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Journal of Applied Mathematics and Stochastic Analysis8, Number 3, 1995, 261-264

RANDOM FIXED POINTS OF WEAKLY INWARDOPERATORS IN CONICAL SHELLS

ISMAT BEGKuwait University

Department of MathematicsP.O. Box 5969

Safat 13050, Kuwait

NASEER SHAHZADQuaid-i-Azam University

Department of MathematicsIslamabad, Pakistan

(Received February, 1995; Revised March, 1995)

ABSTRACT

Conditions for random fixed points of condensing random operators areobtained and subsequently used to prove random fixed point theorems for weaklyinward operators in conical she]Is.

Key words: Banach Space, Inward Operator, Random Fixed Point.

AMS (MOS)subject dassifications:47H10, 60H25, 47H40, 47H04.

1. Introduction

The fixed point theorems play a very important role in the questions of existence, uniquenessand successive approximations for various type of equations. The random fixed point theoremsare useful tools for solving various problems in the theory of random equations, which form a partof random functional analysis. In recent years, there has been exciting interaction betweenanalysis and probability theory, furnishing a rich source of problems for analysis. Bharucha-Reid[2] proved the random version of Schauder’s Fixed Pint Theorem. Random fixed point theory hasfurther developed rapidly in recent years; see e.g. Itoh [4], Sehgal and Waters [8], Sehgal andSingh [7], Papageorgiou [6], Lin [5], Xu [10], Tan and Yuan [9] and Beg and Shahzad [1]. Thispaper is a continuation of these investigations. We have obtained random fixed points of weaklyinward random operators in the shell, a stochastic analogue of the results of Deimling [3].

2. Preliminaries

Let (D, at) be a measurable space with at as a sigma-algebra of subsets of D. Let X be a realBanach space; a map q:--X is called measurable if for each open subset G of X, q-I(G) E at.Let S be a nonempty subset of X; a map f: 12 x S-X is called a random operator if for each fixedx E S, the map f(. ,x): D--+X is measurable. A measurable map :gt--+S is a random fixed pointof the random operator f if f(w, {(w)) (w) for each w e gt.

Let K C X be a cone; that is, K is closed and convex such that AK C K for all A > 0 and

KN(-K)-{0}. Denote {xK: Ilxll <r} by gr and {xK’p< Ilxll <r} by gp, r for

Printed in the U.S.A. ()1995 by North Atlantic Science Publishing Company 261

262 ISMAT BEG and NASEER SHAHZAD

some 0 < p < r. A map f: Kr+X is a-condensing if f is continuous and bounded and a(f(B)) <a(B) for all B C gr with a(B) > 0, where a(B)- inf{d > 0: B can be covered by a finite numberof sets of diameter _< d}. A random operator f:f.x Kr---*X is a-condensing if for each w

f(w,. is a-condensing. An operator f:f x C--,X is said to be weakly inward on a closed convexsubset C of X if for each w f, f(w, x) Ic(x) for all x e C, where Ic(x denotes the closure of{x + A(y x)" A _> 0, y C}. In the case C K, it simply becomes x Oh’. The latter, x* E K*(the dual cone), and x*(x) 0 imply that

x*(f(w, x)) > 0 (1)

for each w f, where X* is the dual space, K* = {x* e X*:x*(x) >_ 0 on K} and OK is the bound-ary of g. Suppose f:f x grx satisfies x OK, [[ x [[ _< r, x* K* and x*(x)= 0 imply thatx*(f(w,x)) >_ 0 for each w E f. Let Pr:ggr be the radial projection; that is, Pr(x)= x forI[ x 1[ <_ r and Pr(x) IlrX II for 1[ z 11 > r. Then for each w e f, f(w,. )o Pr satisfies (1). Since

c(Pr(B)) _< c(B) for all bounded B C K, f(w,. o Pr is a-condensing if f is.

3. Main Results

Theorem 1: Let X be a separable Banach space, K C_ X a cone and f’f Kr---*X an a-con-

densing random operator, such that(i) x e OK, Il x ll - r,x* e g* and x*(x) O imply that x*(f(w,x))

_O for all w , and

(ii) f(w,x) Ax on [[x[[ r and for all w E f and A > l, are satisfied.Then f has a random fixed point.

Proof: Define a random operator g:f x gr--X by g(w,x)- (f(w,.)o Pr)x. It is clear fromDeimlig [3, proof of Theorem 1] that for any w f, g(w,. is a-condensing and weakly inwardon Kr for some r > 0. Further application of Xu [10, Theorem 2] or Tan and Yuan [9, Corollary2.6] gives the desired result. [3

Theorem 2: Let X be a separable Banach space, K C X a cone, f: x Kr---*X an a-condens-ing random operators, such that (i), (ii) (from Theorem 1) and

(iii) there exists p G (0, r) and e K\{O} such that for any

for II II P and A > 0

are satisfied.Then f has a random fixed point in K

Proof: Let Pr be as before and for all w

C(w) sup{ II f(w, x)I1" e Kr}.

We use (ii) to get a "barrier" at II x l] P as follows. Let Cn: x [0, r]--,[0, cx) be a continuousrandom map such that ,(w, t) 0 for t > p and ,(w, t) 5(w) for t < p- and large n, with ameasurable map 5:(0,)such that 5(w) llll > p/C() for each w. Let f,(w,x)=

/ II II Evidently, fr, is a-condensing and weakly inward random opera-tor on K. By Theorem 1, there exists a measurable map ,:K such that (,(w)=f(w,,(w)) for each w E . Fix w arbitrarily. We cannot have II > . Hence,(n(w) f(w, (n(w))+ Ca(w, 11 (n(w)II )" By the choice of , we cannot have IIand we are done if [[(n(w)[[ >p. So assume that p-< II(n(w)[[ <P for all large n. Since{Ca(w, II ((w)[] )} is bounded and f is a-condensing, we have without loss of generality (n(w)-

Random Fixed _Points of Weakly Inward Operators in Conical Shells 263

f(w,(n(w))-Ae and tn(w)o(W) with II 0(w) II-P where A depends on w. Hence, (o(W)-f(w,(o(W)) + Ae and therefore A- 0 by (iii). E]

Theorem 3: Let X be a separable Banach space, K C X a cone such that K1 is not compact,f’ KrX a compact random operator satisfying (i) and (ii) of Theorem 1 and() t (o.) uc tat f(..) . o I111- ad e(0.1) and

Then f has a random fixed point in Kp,r"

Proof: Since for any wgt inf{llf(w,x) ll’llxll -P} >0, and A-{xK: ]lxll -1} isnot compact, we can find e A such that -Ae f(w, pA) for all A _> 0.

Consider a random operator fn: x KX defined by

fn(w,x)

(f(w, .)o Pr)x

z)

0 lii +(’ II

for IIII >_,o

fr 0<Po-< IIII <for II II < po

where tn:f [0, P0]---*[0, cx is a continuous random map, tn(W, p0)- 0 and tn(w,t)- 5(w) fort <_ P0(1- ), with a measurable map 5: fl(0, o) such that

5(w) > Po -4-sup{ II f(, )I1"11 II p) for each

Since fn is compact and a weakly inward random operator, it has a random fixed point n"Kr as before. We cannot have Po<- IIn(w) ll <P or IIn(w) ll -<P0(1-) by (iv) and thechoice of Ca" If, for any w

1such that Po- < II n(w) II < Po for all large n. Then

II ()p II ()_ IIpwith II /,,(w)II- II II ()II ’() II- p, By letting noo we get Yo-f(w, Yo)+ Se with

Yo pA and 0 < A < 5(w) depending on , hence po0 yields oe f(w, pA) for some $o > 0,contradicting the choice of e.

Pemark 4: Theorem 3 does not hold if K1 is compact. For a counterexample, see [3].

Acknowledgement

This work is partially supported by Kuwait University Research Grant No. SM-108.

References

[]

[2]

[3]

Beg, I. and Shahzad, N., Random fixed points of random multivalued operators on Polishspaces, J. Nonlinear Anal. 20 (1993), 835-847.Bharucha-Reid, A.T., Fixed point theorems in probabilistic analysis, Bull. A mer. Math.Soc. 82 (1976), 641-657.Deimling, K., Positive fixed points of weakly inward maps, J. Nonlinear Anal. 12 (1988),223-226.

264 ISMAT BEG and NASEER SHAHZAD

[4]

[5]

[6]

[7]

IS]

[lo]

Itoh, S., Random fixed point theorems with an application to random differential equa-tions in Banach spaces, J. Math. Anal. Appl. 67 (1979), 261-273.Lin, T.C., Random approximation and random fixed point theorems for non-self-maps,Proc. Amer. Math. Soc. 103:4 (1988), 1129-1135.Papageorgiou, N.S., Random fixed point theorems for measurable multifunctions inBanach spaces, Proc. Amer. Math. Soc. 97 (1986), 507-514.Sehgal, V.M. and Singh, S.P., On random approximations and a random fixed pointtheorem for set-valued mappings, Proc. Amer. Math. Soc. 95 (1985), 91-94.Sehgal, V.M. and Waters, C., Some random fixed point theorems for condensing operators,Proc. A mer. Math. Soc. 90 (1984), 425-429.Tan, K.K. and Yuan, X.Z., Random fixed point theorems and approximation, J. Stoch.Anal. Appl., (to appear).Xu, H.K., Some random fixed point theorems for condensing and nonexpansive operators,Proc. Amer. Math. Soc. 110 (1990), 395-400.

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