Xo E X such that Xo E F(xo).
Theorem 7.29 can be derived from Theorem 7.7. A proof is also given in Park [123). As an application of Theorem 7.29, we have the following result which is Fan's geometric lemma [59) in a hyperconvex metric space.
Theorem 7.30 Let X E A(M) be a non-empty compact hyperconvex subset of a hyperconvex metric space M. Suppose C is a non-empty subset of X X X such that:
(1) for each fixed x EX, the set {y EX: (x, y) ¢ C} is empty or admissible;
(2) for each fixed y E X, the set {x EX: (x, y) E C} is closed; and
(3) for each x E X, (x,x) E C.
Then there exists Xo E X such that {xo} X X c C.
Proof. In order to apply Theorem 7.29, we define a set-valued mapping F : X -> 2x by
F(x) := {y EX: (x,y) ¢ C}
for each x EX. Then we can verify that F satisfies all the hypotheses of Theorem 7.29. Since F has no fixed point, by our condition (3), it follows that there must exist
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Xo E X such that F(xo) = 0. Thus we must have {xo} x Xc C by the definition of F. •
As an application of Theorem 7.13, we have the followmg Schauder-Tychonoff fixed point theorem for set-valued mappings.
Theorem 7.31 Let X E F(M) be a compact subset of a hyperconvex metric space M. Suppose F : X -+ A(M) is a set-valued continuous mapping with non-empty closed values such that for each x E X with x rf: F(x), there exists z E X such that d(z,F(x)) < d(x,F(x)). Then F has a fixed point in X. In particular when F takes compact values, the conclusion of Theorem 7.31 still holds if for all x E Bd(X) with x rf: F(x), there exists z E X such that d(z,F(x)) < d(x,F(x)).
Proof. By Theorem 7.13, there exists Xo E X such that
d(xo,F(xo)) = inf d(x,F(xo)). xEX
We claim that xo is a fixed point of F. Indeed, if it were not true, i.e., Xo rf: F(xo), it follows that d(xo, F(xo)) > O. Then by assumption, there exists Zo E X such that d(zo, F(xo)) < d(xo, F(xo)). On the other hand, note that
d(zo,F(xo)):::: d(xo,F(xo)) > O.
This implies that 0 < d(zo,F(xo)) < d(zo, F(xo)), which is impossible and thus Xo must be a fixed point of F. When F takes compact values and Xo rf: F(xo), it follows by the particular case of Theorem 7.13, Xo E Bd(X) and thus the conclusion follows by the same argument above. •
As an immediate consequence of Theorem 7.31, we have the following fixed point theorem which is Corollary 3.5 of Kirk and Shin [95, p.180].
Corollary 7.32 Let X E F(M) be a compact subset of a hyperconvex metric space M. Suppose F : X -+ A(M) is a set-valued continuous mapping with non-empty closed values for which F(x) n X # 0 for all x EX. Then F has a fixed point in X.
Proof. Note that F(x) n X # 0 for each x E X. It follows that for any x E X with x rf: F(x), if we take z E F(x) n X, we have d(z,F(x)) = 0 < d(x,F(x)). Thus all hypotheses of Theorem 7.31 are satisfied and the conclusion follows. •
Remark 7.33 We note that a spacial case of Theorem 7.31 was given by Khamsi [92] for a single-valued mapping.
Before concluding, we prove a version of the Browder-Fan fixed point theorem for setvalued mappings in a hyperconvex space. In order to do so, we first need the following Ky Fan matching theorem for open covers in hyperconvex spaces.
Theorem 7.34 Let X E A(M) be a non-empty compact subset of a hyperconvex space M and A : X -+ 2x be a set-valued mapping such that:
(1) A(x) is open for each x EX; and
Fixed point theorems for set-valued mappings
(2) A(X) := UxEXA(x) = X.
Then there exists a non-empty finite subset {Xl, ... , Xn } C X and an
Xo E cov{ Xl,··· , xn} such that Xo E ni'=l A(Xi).
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Proof. Define a set-valued mapping F : X -> 2x by F(x) := X \ A(x) for each X E X. Suppose the conclusion is false. Then for each finite subset {Xl,··· ,Xn} eX, it follows that COV{XI,··· ,Xn} C Ui'=IF(Xi). Therefore F is a MKKM mapping with closed values. As X is compact, by Theorem 7.4, we have nxEXF(X) # 0. But this implies UXEXA(X) = A(X) # X, which is against our assumption. •
Theorem 7.34 includes Theorem 1 of Park [123]. As another consequence of Theorem 7.34 we have a hyperconvex version of the Browder-Fan theorem.
Theorem 7.35 Let X E A(M) be a compact subset of a hyperconvex metric space M. Suppose there exist two set-valued mappings FI, F2 : X -> 2x \ {0} such that:
(1) FI(X) C F2(X) for each X E Xi
(2) X = U{intFI-I(y) : y E X} (i.e., FI is transfer open inversed valuedi and
(3) F2(X) is admissible for each X EX.
Then there exists Xo E X such that Xo E F2(XO).
Proof. Define a set-valued mapping A : X -> 2x by A(x) := intFI-I(x) for each X E X. Then we have
(a) A(x) is open for earn X E X;
(b) UxEXA(x) = A(X) = X by condition (2); and
(c) A-I(x) C F2(X) for each X E X by condition (1).
By Theorem 7.34, there exists a non-empty finite subset {Xl,··· , Xn} C X and an Xo E COV{XI'··· , xn} such that Xo E n~l A(Xi). Therefore
Xi E A-I(xo) C F2(XO) for i = 1,··· ,n.
By the condition F(xo) is admissible, it follows that Xo E COV{XI'··· ,xn} C F2(XO), which is a fixed point of F2. •
Theorem 7.35 unifies Theorem 3 of Park [123] and Theorem 3.1 of Kirk et al. [96] for the Browder-Fan fixed point theorems in hyperconvex spaces. Using the notion of a measure of noncompactness, a non-compact version of the KKM principle in hyperconvex metric spaces can be established. For more details, see Kirk et al. [96] and thus we omit their details here.
7.5. Existence of Saddle Points and Nash Equilibria
In this section, we use our generalized metric KKM principle in hyperconvex spaces to give some applications to the existence of saddle points, intersection theorems, and the existence of Nash equilibria in game theory. We first have the following result.
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Theorem 7.36 Let X E A(M) be a non-empty compact subset of a hyperconvex metric space M. Suppose 'ljJ : X X X -> (-00, +(0) is a function such that:
(1) for each fixed y E X, 'ljJ(x,y) is lower semicontinuous in x; and for each fixed x E X, 'ljJ(x, y) is hyper O-generalized quasi-concave in y; and
(2) for each fixed x EX, 'ljJ(x, y) is upper semi continuous in y; and for each fixed x EX, 'ljJ(x,y) is hyper O-generalized quasi-convex in x.
Then'ljJ has at least one saddle point in X x X, i.e., there exists (xo, YO) E X x X such that
max inf 'ljJ(x, y) = 'ljJ(xo, yo) = min sup 'ljJ(x, y) = O. yEX xEX xEX yEX
Proof. By our assumptions, since 'ljJ satisfies all the hypotheses of Theorem 7.12, it follows that there exists xo E X such that
sup'ljJ(XO,Y):S; O. yEX
(2.8.4)
If >.(x,y) = -'ljJ(x,y) for each (x,y) E X x X, then>. : X x X -> (-00,+00) is a mapping which also satisfies all hypotheses of Theorem 7.12. By Theorem 7.12, it follows that there exists yo E X such that
sup >'(x, YO) ~ o. (2.8.5) xEX
By combining (2.8.4) and (2.8.5), it follows that
max min 'ljJ(x, y) = 'ljJ(xo, YO) = min sup 'ljJ(x, y) = 0, yEX xEX xEX yEX
thus completing the proof. • In what follows, we shall establish an intersection theorem and the existence of Nash equilibria in hyperconvex metric spaces. We first recall the following notion: Given a Cartesian product X = IT~=1 Xi of topological spaces, let xj = ITih Xi and let Pi : X -> Xi and pi : X -> Xi denote the natural projections; write Pi(X) = Xi and pi(x) = xi. Given X,y E X, we let (Yi,Xi ) = (Xl,··· ,Xi-I,Yi,Xi+I,··· ,Xn).
The following intersection theorem is a hyperconvex version of Fan's intersection theorem.
Theorem 7.37 Let X!,··· ,Xn be non-empty compact sets in the hyperconvex metric spaces Mi such that Xi E A(Mi) for i = 1,··· ,n and let AI, A 2 ,··· ,An be n subsets of X (where, X = IT~=1 Xi) such that:
(1) for each fixed X E X and for each i = 1,··· ,n, the set
Ai(X) := {y EX: (Yi, Xi) E Ai}
is a non-empty and admissible set; and
(2) for each fixed y E X and each i = 1, ... ,n, the set
Ai(y) := {x EX: (Yi, xi) E Ai}
Fixed point theorems for set-valued mappings 679
is open.
Proof. Define a set-valued mapping G : X --> 2x by
for each x EX. Then we can show that G is not a metric KKM mapping. Indeed, by condition (1), for each x EX, there exists Y E X such that
(Yi,Xi)EAi for all i=I,2,··· ,no
This implies that x E nf=1Ai(y), which means that X C uyEx(nf=1Ai(y)). Therefore nyExG(y) = 0 and also by Corollary 7.6, G must not be a metric KKM mapping. Thus there exists a non-empty finite subset {x(I), x(2), ... ,x( m)} C X for which there exists W E cov{x(I), ... ,x(m)} with the property that for all j = 1,··· ,m, W rf. G(x(j)), i.e., W E nf=1Ai(x(j)). Note that the set Ai(W) is admissible and x(j) E A;(w) for all j = 1,2,··· ,m. It follows that W E cov{x(I),x(2),··· ,x(m)} C Ai(W). Thus we have wE Ai(W) for all i = 1,2,··· ,n, i.e., wE nf=1A; =10. •
As an immediate corollary of Theorem 7.37, we have the following existence of Nash equilibria in hyperconvex spaces.
Theorem 7.38 Let X!,··· ,Xn E A(M) be non-empty compact sets in a hyperconvex metric space M. Let iI,··· ,in be n real-valued continuous functions defined on
n
X= II Xi i=1
such that for each Y E X and for each i = 1,··· ,n, the function Xi ....... fi(Xi, yi) is hyper quasi· concave on Xi. Then there exists a point Yo E X such that
/;(yo) = max fi(Xi, yb) for i = 1,··· ,n. xiEXi
Proof. For any given I: > 0, define for each i = 1,2,· .. ,n,
A; = {Z EX: fi(Z) > max fi(Xi,yi) - I:}. xiEXi
/
It follows that for each fixed x EX, there exists y E X such that (Yi, xi) E A~ for all i = 1,··· ,n and the set A~ is non-empty and admissible, by the hyper quasi-concavity of the mapping Xi ....... /;(Xi,yi) for each y EX. Secondly, for each y E X and for each i = 1, ... ,n, we note that x E AHy) if and only if (Yi, xi) E A~. Further A~ is open by the upper semicontinuity of the mapping yi ....... fi(Xi,yi) for each fixed x E X. Thus all the hypotheses of Theorem 7.37 are satisfied and so there exists x(l:) E X such that x(l:) E n~1A~. As X is compact, without loss of the generality, we may assume that x(l:) I-> yO E X as I: ....... 0+. The conclusion then follows by the continuity of fi for i = 1,2,··· ,no •
Remark 7.39 By following methodology and arguments similar to those used by Fan [69] (see also Park [122] and references therein for more recent study), many variations
680
and generalizations of the results given in this section can be established. Since the ideas are basically the same as those used in Fan [69], we will not include further discussion here. Finally we point out that by using the notion of noncompactness measure, noncompact versions of results in this section can be derived. As their arguments are not particularly difficult we will omit their details.
7.6. Generic Stability of Fixed Points in Hyperconvex Metric Spaces
The aim of this section is to establish generic stability of fixed points for upper semicontinuous set-valued mappings in hyperconvex metric spaces. Our results show that almost all upper semicontinuous set-valued mappings defined in compact hyperconvex metric spaces have stable fixed points from viewpoint of Baire category theory.
Let (X, d) be a metric space and K(X) be the space of all non-empty compact subsets of X equipped with the Hausdorff metric h which is induced by the metric d. For any € > 0, Xo E X and A E K(X), let U(€,A) = {x EX: d(u,x) < € for some u E A} and O(Xo,€) = {x EX: d(xo,Y) < €}. Let Y be a topological space. Recall that a subset Q c Y is called a residual set if it is a countable intersection of open dense subsets of Y.
Throughout the rest of this section, X denotes a hyperconvex metric space (and thus it is complete) and A(X) denotes the collection of all ball intersections (Le., admissible sets) in X. By Theorem 15 of Sine [142) (see also Lemma 2 of Kirk [94)), we have the following fact.
Lemma 7.40 Let X be a hyperconvex metric space. Then the space (A(X), h) is a hyperconvex metric space with the Hausdorff metric h (thus (A(X), h) is a complete metric space).
Let K A(X) be the collection of all compact admissible sets of X. By Theorem 4.3.9 of Klein and Thompson [98)' it follows that (KA(X), h) is still a complete metric space.
Let C := {f : X -+ KA(X) : I is upper semicontinuous with non-empty compact and admissible values on X} and for each I, f' E C, define
p(f,J'):= suph(f(x),J'(x)). xEX
Clearly, p is a metric on C and we also have the following result.
Lemma 7.41 The space (C,p) is complete.
Proof. If {fn}~=l is a Cauchy sequence in C, then for any € > 0, there is a positive integer N(€) such that p(fn,/m) = sUPxEX h(fn(x), Im(x)) < € for any n, m 2 N(€). It follows that for each x E X, {fn(x)}~=l is a Cauchy sequence in A(X). Note that since KA(X) is complete by Lemma 7.40 and Theorem 4.3.9 of Klein and Thompson [98), it follows that there is I : X -+ A(X) such that liffim-+oo Im(x) = I(x). for all x E X. Now for each given x E X, each € > 0 and each n 2 N(€), the inequality above implies that In(x) C U(2€,/(x)) and I(x) C U(2f'/n(X)). Fix n 2 N(€); since In E C, there is 6 > 0 such that In(x') C U(€'/n(x)) whenever d(x,x') < 6. Thus, I(x') c U(2€,ln(x')) C U(3f'/n(X)) C U(5f, f(x)) whenever d(x, x') < 6. Therefore I
Fixed point theorems for set-valued mappings 681
is upper semicontinuous on X, so that fEe and that fn ---- f. Hence C is complete .
• Throughout this section, we set Y := C x KA(X) and for each y y' = (f', A') E Y, define a metric D on Y by
D(y, y') = p(f, f') + h(A, A').
(f,A) E Y,
Then it follows by Lemma 7.41, Lemma 7.40 and Theorem 4.3.9 of Klein and Thompson [98] again that Y is a complete metric space. Define
M:= {y = (f,A) E Y: there is x E A such that x E f(x)}.
Then M is, indeed, a subset of Y and we have the following result.
Lemma 7.42 The metric space (M, D) is complete.
Proof. Since M c Y and Y is complete, it is sufficient to prove that M is closed in Y. Let {yn}~l be a sequence in M and Yn ---- Y E Y. Set Yn = (fn, An), n = 1,2" .. and y = (f, A); then fn ---- f and An ---- A. For each n = 1,2,"" since Yn E M, there is Xn E An such that Xn E fn(xn). Since An and A are compact and An ---- A, by A.5.1 (ii) of Mas-Colell [111, p.lO], U~=lAn U A is compact. Since Xn E An C U~=lAn U A, we may assume without loss of generality that Xn ---- x E U~l An U A. If x ¢ A, since A is compact, there is a > 0 such that U(a, A) n O(x,a) = 0. Since An ---- A and Xn ---- x, there is Nl such that An C U(a,A) and Xn E O(x,a) for all n 2: Nl, which contradicts the assumption that Xn E An. Hence we must have x E A. If x ¢ f(x), since f(x) is compact, there is b > 0 such that U(b,f(x» n U(b,x) = 0. Since fn ---- f, there is N2 such that fn(u) C U(b/2, feu»~ for all n 2: N2 and for all u EX. Since f is upper semicontinuous at x and Xn ---- x, there is Na 2: N2 such that f(xn) C U(b/2, f(x» and Xn C U(b/2, x) for all n 2: N a. Thus for all n 2: N a, fn(xn) C U(b/2,J(xn» C U(b, f(x» and Xn C U(b, x) which contradicts the assumption that fn(xn) n Xn t- 0. Hence we must also have x E f(x). Therefore y = (f, A) E M so that M is closed in Y. •
Now for each y = (f, A) E M, define a fixed point mapping S : M ---- K(X) by S(y) := {x E A : x E f(x)}. Then S(y) t- 0 and following fact is true.
Lemma 7.43 The fixed point set S(y) E K(X) for each y E M.
Proof. Let y = (f, A) E M be given. Note that since f is a set-valued upper semicontinuous mapping with non-empty closed values, its graph is closed and thus the set of fixed points for f in A is closed. Note the A is compact. It follows that S(y) is a non-empty compact subset of X, i.e., we have S(y) E K(X). •
By Lemma 7.43, the fixed point mapping y ...... S(y) defines a set-valued correspondence from M to K(X); moreover, S is upper semicontinuous.
Lemma 7.44 The mapping S is upper semicontinuous on M.
Proof. Suppose S is not upper semicontinuous at y E M. Then there exist EO > 0 and a sequence {yn}~l in M with Yn ____ Y such that for each n = 1,2", " there exists
682
xn E S(Yn) with Xn ¢ U(fO,S(y)). Let Yn = (fn,An) and y = (f,A). Then fn --> f and An --> A. Since Xn E An C U;:::'=I An U A and U;:::'=I An U A is compact, we may assume without loss of generality that Xn --> x E U;:::'=I An U A. Note that we must have x ¢ U(fO,S(y)). Now the same argument as in the proof of Lemma 7.42 shows that x E A and x E f(x) so that x E S(y); this contradicts that x ¢ U(fO,S(y)). Therefore S must be upper semicontinuous. •
In order to study the stability of fixed points in hyperconvex spaces, we need the following notion.
Definition 7.45 Let MI be a non-empty closed subset of M (since M is complete, so is MI). For any given point YI E Ml, a point x in S(y) is said to be an essential fixed point of YI with respect to MI provided that for any f > 0, there is 8 > 0 such that for any y' E MI with D(Yb y') < 8, there exists x' E S(y') with d(x, x') < f. The point y is said to be essential with respect to MI if every x E S(y) is an essential fixed point of y with respect to MI.
Although, in general the fixed point set of any given mapping f E C is not stable, we do have the following characteristic of stability for fixed points of upper semicoIitinuous set-valued mappings in C.
Theorem 7.46 The mapping S is lower semicontinuous at y E MI if and only if y is essential with respect to MI.
Proof. Suppose S is lower semicontinuous at y E MI. Then for any f > 0, there is 8 > 0 such that for any y' E MI with d(y,y') < 8, we have S(y) c U(f,S(y')) so that for any x E S(y), there is x' E S(y') with d(x,x') < f. Thus every x E S(y) is an essential fixed point of y with respect to MI and hence y is essential with respect to MI·
Conversely, suppose that y is essential with respect to MI. If S were not lower semicontinuous at y E M I , then there exist fa > 0 and a sequence {Yn};:::'=1 in M with Yn --> Y such that for each n = 1,2,···, there is Xn E S(y) with Xn ¢ U(fQ,S(Yn)). Since S(y) is compact, we may assume that Xn --> x E S(y). Since x is an essential fixed point of Y with respect to M I , Yn -+ Y and Xn -+ x, there is N such that d(xn,x) < fo/2 and x E U(fo/2,S(Yn)) for all n ~ N. Hence Xn E O(x,€o/2) C U(fO,S(Yn)) for all n ~ N which contradicts the assumption that Xn ¢ U(fO, S(Yn)) for all n = 1,2,···. Hence S must be lower semicontinuous at y. •
The following result says that if an upper semicontinuous set-valued mapping defined in a hyperconvex metric space and having compact values has fixed points, then it can be approximated arbitrarily by an upper semicontinuous set-valued mapping with compact and admissible values whose fixed point set is stable.
Theorem 7.47 The set of essential points with respect to MI is a dense residual set in MI. In particular, every point in MI can be arbitmrily approximated by essential points in MI.
Proof. By Lemma 7.43 and Lemma 7.44, the mapping S : M -+ K(X) is an liSCO
mapping. Since MI is complete, by Theorem 2 of Fort [72], the set of points where
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S is lower semicontinuous is a dense residual set in MI. By Theorem 7.46, the set of essential points in Ml is a dense residual set in MI. •
By Lemma 7.44, Theorems 7.46 and 7.47, we have the following:
Theorem 7.48 The mapping S is continuous at y E Ml if and only if y is essential with respect to MI. Moreover, the set of points at which S is continuous is a dense residual set in MI.
Remark 7.49 We remark that S is continuous at y E MI, if and only iffor each E > 0, there is 6 > ° such that h(S(y),S(y')) < E for each y' E M with D(y,y') < 6. Theorem 7.48 implies that if y = (f,A) E Ml, tl}en y is essential with respect to Ml if and only if its set sty) of fixed points is stable in the sense that: sty') is close to sty) whenever y' is close to y.
We shall now give a sufficient condition that y'E Ml is essential with respect to MI.
Theorem 7.50 If y E Ml is such that sty) is a singleton set, then y is essential with respect to MI.
Proof. Suppose sty) = {x}, It follows by Lemma 7.44 that S is upper semic~tinuous at y. Thus for any E > 0, there is 6 > ° such that for each y' E M 1 , D(y, y') <15 implies sty') c U(e,S(y)) = O(X,E), so that sty) = {x} c U(E,S(y')). This shows that Sis also lower semicontinuous at y. By Theorem 7.46, y is essential with respect to MI .•
Finally, as an application of Theorem 7.48, we have the following stability result for fixed points of upper semicontinuous set-valued mappings on hyperconvex metric spaces.
Theorem 7.51 Let X be a compact hyperconvex metric space and C1 be the space consisting of all upper semicontinuous set-valued mappings from X to itself with nonempty admissible values. Then there exists a dense residual subset C1 of C such that:
(1) the fixed point set of each mapping in C1 is essential; and
(2) any given J E C and E > 0, there exists 6 > ° such that Jar each J' E Cl with sUP",ex h(f(x),!,(x)) < 6, we have h(S(f),S(f')) < Eo
Proof. By Theorem 7.2, it follows that C1 x {X} is a non-empty closed subset of M. Thus the conclusion follows by Theorem 7.48. •
In conclusion, we remark that by introducing so-called 'locally generalized convex spaces' -which is another kind of general abstract convexity due to Ben-El-Mechaiekh et a1. [17]- Himmelberg's fixed point theorem [80] has been generalized by replacing the usual convexity in topological vector spaces with an abstract topological notion of convexity that generalizes classic convexity as well as several metric convexity structures found in the literature [17].
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Index
Absolutely continuous, 369 Adjoint, 282 Admissible curve, 441 Admissible subset, 397, 438 Affine, 295 Algebra
C*-, 522 Fourier Stieltjes, 532 von Neumann, 288, 522
Almost-orbit, 545-546 Almost isometric, 270 Alspach's example, 43, 189, 194
Sine's modification, 45 Amann's proof, 613, 615 Amir's lower bound, 159 Antipodal points, 328 Application, 410 Approximate fixed point property, 63, 507 Approximate fixed point sequence, 188 Approximate quantities
collection of, 22 Approximate quantity, 22 Approximating sequence
in the Hilbert ball, 472 Approximation, 69 Asymptotic center 1 77 Asymptotic center method, 462 Asymptotic center technique, 57, 76-77, 80 Asymptotic diameter of a sequence, 204 Asymptotic equidistant sequence, 380 Asymptotic normal structure, 52, 169, 187-188 Asymptotic normal structure coefficient
AN(X),232 with respect to the weak topology (w - AN(X)),
232 Asymptotic radius, 76, 257 Asymptotic radius of a sequence, 204 Asymptotic regularity, 69 Asymptotically regular, 474 Averaged mapping, 475, 486
of the first kind, 473, 478 of the second kind, 474
Axiom of Choice, 613, 619-620, 622-623, 626, 628-631, 633
countable form of, 617, 625 weaker forms, 625
Axiom of Dependent Choices (DC), 625--B26 B-FPP,50 Baker transform, 41 Baker transformation, 68 Ball,54
c-regular, 57 closed unit, 277 unit, 50, 55, 59, 282, 295
Banach's Contraction Mapping Principle, 1-4, 6, 21, 61, 259, 324, 617, 619-620, 623, 625, 627,
634--B35 converse of, 23
Bessaga,23 Janos, 25 Leader, 25 Meyers, 25 topological variant, 29
extensions, 7, 13, 617 Boyd and Wong, 7 Browder, 8 Caristi's theorem, 14 for continuous mappings, 29 for local contractions, 28 for mappings with koo < 1, 12-13 for random operators, 29 for set-valued mappings, 15-17, 26" for topological vector spaces, 26 Geraghty, 10 Hybrid,28 Matkowski, 8-9 Meir and Keeler, 9 Rakotch,7 uniform space setting, 26
generalizations, 18-19, 26 for fuzzy settings, 22,-23 for probabilistic metric spaces, 20-21 Jachymski and Stein, 20 Walter, 18
Nadler's extension, 621 Banach-Mazur distance, 60, 155, 159-160, 257, 265 Banach algebra, 522
semi-simple commutative, 531 Banach lattice, 270-271
weakly orthogonal, 164 Banach space ultrapower, 179 Banach space(s)
691
k-UC, 104, 111 k-uniformly convex (k-UC), 105, 144 k-uniformly rotund (k-UR), 101, 143 k-uniformly smooth (k-US), 106; 145 k-UR,105 L-embedded, 295 M -embedded, 295 «(00(S)),518 containing an asymptotically isometric copy of £1,
272, 278-279, 285, 289, 294 containing an asymptotically isometric copy of £1 or
cO, 274, 284, 293 containing an asymptotically isometric copy of CO,
272-273, 279, 289, 292-294 containing an isomorphic copy of £1, 270-271, 278,
285, 289, 292 containing an isomorphic copy of Co, 270, 289,
291-292 finite dimensional, 123 infinite dimensional, 125
692
isomorphic, 161, 165, 170-171 locally uniformly rotund (LUR), 137 LUR,137 nearly uniformly convex (NUC), 1l0, 146, 155 nearly uniformly smooth (NUS), 112 nearly uniformly smooth with respect to T
(NUS(r», 113 non-SchuT, 162 nonreflexive, 158, 265, 269---270 not reflexive, 111 NUC, Ill, 249 NUS, 113, 260 orthogonally convex, 54 reflexive, 51-52,63-64,137, 148, 158, 161-162, 166,
169, 246, 265, 269, 314, 538 SC not WLUR, 140 uniform normal structure, 134 with normal structure, 64, 546
separable, 64-65, 99, 246, 254, 256, 282, 292 smooth, 96, 98 strictly convex (or rotund), 94 strictly convex (SC), 134 strictly convex, 94-95, 440, 493, 495, 501, 532 superreflexive, 61, 103, 134
and has UNS, 140 UC,104 UCED,lOO ultrapower, 55 uniformly convex (UC), 134, 136-138, 142, 156 uniformly convex, 50, 55, 57, 59, 62, 73-74, 76, 78,
302, 309, 319, 529, 532, 534, 538-539, 541, 543, 546-550, 568
uniformly convex in every direction (UeED), 99, 141
unifurmly noncreasy (UN C), 55 uniformly nonsquare, 134 uniformly smooth, 71, 140, 309, 320 US, 107, 140 weak' uniformly rotund (W'UR), 139 weakly compactly generated, 246 weakly locally uniformly rotund (WLUR), 139 weakly uniformly rotund (WUR), 138 which contain copies of £1 or CO, 289 which fail to have the fixed point property, 284 with dim(X) 2: 2, 100, 102 with normal structure, 527 with property L(p,r), 115 with the fixed point property, 295 with unconditional basis, 270-271
Base, 558, 565, 567, 570 Basis
canonical, 277 Schauder, 55, 110 standard, 126 standard unit vector, 274 unconditional, 52 unit vector, 292-293
Bernal-Sullivan convexity property, 212 Biorthogonal system, 290 Boundary condition, 299, 301, 303, 306, 308, 314
A, 301, 309-310, 313 A" 302 A w, 313--314 Leray-Schauder, 299, 301, 307-308, 310, 313, 315,
318 Nagumo, 300
Boundary of a set, 299
Bruck's metric method, 482, 498 Bynum's lower bound, 159
Canonical image, 282 Caristi's theorem, 14-15 Cayley's transform, 482 Cayley transform, 486 Center, 396 Cesaro averages, 498 Cesaro Sequence Spaces
cesp , 378 Chain, 614, 616-617, 620, 622-624, 627-628, 630, 635
w-complete, 630 complete, 614 countable, 627-628, 630-631, 637
Characteristic of uniform convexity, 184 Characteristic
of convexity, Clarkson's, 134 Lifschitz, 57 of k-convexity, 103 of k-convexity, Geremia-Sullivan, 143 of convexity, 57, 77, 103 of convexity, Lovaglia, 137 of noncompact convexity, 146
associated to the Hausdorff measure of noncompactness, 148
associated to the separation measure of noncompactness, 149
of smoothness, 107 Characterization
generalized metric KKM, 668 Chebyshev center, 51, 83, 118, 123, 127, 396, 604, 629 Chebyshev radius, 51, 57-58, 83, 118, 252, 254, 396,
587 relative, 157
Chebyshev self-radius, 158 Chebyshev set, 449 Class
of (F)-attracting mappings, 567 of u-porous sets, 571 of contractive mappings, 61 of hyperbolic spaces, 557 of isometries, 61 of nonexpansive mappings, 67 of nonexpansive self-maps, 271 of rotative mappings, 63 of sets which have measure 0 and are of the first
category, 571 of uniformly lipschitzian mappings, 56 of uniformly lipschitzian maps, 271 trace, 288
Classes isomorphism, 270 of Banach spaces, 270
Clm-NS, 121 Closed subalgebra
of C~(S) (GB(S», 518 Coefficient(s), 133
I«M),167 1<,(X),266 I<w(X),266 po(X),128 T-convergent sequences (rCS(X», 121 rGS(X), 255, 266 BS(X),159 M(X),265 R(X), 259, 265 Ro(X), R~(X), R~(X), 152 WGS(X),264 for asymptotic normal structure, 169 for asymptotic normal structure with respect to the
weak topology, 170 for normal structure, Bynum's, 169
INDEX
for semi-Opial property, 171 for semi-Opial property with respect to the weak
topology, 171 for uniformly lipschitzian mappings, Lifschitz and
Dominguez-Xu, 167 Gao-Lau, 163 Lifschitz's, 167 normal structure (N(X}), 121 normal structure, 58, 122-123, 125, 128, 204, 265 of (uniformly) normal structure, Bynum's, 158 of convexity corresponding to D x, 156 of noncompact convexity corresponding to DX,I.II
155 related to normal structure, 120 related to the modulus bx TI 128 related with (NUS) properly, 164 related with (NUS) property, Dominguez
generalization, 165 scaling, 93 weakly convergent sequence, 204 weakly convergent sequence, Bynum's, 160 weakly convergent sequences (WCS(X)), 121
Common fixed point, 457, 459, 462--463 Commuting family
common fixed point, 412 of kBJi -nonexpansive mappings, 462 of holomorphic mappings, 507 of mappings, 463
Compactness, 635, 637 r-,254 T-sequential, 254 countable, 637 weak, 252, 295 weak sequential, 252
Complementary function Young conjugate, 340
Complete normal structure, 528 Completeness, 631, 635 Complex extreme point, 440 Component, 316, 318
disjoint nontrivial, 316 Condition
Lla-, 287-288 boundedness, 617, 624 Bynum's, 160 contractive, 617, 621, 625-626 Hausdorff, 617 linear contractive, 634 non-strict Opial, 266 nonlinear contractive, 634 nonstrict Opial, 148 Opial, 161-162 range, 336 uniform Opial, 168
Constant, 133 contractive, 633 Jung J(X}, 158 Lipschitz, 637 self-Jung J.(X}, 158 stability, 60
Contraction, 36 k-set-,609 Banach, 625, 633-636 diametric, 613, 615 set-valued, 613, 618, 627
Nadler's, 626 Walter's, 615
Contractive gauge function, 18 Convergence
T-,497
clm-, 251-252 Convex modular I.z,(x}, 341 Convex
o-paraconvex, 27 metrically, 8
Convexity, 114, 286 Convexity structure, 80, 82, 630
admissible, 81 admissible compact, 82 closed,80 compact, 82 countably compact, 82 metric, 80-81, 83
normal, 83 normal compact, 83 normal countably compact, 84 uniformly normal, 83
Convexity structures, 79-80 metric, 80
Convexity B-,358 k-nearly uniform, 362 k-uniform, 127, 144 general set-theoretic, 80 Menger, 394 of balls, 101 of balls in Banach spaces, 105 orthogonal, 54 strict, 93-96, 98
partially dual to, 96 uniform versions of, 127
Countably incomplete ultrafilter, 178 Crease, 55 Curve
lipschitzian, 328 Decomposition
Schauder finite dimensional, 133 Dentable, 531 Derivative
of the norm, 127 Diagonal argument, 261 Diameter, 396
asymptotic, 159-160, 265 of a set, 240
Diameterizing sequence, 188 Diametral, 119, 126, 258 Diametral point, 53 Diametral set, 187 Differentiability, 97, 127 Direct sum
(l-,265 Directional curve, 63 Directional derivative, 107 Directional modulus
of rotundity, 141 Distribution function, 20, 22 Domain, 437 Drop, 151 Dual space
693
contains an asymptotically isometric copy of £1, 293 contains an asymptotically isometric copy of co, 279 contains an isometric copy of Ll[O,l), 294
Dual of an arbitrary C'-algebra, 295 topological, 518
Duality partial, 294
Eigenvectors, 317 of pseudocontractive mappings, 310
Embedding
694
metric, 558 Embeds
£= in (L"'(",), II· II",), 287 isometrically, 287 isomorphically, 287
Epigraph, 97 Equivalent metrics, 12-13 Ergodic transformation, 46 Essential fixed point, 682 Euclidean space
k-dimensional, 101 Example
Alspach '5, 68 Extension problem, 469 Extension
uniformly lipschitzian, 57 Extensions
Hahn-Banach, 293 Ext.ernally hyperconvex, 397 Extremal functions, 419 F.p.p.,51 Failure of the fpp
in Co, 36-37 in CO with a contraction, 36 in Co with a non-affine map, 36 in a topology coarser than the weak, 37
Failure of the w-fpp, 43 Failure of the w'-fpp
in £1 = c*, 41 in £1 = c* with a non-affine map, 43 in a renorming of l\ = co' 40
Family normal, 118 of k-convex functions, 102 of all k dimensional subspaces, 105 of all admissible subsets, 81 of all chains, 628 of all closed and convex subsets, 242 of all nonempty compact subsets, 634 of all nonempty compact subsets of IR+., 621 of all nonempty well-ordered subsets, 628 of all nonempty, dosed subsets, 637 of all subsets, 614 of all subsets of a set, 627 of Banach contractions, 636 of convex functions, 100 of convex uniformly bounded functions, 98 of metric lines, 558 of nonexpansive mappings, 56 of pseudometrics, 617, 621, 631
P',621 of pseudonorms {PA : A E K(IR't-)}, 621 uniformly normal, 119
Fan geometric lemma, 643
Fatou property, 369 Filter, 177
non-free, 178 trivial, 178
Finite intersection property, 54, 82, 241, 517, 617, 630, 649, 667
for closed left ideals, 537 Finitely representable, 52, 103, 117, 181 Fixed point, 450, 452, 474-475 Fixed point property, 35, 50, 61, 67, 240, 269-271,
274, 276, 279, 284-285, 287-289, 294-295, 531 for A(G), 531 for k-uniformly lipschitz ian mappings, 271 for affine asymptotically nonexpansive mappings,
292
for asymptotically nonexpansive mappings, 137, 291 for closed convex sets, 325 for continuous functions, 239 for contractive mappings, 62 for firmly nonexpansive mappings, 62 for isometries, 195 for nonexpansive affine self-maps, 295 for nonexpansive mappings, 63, 272, 305-306, 308,
318-320, 323 for nonexpansive maps, 295 for rotative non expansive mappings, 63 for uniformly k-lipschitzian mappings (FPP(k)), 59 for uniformly lipschitzian mappings, 271, 291 in hyperconvex metric spaces, 411 topological, 577 with respect to T (T-FPP), 252 with respect to Fo, 61
Fixed point set, 35, 186 Fixed point
Browder-Fan in hyperconvex spaces, 673 Fan-Glicksberg in locally G-convex spaces, 664 in finite powers of the Hilbert ball, 460 of a semigroup, 502 of continuous semigroups, 502
Formula constructive, 631 Figiel's, 143
Fourier algebra, 531 Fpp, 35 FPP, 50-52, 55, 61-{j2, 64, 72, 138, 164, 257, 265
with respect to Fo, 61 Fnkhet derivative
of f at x, 96 Frechet differentiable, 96
uniformly, 96-97, 107 Function(s)
dm-Iower semi-continuous, 266 k-convex, 102 conjugate Young, 287-288 double dual Young's, 104, 126 dual Young's, 98, 107, 113 Fhkhet differentiable at x, 96 Lempert, 441 Lipschitzian, 98 lower semicontinuous, 241, 625 non-decreasing, 615 nondecreasing, 316 nondecreasing subadditive, 634 Rademacher-like, 115, 283 Rademacher, 121, 125, 282-283 scaling, 127 subhomogeneous, 148 weakly lower sequentially semi-continuous, 266 weight, 97, 288 Young, 287-288
Fuzzy set, 20, 22 collection of, 22
Gateaux derivative, 309 Gateaux differentiable, 96
at x, 96-97 Garcia-Falset coefficient R(X), 192 Garcia - Falset coefficient
R(X),377 Genel-Lindenstrauss example, 475 Generalized Gossez-Lami Dozo property, 208 Generic seq uence
of operators, 566 Geodesic line, 447 Geodesic segment, 447 GGLD, 160, 292
INDEX
GGLD property, 257, 260 Girth, 328
of sphere, 337 of the unit ball, 163
Global modulus of convexity, 101
Goebel-Karlovitz lemma, 188 Lin's generalization, 191
Grade of membership, 22 Graph, 330, 332-333 Group, 518
[INj-group,531 abelian, 531 compact, 531 connected IN-group, 531 discrete, 531 discrete amenable, 522 locally compact, 522, 531 locally compact abelian, 532
Haar system, 264 Hardy space H', 195 Hausdorff, 518 Hausdorff distance, 15, 22, 69, 79 Hausdorff metric, 15, 26, 78-79 Hausdorff uniform space
sequentially complete, 632 Hausdorff uniform structure, 621 Heine-Borel metric space, 637 Hereditary fixed point property
for nonexpansive mappings, 64 HFPP, 65, 67 Hilbert ball, 558
infinite powers, 481 Hilbert space, 50, 58--60, 62--{l4, 77, 97, 100, 102, 105,
117,127,135--136,140--141,144,159,161,166, 246-247, 256, 325, 328, 330, 332-333, 529, 535, 543-544, 550, 562, 568
n-dimensional, 538 infinite dimensional, 125, 168, 247 separable, 156 separable and infinite dimensional, 155 with dim(X) ;::: 2, 106
Homotopy, 618 Horosphere, 491-493 Hyperbolic metric, 442 Hyperbolic space, 557-558
complete, 558, 561-562, 564, 567, 571 Hyperconvex, 394 Hyperplane, 55 IFS,637 Inequalities
Clarkson's and Hanner's, 102, 106 Infinite product
of a sequence of mappings, 568 Infinite products, 564
random, 564 Injective envelope, 418 Inquadrate, 184 Invariant subspace
ofL=(S),517 Involution
k-lipschitzian, 328-329, 336 2-periodic, 596
Inward set, 300 Inwardness condition, 299 Isometric embedding of X into (X)u, 180 Isometry, 61
0-, 180 Edelstein's, 485 linear, 527
nonlinear, 52 on the Hilbert ball, 465
Isomorphism bicontinuous, 163
Iterated function systems, 614 Iterates, 325-326, 336
of an operator, 563 of holomorphic mappings, 507
Jordan triple, 295 K-UC,144 K-US,145 Kothe sequence space
Banach sequence lattice, 367 Kadec-Klee property, 495
with respect to u(x,N), 445 with respect to T (KK(T)), 108
Kakutaoi's Examples, 580 KK-space, 69 KKM
dual form, 645 Kobayashi distance, 441, 481, 507
on the Hilbert ball, 447 Kobayashi metric, 460 Kottman constant, 145 Kuratowski measure of non-compactness, 362 Lattice
complete, 614 Left ideal orbits, 532, 537 Left ideals
closed, 532 Left reversibility, 517 Lemma
Goebel-Karlovitz, 53, 56, 257-258 Lemma
Zhou and Tian, 648 Limit affine sequence, 343 Limit along an ultrafilter, 179 Limit constant sequence, 343 Lipschitz condition, 105, 111
695
Lipschitz constant, 3--4, 11, 27, 56, 149, 185, 261, 309 k=,12-13 exact, 150--151, 169 for set-valued mappings, 15-17 uniform, 271
Lipschitz extension, 583 Lipschitz mapping, 185 Local character, 101 Local modulus
of convexity, Lovaglia, 137 Locally uniform convergence, 497 Lower semi-continuity
T -sequential, 254, 264 LUR, 137-139 Luxemburg norm, 369 Mobius transformation, 441, 447 Maluta coefficient, 380 Mapping(s)
(1 + l,2)-rotative, 329 k-lipschitzian, 331
(1 + f )-lipschitzian, 334 Mapping(s)
( a,2)-rotative k-lipschitziao, 331 firmly k-lipschitzian, 333 monotone, 333
(a,n)-rotative, 324 k-lipschitzian, 326
(F)-attracting, 567-568 (F)-attracting w.r.t. e, 568 (F)-attracting with respect to e, 568
696
(m - n + a,m)-rotative, 324 (n,a i-rotative, 63 a-almost convex, 74 a-condensing, 242-243 a-lipschitzian, 324 a-strongly accretive, 310
zero of, 311 a-strongly pseudocontractive, 31()-313
Mapping(s) II . II-nonexpansive
lacks fixed points, 138 ,),-condensing, 415 q,.contractive, 634 :5-continuous, 63Q--631 l?-lipschitzian, 56 j-contractive, 613, 618, 621 j-Lipschitzian, 618, 632 k-lipschitzian, 323, 327, 329
with constant displacement, 333, 335 k-pseudocontractive, 316 k-set contraction, 28 k-uniformly lipschitzian, 160, 169 n-periodic, 324, 336 n-rotative, 324 r-nonexpansive, 56 t-norm, 2()-21
continuous, 22 accretive, 305, 310
zeros of, 305 affine nonexpansive, 62 affine, uniformly lipschitzian, 271 asymptotically nonexpansive, 74, 151, 291, 413, 597
affine, 291 asymptotically regular, 60, 69, 149-151, 163, 169,
26()-261, 266 averaged, 69, 568 closed,320 closed graph, 26 commuting, 501 condensing, 28, 242, 244, 439, 494-495 continuous, 335-336 contractifiable, 25 contraction, 1, 3-5, 13, 18, 22, 27, 53, 61, 71, 240,
301, 303-305, 307-308, 324 generalized, 18-22, 27 local,28 local radial, 28 on separable metric spaces, 28 on topological vector spaces, 26 topological, 29
contractive, 6, 9, 28, 62, 259, 561-562 on probabilistic metric spaces, 22 set-valued, 568-569
demiclosed, 73 diametric 4'-contractioD, 615 dissipative, 628 duality, 117, 127, 518, 539
sequentially T-continuous, 117 with the weight <1>, 97
eventually nonexpansive, 67 firmly k-lipschitzian, 331-332 firmly kBH-nonexpansive, of the first kind, 489 firmly kBH-nonexpansive, of the second kind, 489 firmly lipschitzian, 337 firmly nonexpansive, 62, 337, 490 fuzzy, 22 fuzzy set-valued, 22 generalized HKKM, 649 generalized metric KKM, 668 generic, 568
generic nonexpansive, 570 globally a-strongly accretive, 310 globally nonexpansive, 319 globally one-to-one, 318-319 globally strongly pseudocontractive, 315 HKKM,648 holomorphic, 437 hyper ,),-diagonal quasi-convex (concave), 669 hyper ')'-generalized quasi-convex (concave), 669 hyper diagonal quasi-convex (concave), 669 hyper quasi-convex (concave), 669 increasing, 614 inward,3oo inwardly directed, 301 isotone, 614, 618, 622
single-valued, 622 Lipschitz, 318 lipschitzian, 3, 5, 11-12,56,305,309,335 local contraction, 309 local contractive, 299 local radial contraction, 309 locally ",-strongly accretive, 310 locally k-lipschitzian, 308 locally accretive, 320 locally expansive, 320 locally lipschitzian, 320 locally nonexpansive, 302, 309, 319-320 locally pseudocontractive, 309, 318-320 locally strongly accretive, 318 locally strongly pseudocontractive, 315, 317, 319 lower semicontinnQus, 14-15 metric embedding, 557 monotone, 50 multivalued contraction, 301 Nagumo inward, 300 nonexpansive, 5()-51, 53, 55-57, 61-tl4, 67-tl9, 71,
73-74, 76-77, 79, 81, 83, 86, 95, 118, 240, 252-255, 257-258, 260, 299, 302, 304-306, 308, 311, 313-314, 323-324, 336, 438, 517-518, 527-528, 530, 532, 543-544, 549, 562, 629
nonexpansive extension, 469, 471-472 nonexpansive multivalued, 77
Mapping(s) nonexpansive
on a metric space, 630 set-valued, 569
normalized duality, 72, 304 of convex type, 74 periodic, 336 periodic lipschitzian, 57 power convergent, 497 progressive, 623, 625, 628-629 projective w.r.t. e, 568 pseudocontractive, 299, 305, 310, 314 random operator, 29
random fixed point of, 29 set-valued contraction, 29
regular, 559 rotative, 63, 324, 336
k-lipschitzian, 325, 333 nonexpansive, 324
sernicontinuous set-valued, 79 set-valued, 15, 29, 570 set-valued contraction, 15-17, 26-27,78 set-valued. nonexpansive, 78
Mapping(s) set-valued
order preserving, 621 strongly accretive, 305-307, 310, 313 strongly pseudocontractive, 305, 310, 312, 318
INDEX
super-regular, 559-560 transfer closed valued, 648 uniformly k-lipschitzian, 56-57, 59, 159 uniformly Lipschitzian, 56, 58, 136, 169, 271, 595
fixed point free, 60 upper semicontinuous, 7-8 weakly (F)-attracting, 567 weakly continuous Q-nonexpansive, 530 weakly inward a-strongly pseudocontractive, 311 weakly inward, 300, 306 weakly inward pseudocontractive, 305 weakly lower semicontinuQus, 524 weakly sequentially continuous duality, 314 with constant displacement, 323-324, 334-335, 337
Maurey's results, 47, 193 Maximality, 319 Maximum Value Theorem, 6 Mean, 519, 522, 544
Cesaro, 543 invariant, 517, 528, 539, 542, 544 left invariant, 519 non-linear, 517 right invariant, 538
Measure of noncompactness, 240, 248, 415 a-,241 (3(B),248 Hausdorff, 148, 243, 266, 370, 409 Istra\escu (or separation), 149 Kuratowski, 145, 240, 242, 409, 494 separation, 244, 248
Measure preserving transformation, 46 Measure space, 97, 103, 106, 115, 135, 140
u-finite, 108, 121, 163 finite and not purely atomic, 287-288 finite nonatomic, 287 purely atomic, 124
Measure X-,243 counting, 108, 251 Hausdorff, 243 Kuratowski, 243 Lebesgue, 550 of the degree of WORTHwileness, 164 purely atomic, 103 separation, 151
Measures of noncompactness a, (3, S, 151
Meir-Keeler condition, 9 Metric convexity
of fixed point set for T, 191 Metric embedding, 472 Metric line, 473, 558 Metric midpoint, 95 Metric segment, 473, 558 Metric space
abstract hyperbolic, 128 bounded,86 bounded complete, 84 bounded hyperconvex, 82 compact, 123 convex, 320 hyperbolic, 127 hyperconvex, 394 injective, 399 metrica.lly convex, 320 probabilistic, 628 separable complete, 65 uniformly convex, 448
Metric Euclidean, 616, 624
generalized Hausdorff, 622 Hausdorff, 51, 635
Metrical line, 324 Metrically convex, 69, 394 Metrically parallel, 69 Metrics
equivalent, 56 Metrizable, 558, 565, 567, 570 Minimal displacement, 587 Minimal invariant set, 40, 43, 45, 186 Minimal
Lipschitz constant, 329 Minimizer, 534-535, 538 MNC
a,244 X,243-244 Hausdorff, 243-244 Kuratowski, 243-244, 246, 248
Modular sequence spaces, 368 Moduli, 98, 115, 118, 127, 133, 250
.o.x.lI(e),249
.o.x.x(e),249 bx.w and bx, 113 bx.w and dx •• w, 113 dX,T and KX,TI 111 RXI R'x, R'Jc, 152 continuity of, 127 for the nearly uniform convexity, 155 of k uniform convexity, 248 of convexity and smoothness, 106
697
of noncompact convexity, 244, 248, 252, 255, 266 associated to 1>, 248
of smoothness, Milman k-dimensional, 145 Modulus, 101, 105, 127, 133, 251 Modulus of noncompact convexity, 370 Modulus of uniform convexity, 184 Modulus
of convexity, Clarkson, 252 0,102 .o.X.II.,O, 255, 264 ox(x;E),99 ox(z;E),100 p,107 px,106 bx ... 113 Kx.<> 109 Kx,109 corresponding to the norm, 104 corresponding to UKK(r) property, 111 for the UKK property, 250 for uniform convexity, Dx, 156 of k-convexity 01:(E), 101 of k-convexity, 127 of k-rotundity, Geremi .... Sullivan, 143 of k-uniform smoothness .B~, 105 of k-unifurm smoothness, 265 of "UKK-ness", Partington, 145 of convexity, 57, 102-103, 107, 127, 133 of convexity, Clarkson's, 134 of convexity, Gurarii, 142 of convexity, Milman k-dimensional, 144 of convexity, Milman, 142 of nearly uniform smoothness, 265 of noncompact convexity, 146, 261, 266
.... ociated to (3, 150, 250 of lp." 147
of noncompact convexity, Banas, 148 of noncompact convexity, Dominguez-LOpez, 149 of noncompact convexity, Goebel-Sekowski, 145 of NUC, 260
698
of NUS, Dominguez, 153 of smoothness, 125, 127, 140, 155 of smoothness at X, k-dimensional, 145 of smoothness, Lindestrauss, 140 of smoothness, Milman's k-dimensional, 145 of squareness, a universal one, 154 of uniform "nonoctahedralness" 1 156 of uniform smoothness PX, 106 of weak local uniform rotundity, Lovagia, 139 of weak uniform rotundity, Smulian '5, 138 of weak* uniform rotundity, V.L. Smulian, 139 apial's, 150
with respect to T, 115 apia!, 128, 260--261, 266 Partington '5, 150, 250 uniform apial, 263
Monotonicity, 616 Near uniform convexity, 152 Nearly uniform convexity, 127, 362 Nearly uniform smoothness, 128, 362 Nearly uniformly smooth, 193 Net
asymptotically invariant, 540, 543-544 ergodic, 544 increasing, 622, 624 nonincreasing, 624 of finite averages, 542, 544 of finite means, 542 strongly regular, 540
Non-Contractibility of spheres, 578 Nondiametral point, 50, 252 Nonexpansive, 35 Nonlinear resolvent, 453 Nonreflexive, 170--171, 276, 294-295 Nonstandard methods, 60 Nonstrict Opial condition
with respect to T, 256 Nonstrict Opial property
with respect to r, 256 Norm
r-sequentially lower semicontinuous (r-slsc), 253 r-sequentially lower semicontinuous, 261 T-slsc, 253, 255-256 w-slsc, 254 elm-sequentially lower semicontinuous, 108 Day's, 138 differentiability of, 96, 107 dual,293 equivalent, 96, 99, 103, 107, 126, 264, 284-285, 292 euclidean, 157, 162 Fhkhet differentiable, 76, 518-519, 537-538, 540,
542-543, 548-549 Gateaux differentiable, 96, 98, 107 Gateaux differentiable, 519, 550 Luxemburg, 287-288 monotone, 162 arlicz, 287-288 strictly convex, 64 uniformly convex, 61, 78, 134, 137 uniformly Frechet differentiable, 107 uniformly Frechet differentiable, 72 uniformly Gateaux differentiable, 519 uniformly nonsquare, 134
Normal structure, 50--51, 57, 61, 67, 72,85, 118-119, 126, 128, 141, 146-148, 152, 155-156, 187, 252, 257, 265, 269, 343, 517, 523, 525-528, 531, 629-630
with respect to T (T-NS), 119, 252, 254 with respect to an arbitrary topology, 128
Norming set, 439
Norms equivalent, 136, 147
Notation U,665 L'>.~, 646 8L'>.N,646 j-complete, 630 ~-completeness, 630 j-continuity, 630 2-completeness, 637 T-FPP, 253, 255-256 T-NS, 120--121, 253-254 TGS(X), 128, 262 €-fixed point, 413 A(M),397 t:(M),397 1i(M),397 W(M),397 D[(xn)J, 205 Hco, 646 k-face, 644 k-UC, 104-105, 150 k-UR,143 k-US, 106, 114 KA(X),680 N(V(fl)), 124 N(X), 123-124, 265 R(X),215 w-FPP, 257, 259-260, 264-265 wc-FPP, 50, 52, 54, 60 WGS(X),128 WGS(lP(r)), 124 WGS(LP(fl)), 124 WGS(lP), 128 WGS(X), 153, 160, 265 weak-FPP, 255 clm-FPP, 255 GHKKM,649 GMKKM,668
NS, 119-121, 126, 128 NUC-characteristic
associated to 4>, 248 NUC, 111, 114, 117, 127, 146, 148, 150, 152 Numerical range, 454 NUS, 112-114, 153 NUS(T),118 NUS(w), 113 NUS(clm), 116 Open questions, 194 Operator
m-accretive, 545 accretive, 305 Chebyshev, 629-630 closure, 615 compact, 242 condensing, 242 continuous, 564 contractive, 561
set-valued, 569 with respect to F, 563
Hutchinson-Barnsley, 634, 636-637 limit, 628 limit compact, 416-417 locally k-pseudocontractive, 314 locally strongly accretive, 318 nonexpansive, 564, 566 nonexpansive retraction, 566 resolvent, 305 shift,60 ultimately compact, 416-417
INDEX
uniformly continuous, 564 Operators
commuting nonexpansive, 528 Opial's condition, 74, 79 Opial property, 368 Optimal retraction in Hilbert space, 604 Orbit, 47, 58, 619, 633
left ideal, 518 Ordering
Amann's, 615-616 Br~ndsted 's, 625 Ekeland's, 622, 632
Orlicz function, 340 Orlicz sequence space, 341
g",341 hif?,341
Orlicz space, 340 Orthogonal convexity, 197 Packing rate, 380 Parallelogram law, 100, 102
generalized, 105 Partial ordering, 613 Path, 318, 320 Perimeter
of the unit sphere, 163 Periodic point, 335-336
of a semigroup, 502 Periodicity, 323 Picard iterates, 2, 7, 18, 21 Picard sequence, 1
error estimates, 2 rate of convergence of, 2, 11
Poincare distance, 437, 440 Point
diametral, 293 Polish space, 68 Porosity, 570 Porous, 573
a-porous, 573 Power convergence, 559
locally uniform, 559 Predual(s), 277, 288
of von Neumann algebras, 295 Principle
demiclosedness, 73-74, 79 dual form in hyperconvex space, 671 Ekeland's, 625-626 FKKM,651 KKM,645 maximality, 625 ordering, 614 variational, 618 well-ordering, 623
Probabilistic metric space, 20-21 complete, 20-22 Hausdorff topology on, 21
Problem convex feasibility, 568 minimal displacement, 586, 604 optimal retraction, 597
Products, 568 Projection, 171, 544
metric,426 nearest point, 64, 604
Proper filter, 178 Property
(Fr) with respect to M, 266 T-GGLD, 254-255 L(pp,clm), 116 L(p,T), 115, 117, 121
L(p,w), 117 w-GGLD,264 w-USO,l71 ({3),151-152
of Rolewicz, 152 (H), 127 (k), 164 (KK),165 (P) with respect to M, 167 (WORTH), 157, 165 absolute I-local retract, 401 absolute retract, 401 asymptotic (P), 292 ball intersection, 395 convex approximation, 544 Denjoy-Wolff, 504 dual to k-UC, 105 dual to NUC, 112
699
generalized Gossez-Lami Dozo (GGLD), 160, 292 generalized Gossez-Lami Dozo, 292 geometrical, 120 GGLD,161 Hahn-Banach extension, 282 Kadec-Klee, 127-128 KK,109 KK(T),109 KK(w),108
Property nonstrict (or weak) Opial
with respect to T, 114 nonstrict Opial, 115, 15D-151, 162, 171
with respect to T, 115 Opial, 128, 448
with respect to T, 114-115, 120 Radon-Nikodym, 531 Radon-Riesz, 127 Schur, 109, 116 semi-Opial, 171 UKK, 109-111, 117, 121 UKK(T), 109, 111-112, 118, 121 UKK(clm), 116 uniform Kadec-Klee, 127 uniform Opial, 121, 150-151
with respect to T, 115, 118, 121 with respect to the elm topology, 116 with respect to the weak topology, 128
Property uniform semi-Opial
with respect to the weak topulogy, 171 weak Banach-Saks, 165 weak semi-Opial, 171
Prus' lower bound, 159 Quasi-metrics, 28 Quasi-regular point, 497 Rademacher function, 47 Rademacher system, 125 Radius, 396
asymptotic, 160, 169 Random products
of operators, 566 Reflexive, 96, 103, 11D-111, 113, 117, 123, 155-156,
159, 164, 168, 170, 270, 274, 276, 287-288 Reflexive subspace of Ll, 194 Reflexivity, 52, 58, 65, 269-270, 287-288, 295 Regular fixed point, 497 Relation, 617
=" 614, 617 =,., 615 ='¢' 624
Renorming
700
equivalent, 146, 167 of (00, 285, 287 oU', 276, 278-279 of £1 and co, 270, 276 of e' or CO, 274 of £1 (r), 284 of CO, 276, 279, 292 of CO or (1, 270 of co(r), 284, 293
Representation, 517-518, 528-530, 544 <I>-admissible, 525 X-admissible, 525 bounded, 544 continuous, 527, 529, 532, 535, 538 jointly continuous, 529 norm nonexpansive, 530 uniformly k-Lipschitzian, 529
Resolvent, 305 Retract, 400
A-local, 401 I-local, 84-85 I-local, 401 holomorphic, 482, 498, 501 lipschitzian, 57 nonexpansive, 64--65, 67, 78, 85 proximal nonexpansiv€, 426
Retraction, 400, 507, 539, 563, 568, 630 k-lipschitzian, 329, 334 holomorphic, 497 lipschitz ian, 57, 337 nonexpansive, 64, 67, 539-540, 542-543, 566, 568 onto fixed point sets, 496
Riesz angle, 164, 213, 355 Right-inverse function, 340 Root Test for convergence, 11 Rotativeness, 323-325, 334, 336 Rotund, 294 Rot und dual renorming
off 00, 279 Rotund renorming
of Coo, 279 Rotundity
k-uniform, 144 se, 138-139, 141 Schauder's conjecture, 666 Schauder basis
in L,([O,I]), 264 Schechtman's construction, 45 SchrOder functional equation, 621 Schur property, 55, 74, 265 Schur space, 165, 170--171 Schwarz-Pick system, 437, 442, 507 Segment, 95
affine, 95 metric, 95 nontrivial, 94
Self-mapping(s) <,b-contractive, 634 k-uniformly Lipschitzian, 167 affine nonexpansive, 295 continuous, of X, 636 generic nonexpansive, 559 isotone single-valued, of a complete chain (P, ::5),
622 isotone, of C,(X), 616 nonexpansive, 295
set-valued, 570 Self-mapping(s)
of (X,P) i-contractive, 619
j-Lipschitzian, 618 Self-mapping(s)
of a partially ordered set :j-continuous, 630 progressive, 627
of a uniform space (X,'P), 618 of an abstract set, 620 of the index set, 618 progressive, of M f' 630
Semi-Fatou property, 377 Semi-Opial coefficient
with respect to the weak topology (w - SOC(X», 233
Semigroup, 65, 507-508, 517-519, 525, 539, 544-545 amenable, 541, 543-545 asymptotically regular, 151 commutative, 530--531 commuting, 530 continuous, 502 discrete, 502, 521-522 discrete left amenable, 530 discrete left reversible, 530 left reversible, 67 locally uniformly continuous, 505 nonexpansive, 538-539, 542-543, 548, 550 of nonexpansive mappings, 65, 517, 527-528 one-parameter nonexpansive, 545, 550-551 uniformly k-Lipschitzian, 529
Semigroups left reversible, 518 nonexpansive rotative, 336 nonlinear, 336
Semitopological scmigroup, 518, 520, 522, 527-530, 532, 538, 542
Q-nonexpansive, 529 commutative, 540 left reversible, 519-520, 522, 528 left subamenable, 520 right reversible, 536-537, 539
Separability, 65, 285, 294 Separation constant, 110 Sequence
a-minimal, 247 approximate fixed point, 257 approximated fixed point, 257-260
weakly null, 258 asymptotically invariant, 545 asymptotically regular (a.r.), 170 basic, 111 center of, 257 elm-convergent, 251 elm-null, 251 diametral, 523 disjointly supported, 277 ergodic, 544-545 of Banach spaces, 159 of norm-one functions, 287 of reflexive Banach spaces, 162 regular, 78--79
Set of mappings uniformly (F)-attracting with respect to ~, 567
Set a-Ievel,22 a-minimal, 380 ,9-minimal, 380 4>-minimal, 244-245 p-convex, 558 F-,646 H-compact, 646 H -convex hull, 646
INDEX
r-net, 243 r-separated, 248 r-separation, 248 'Imore compact", 240 admissible, 81 approximate fixed point, 67, 69 bounded non precompact, 240 contractible, 644 diametral, 118, 121, 258 finitely metrically closed, 667 fixed point, 16, 29, 64-65, 67-69, 86, 518, 532, 534,
537, 543, 567, 635 disconnected, 16 structure of, 27
fractal in the sense of Barnsley, 634 minimal, 245, 248, 258 of all fixed points, 95, 621 of all nonexpansive operators, 569 of all nonexpansive self-mappings, 558, 571
Set of operators
uniformly equicontinuQus, 565 of standard basis vectors, 244 of strict contractions, 562 partially ordered, 614 weakly H -convex, 646
Sets a-porous, 571 nowhere dense, 571 porous, 571 uniformly (F)-attracting, 568
Shapley-selection, 645 Simplex
n-dimensional, 644 Sine's modification
of Alspach's example, 45 Slice, 55 Smooth, 98, 519 Smoothness, 96, 98, 114, 127 SO, 171
w-,I71 Space(s)
.6.-uniformly convex, 146
.6.x -uniformly convex, 148 C-,646 G-convex, 662 H-,646 H-structure, 644, 646 k-uniformly convex, 248 k-uniformly rotund, 77, 79 k-UR,110 k-US, 106 l.c.-,665 n-dimensional, 158 NUS(T),265 Bynum's, 126 classical Orlicz, 288 Day, 146 Euclidean, 637 finite dimensional, 55, 104, 125 finite dimensional Euclidean, 571 Hardy, 666 hyperbolic, 473 infinite dimensional, 158 infinite dimensional NUS, 112 inner product, 142 James quasi-reflexive, 496 locally G-convex, 663 locally convex H-, 665 locally convex, 529
locally generalized convex, 683 locally metrically convex H -, 665 Lorentz, 288 nearly uniformly convex, 248 noncreasy, 55 nonreflexive, 52 NUC, 121, 147 NUS, 257 of almost periodic functions, 518 of compact operators on a Hilbert space, 289 of left uniformly continuous functions, 518 of right uniformly continuous functions, 518 of weakly almost periodic functions, 518 Orlicz, 287-288 positive a-finite measure, 250
701
reflexive, 96, 109, 111, 113, 119, 123, 150, 249-250 normal structure, 265 uniformly convex in every direction, 78
separable, 52, 96, 109 separable reflexive, 52 separated locally convex, 529-530 sequential, 637 strictly convex, 55, 95, 104 superreflexive, 57, 103, 106-107 UC,104 UCED, 104, 265 uniform noncreasy, 64 uniformly convex, 103 uniformly nonsquare, 103 uniformly smooth, 55
Stability, 265, 559 T-FPP,228 "x(X) < ~, 212 "1 (X) < 1, 211 M(X),217 asymptotic normal structure, 232 Hilbert spaces, 223 normal structure, 204 of condition "",(X) < ~, 149 of condition "o(X) < 1, 136 of condition 10, (X) < 1, 147 orthogonal convexity, 234 results for GGLD, 160 semi-Opial property, 233 uniformly convex spaces, 231 weakly orthogonal Banach lattices, 212
Standard bases, 245 Stationary point
of a semigroup, 502 Strict convexity, 104, 127 Strictly convex, 137, 142 Strong convergence, 71 Strong weakly orthogonal, 214 Structure, 613
face, 646 Subdifferential, 97, 123, 125
of f at x, 97 of the norm at nonzero points, 98 of the norm at zero, 98
Subdifferentials, 122 Submean, 517, 519-520, 522-523, 526
invariant, 517 left invariant, 517, 519 left subinvariant, 525, 527 right invariant, 519
Subsequence ~minimal, 245
Subset a-minimal, 246-247 x-minimal, 245
702
q,..minimal, 244-245 IT-porous, 571 admissible, 397 diametral, 119 directionally bounded, 63 externally hyperconvex, 397 linearly bounded, 63 porous, 571 proximinal, 398 weakly externally hyperconvex, 397
Subspace closed left translation invariant, 525 translation invariant, 518
Successive approximations, 558 Sum property, 343 Super-property, 183 Super fixed point property, 195 Superreflexive, 103, 107, 117, 128, 183 Theorem
Abian-Brown-Pe1czar, 614 Alexander's subbase, 82 Alexandroff-Pasynkoff, 674 Amann, 614 Angelov's, 614, 619, 621, 631, 633 Baillon's, 72, 543 Banach, 240, 303 Bessaga-Pelczyr'iski, 277 Borsuk-Lyusternik-Shnirel'man on antipodes, 244 Bourbaki-Kneser, 623 Brouwer's fixed point, 239, 577, 608 Browder-Gohde-Kirk, 628 Calka's, 438 Cantor intersection, 241, 625, 628 Caristi's, 301, 613-614, 624-625, 627-628
restriction to continuous functions, 625 restriction to Lipschitzian functions, 626
Chu-Mellon, 490 classical fixed point, of Lim, 518 convergence, 566 Darbo-Sadovskii, 415 Darbo and Sadovski, 242 Day's ergodic, 541 Denjoy-Wolff, 483 Denjoy-Wolff, for isometries, 487 Denjoy-Wolff, in Banach spaces, 490 Domain Invariance, 306 dominated convergence, 116-117 Drop, 626 duality, 106 Earle-Hamilton, 450 Eberlein-Smulian, 252 Eilenberg and Montgomery, 663 ergodic, 518 Fan's, 527 fixed point, 128
for <;f>-contractive mappings, 634 inL' (n) with elm the topology, 127
Fuchssteiner iteration, 628 Theorem
fundamental for nonexpansive mappings, 614, 623, 629 Hutchinson-Barnsley, 634
Gorniewicz, 663 Generalized Hartogs" 481 Goldstines's, 282 Hahn-Banach, 72, 393, 518 Hausdorff maximal chain, 625 Hutchinson-Barnsley, 614, 636-637 intersection in hyperconvex spaces, 671, 674
Theorem
James's distortion improved version for co, 291 stronger version, 290 stronger version for £1, 271, 289 stronger version for co, 290
K-T, 614, 617, 619, 621-622 dual version, 614 set-valued version, 621
Kadec-Pelczynski, 276 Khamsi's structure, 84 Kirk's, 254, 62!Hl30 Kleene's fixed point, 630 Knaster-Tarski, 613-614, 631 Kneser's, 623 Kordylewski and Kuczma, 621 Krein-Smulian, 293 Kuratowski, 241, 617
generalization of, 241 Ky Fan type, 427 Lin's, 223 Luxemburg's, 287 Mackey-Arens, 542 Maurey's, 276 Mazur's, 111, 120, 542 Monotone Convergence, 522 Nadler's, 613, 622, 626--627
more general form, 622 Nash-Moser, 614 nonlinear ergodic, 543 nonretraction, 578, 608 on j-continuity, 635 Pelczynski and Hagler, 285 Rakotch's fixed point, 634 Ramsey's, 120, 246 Schauder's fixed point, 239, 242-243, 334, 336--337,
580,608 Schur's, 109 Smithson, 621-622 Soardi's fixed popint, 630 T-K, 630-631, 633--634, 636-637
restricted to Ekeland '5 ordering, 634 Tan-Tarafdar, 619, 621, 633
on the uniform space (X,P'), 621 Tarski's fixed point, 635 Tarski-Kantorovitch, 613-614, 630 Turinici's fixed point, 627 Tychonoff's, 65, 522 weak ergodic, 565 Yuan, 664 Zermelo's, 613, 623, 625-626, 628-630
extension of, 627 generalization of, 627 special form of, 627 variant of, 627
Zhou,650 Theory
neutral functional differential equations, 618 nonlinear ergodic, 547 nonlinear semigroups, 614 of amenability, 522 of IFS, 634, 636-637 of iterated function systems (IFS), 634
Topological groups, 518 Topological semigroup, 522 Topological sub-semigroup, 522 Topology
clm-,255 elm, 108, 118, 127, 256 Hausdorff T, 108 Hausdorff, 518
INDEX
linear, 108, 254-256 Mackey,542 norm, 108, 518, 569 of convergence in measure, 108 of convergence locally in measure (elm), 108, 250 relative, 563 strong, 565-566 weak, 108-109, 116, 128, 252, 255, 518, 542,
565-566 weak star, 266 weak',108
Trace faithful, normal and finite, 288
Triangulation, 644 UC, 104, 127, 135, 138, 141, 143, 152
in metric spaces, 127 UCED, 99--100, 120, 141, 159 UKK(T),256 UKK( T) property, 250 Ultrafilter, 55, 60, 178
countably incomplete, 178 free, 178 limit along, 179 nontrivial, 163, 178
Ultrametric space, 27-28 spherically complete, 27-28
Ultrapower, 413 finitely represented in the space, 181 of a Banach lattice, 183 of a Banach space, 179 of an Lp-space, 186
Ultraproducts, 54 Ultraproducts of maps, 184 Unconditional basis, 195 Uniform asymptotic normal structure, 170
with respect to the weak topology (w-UAN), 170 Uniform convexity (UC), 102 Uniform convexity, 54, 74, 127, 152, 184
Clarkson's, 127, 143 finite dimensional, 127 in every direction, 127 infinite dimensional generalization, 110 multi-dimensional, 127 noncompact, 127
Uniform Kadec-Klee property, 362 with respect to T (UKK(T)), 109
Uniform monotonicity, 358 Uniform nonsquareness, 367 Uniform normal structure, 58, 119, 128, 134, 146, 154,
159, 163, 343 Uniform Opia! condition, 168 Uniform Opial property, 368 Uniform rotundity (UR), 102 Uniform rotundity, 269 Uniform rotundity in every direction, 356
Uniform semi-Opia! property, 233 Uniform smoothness, 106-107, 127, 269
multi-dimensional, 126, 128 Uniform space, 613
(X,P),633 complete Hausdorff, 617 Hausdorff, 631 sequentially complete, 631
Uniformity, 558, 562, 565, 567, 570 Uniformly (F)-attracting with respect to e, 568 Uniformly convex, 103, 142, 154, 529 Uniformly monotone norm, 194 Uniformly nonsquare, 142, 163, 184 UnifOImly rotund
in every direction, 292-293 Uniformly smooth, 154, 529 UNS, 121-122, 125-126, 128, 143, 157 US, 106-107, 127, 141, 153 USO
w-,l71 Vertices of the simplex f),.N, 644 Volterra integral equation, 2 Volume
k-dimensional, 101 Von Neumann algebra, 531 W-fpp, 35, 186 W'-fpp,35 W'UR,139 Weak convergence, 73
703
Weak fixed point property, 35, 186, 252, 259, 265, 279 for co, 193
Weak lower sequential semi-continuity, 265 Weak normal structure, 119, 128, 148-150, 187, 292,
343 Weak null type, 198 Weak orthogonality property, 355 Weak sum property, 343 Weak uniform normal structure, 160,380 Weak'-topology, 518 Weak'-f.p.p., 532 Weak' fixed point property, 35 Weak' uniformly rotund, 139 Weakly convergence sequence coefficient
WGS(X),380 Weakly externally hyperconvex, 397 Weakly orthogonal, 213 Weakly orthogonal Banach lattice, 193 Weight, 98 WFPP, 157, 165, 167, 170 WLUR,139 WNS, 119, 121, 141, 160, 164-165 WNUS, 164, 362 WORTH,367 WUR, 138-139, 141 Zorn's lemma, 51, 53, 55, 66, 82, 528, 620, 623, 625,
628