Available at: http://www.ictp.trieste.it/~pub-off IC/98/214

United Nations Educational Scientific and Cultural Organizationand

International Atomic Energy Agency

THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

ITERATIVE APPROXIMATIONOF FIXED POINTS OF DEMICONTRACTIVE MAPS

Chika Moore1

Department of Mathematics and Computer Science, Nnamdi Azikiwe University,P.M.B. 5025, Awka, Anambra State, Nigeria2

andThe Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.

Abstract

Let K be a closed convex nonempty subset of a real q-uniformly smooth Banach space and

let T : K i—> K be demicontractive with (I — T) demiclosed at 0 G K. It is proved that the

Mann and Ishikawa iteration processes with errors (in the sense of Xu [24]) converge weakly to

a fixed point of T. If K is compact and convex, then the convergence is strong. Related results

deal with other sufficient conditions for strong or weak convergence of the iteration processes

used.

MIRAMARE - TRIESTE

November 1998

1 Regular Associate of the Abdus Salam ICTP.This research was supported in part by Research Grant RG/MATHS/AF/AC No.97-210 from the Third worldAcademy of Sciences (TWAS).

2Permanent address.

1 Introduction:

Let H be a Hilbert space. In [1,7] the following class of maps was introduced. A map T with

domain D(T) and range R(T) in H is said to be strictly pseudocontractive (in the sense of

[1,7,9,13]) if there exists a constant k G [0,1) such that Vs,j/e D(T) we have that

2 < ||a; - y| |2 + k\\x - y - (Tx - Ty)\\2 (1)

Let F(T) = {x G D(T) : Tx = x}. If F(T) / 0 and there exists a constant k G [0,1) such that

\fx G D(T) and x* G F(T) the following inequality holds

| |Ta;-a;* | | 2 < \\x - x*\\2 + k\\x-Txf (2)

then, T is said to be demicontractive. It is easy to see that a strictly pseudocontractive map

with a fixed point is demicontractive. Thus, the class of demicontractive maps properly includes

the class of strictly pseudocontractive maps with fixed points (see e.g., [1,7,17]). The important

class of quasi-nonexpansive maps (where a map T is said to be quasi-nonexpansive if F(T) = 0

and 11To; - x*\\ < \\x - x*\\; V s e D(T) and Va;* G F(T)) is also a subclass of this class of

maps. Moreover, if the fixed point set F(T) is nonempty, then the classes of maps studied by

such authors as Goebel et al [6], Kannan [9] and Wong [21] are special subclasses of the class of

demicontractive maps. Furthermore, If we set λ = 2k) , then it is routine to see that, in Hilbert

spaces, (??) is equivalent to the condition: there exists a constant λ > 0 such that M x G D(T)

and x* G F(T) we have that

-Tx, x-x*)>X\\x-Tx\\2 (3)

which is the condition satisfied by the class of maps introduced by Maruster [12]. Thus, the class

of nonlinear maps introduced in 1977 by Hicks and Kubicek [7] and Maruster [12] independently

coincide in Hilbert spaces. It is easy to observe from (??) that

||Ta?-a?*|| < ||aj-aj*|| + \/fc||a;-ra;||

< (l + \/fc)||a;-a;*|| + \/fc||ra;-a;*||

(l-\/fc)||Ta;-a;*|| < (1 + Vk)\\x - x*\\

\\Tx-x*\\

Wth ^ ( l ± | ) (4,

and from (??) that

\\x-x*\\ > A||a:-Ta:| | > A(||Ta: - x*\\ - | |a;-a;*| |)

so that

| |Ta:-a:*| | < L | | a : - a : * | | ; with L = l + A"1 (5)

Several authors have studied this class of nonlinear maps (see e.g., [1-9, 10, 12-20]) and conver-

gence theorems established for iteration processes of the Mann-type (see e.g., [11]).

It is our purpose in this paper to establish strong and weak convergence of both the Mann

and Ishikawa iteration processes (with errors) to a fixed point of a demicontractive map in real

q-uniformly smooth Banach spaces.

2

2 Preliminaries

Let E be a normed linear space of dimension dim.E > 2. The modulus of smoothness of E is

defined by

1 : ||x|| l , | W

The linear space E is said to be smooth if ΡE(Τ) > 0 Vr > 0. If there exists a constant c > 0

and a real number 1 < q < oo such that

then E is said to be q-uniformly smooth. Typical examples of such spaces are the Lesbesgue Lp,

the sequence £p and the Sobolev Wpm spaces for 1 < p < oo. In fact, these spaces are p-uniformly

smooth if 1 < p < 2 and 2-uniformly smooth for p > 2.

Let E* denote the dual of E and let Jp : E i-> 2E* denote the generalized duality mapping

defined by

/ r — { f* (z w* • I T f*\ — \\r\\P- II f*ll — H r P " 1 !,Jp.L •— \J t -̂ • \-L> J I — II-HI ) IIJ II — II-HI J

where (.,.) denotes the generalized duality pairing. It is known (see e.g., [22,23]) that Jp is

single-valued (and denoted by jp) and 'Lipschitz' Holder-continuous with constant L* > 0 HE

is p-uniformly smooth. That is,

WjpX-jpVW^LtWx-yW"-1, Vx,yeE

Moreover, Mo / x G E, Jpx = \\x\\p~2J2X where J = J2 is the normalized duality map.

If E is a real q-uniformly smooth Banach space, then (see e.g., [22]) the following inequality

holds: V x,y G E and some positive constant Cq we have that

\\x + y\\q<\\xr + q(y,jp{x))+Cq\\y\\q (6)

In [2], Chidume extended the condition (??) to arbitrary real Banach spaces thus: Vx G

D(T), x* G F(T) and j(x — x*) G J(x — x*) there exists a positive constant λ such that

(x-Tx,j(x-x*))>X\\x-Tx\\2 (7)

If E is q-uniformly smooth, then condition (??) becomes

(x-Tx,jq(x-x*)) > \\x-x*\r2(x-Tx,j(x-x*))

> \\x-x*\\q-2.X\\x-Tx\\2

> \q-l\\x-Tx\\q

So that we now have the following inequality

(Tx - x*, jq(x - x*)) < \\x-x*\\q -Xq-l\\x-Tx\\q (8)

We shall need the following definitions in the sequel: Let K be a subset of a Banach space

E. A map T : K i-> E is said to be demiclosed at z G E if the weak convergence of {xn} C K

to some point p G K and the strong convergence of {Txn} to z implies that Tp = z. A map

T : K <-^ E is said to be demicompact at z G K if for any bounded sequence {xn} C K such

that (I — T)xn -^ z as n —> 00 then there exist a subsequence {xn j} and a point p e K such

that x n j —> p as j —>• 00 and (I — T)p = z.

We now prove the following theorems.

3

3 Main Theorems

Theorem 1 Let K be a closed convex nonempty subset of a real q-uniformly smooth Banach

space and let T : K i—> K be demicontractive with (I — T) demiclosed at o e K. Let {un}, {vn}

be bounded sequences in K and let the real sequences {an}, {bn}, {cn}, {a'n}, {b'n}, {c'n} C [0,1]

satisfy the following conditions:

(i) an + bn + cn = 1 = a'n + b'n + c'n; n > 0

(ii) ^ bn = oo and ^ cn < oon>0 n>0

(iii) ^ bnb^ < oo and ^ bncn < s<n>0 n>0

where s = min{1, q — 1}. Then, the sequence {xn} generated from an arbitrary X0 G K by

xn+1 = anxn + bnTyn + cnvn; n > 0 (9)

yn = <4a:ra + ft^T^ + c'nun; n>0 (10)

converges weakly to a fixed point ofT.

Proof. Let x* G K be a fixed point of T. Then, we have the following estimates:

\\h'(Tr — r*) 4- r'(v — r*)\\q < b'q\\Tr — r*\\q 4- C rqhi — r*\\q\\unyi xn x ) -|- cn\u,n x ) \ \ ^ wn ii-i xn x || -|- OgCra \\an x \\

-\-ab'(q~l^r' \\v — r*\\ WTr — r*\\q~l

i<lun cn\\u'n -L W-W1-Ln X ||

< Tqh'q\\<r — <r*\\q -i- C r'q\\ii — r*\\q

— T*) — i (v — T*)\\ < L \W —II I I 9 " 1x ) Jq\yn X ) \ \ L ±j*\\£n yn\\

< L , {[(L + l)b'n + dn\ \\xn -x*\\+ dn\\un - x*

\Vn-x*\\q

r' (v — r* i (r — r*)\ 4- C, \\b' (Tr — r*\ -\- r' (11 — r*)cn\u"n. -L i Jq\-Ln X )) i~ ̂ q\\Un[-L Xn X ) -\- Cn[U,n X )

— rx\ an \\xn x || q+qan un' (\\r — r* \\q — \q~l I IT —Trn i ||xra x \\ A \\xn J-xn

ln Cn\\Un' Hi/ — o™* II 11 o™ — T * \\q~l A- C \\h (Tv _ T * U ^ (11 — <r*\\\q

n\\Un -h \\-\\-hn X || \ ^qW^ny1-^n X ) \ (^n\U>n X ) \ \

~l) (n' 4- / I / )MHT T * I | 9 n\q~ln'(q~l)h' I IT T T \\q

^ / ) ' ( < ? - ! ) • / II , , _ ™ * | | | | T T — T * ! ! 9 " 1 -I- I ^ V ^ H i / — T * l l g

-^ q n n 11 n I r l l ^ i II ' q n 11 ^ 11

-Vi ) ll^ra X \\ qA (l,n Vnv\Xn J- %n\\

qc'n ( a ^ 1 ) + CqLilb^) \\un - x*\\.\\xn - x*

+qc'n I a ra« ^ + G ' g ^ ^ ^ ^ I \\Un - x'-\\.\\xn - x"

The last inequality follows from the fact that for x, y G [0,1] the following estimate holds

f(x, y) = il-x-y)q + q i l - x - y)q~lx + CqLqxq < (1 - x)q + q(1 - a;)"-1^ + C q L q x q

Now,

3J 11 S : Q"n 11 ^"ra ^ II ~T~ ^ q 11 ^ n \-L y n X ) -\- Cn (Vn X )\\

+qaql~

1bn{Tyn - x*,jqixn - x*)) + qa^lCn{vn - x*,jqixn - x*))

< aqjxn - x*\\q + qatlbn (\\yn ~ x*\\q - \q-l\\yn - Tyn

+qaql~

1bn(Tyn - x*,jq(xn - x*) - jq(yn - x*))

+qaqr1cn{vn - x*,jqixn - x*)) + Cqbqn\\Tyn - x*\\q

+C2qc

qn\\vn - x*\\q + qCqb

qn-

lcn\\vn - x*\\.\\Tyn - x*\\q~l

< Inq A- nnq~lh A- C Tqhq\ \\v — T* 11̂ — n\q~lnq~lh Hi/ — Tii II— X^n'q^n " n T W » J J Un I \\Xn X || qA U,n Un\\yn -1 yn\\

— n \ q ~ l n ' ^ q ~ ^ > f n n q ~ 1 A - C T q b q ~ 1 ^ b ft' I I T — T T II9

qA Un \qun -\- K^/qLi Un I UnUn\\Xn J- Xn\\

\ n (n'(q-l) I n r<?-l?,/(<?-l)A\ (nnq~l 4- H Tqhq~l\h o1 Wti T * I I I I T-\-q \Un -\-lsqL/ Un I \qtln ~\- isqLz Un I UnCn\\Un — X | | . | |Xn — .

C2q ( g a r 1 + CqL%tl) bnc'«\\un - x*\\q + C&\\vn - x*\\q

tlbn ([(L + l)b'n + c'n]\\xn -x*\\ + c'Jun - x*\\)q~l x

\\xn-x*\\+c'n\\un-x*

1bt1Cn\\Vn ~X*\\. ([(L + l)b'n + C'n + l]\\xn -X*\\+ C'jun -

-nnq~lr Wit —r*\\ \\r — v*\\q~lqan tn\\vn x \\.\\xn x 11

Let

M = max<sup{||-ura - a ;* | | } , sup{||vra - x*n>0 n>0

We consider the following two cases:

case 1: all n > 0 such that \\xn — x*\\ > M: In this case, we have that

\\xn+i-x*\\q <(l+-fn)\\xn-x*\\q, with ^ 7 r a < o o

case 2: all n > 0 such that \\xn — x*\\ < M In this case, we have that for some constant

M1 > Mq the following inequality holds

Hi- , 1 — 1-*II* < Hi- — 1-*II* -I- M . ' vH-^ri+l X || \ \\Xn X || -f- IVIl /n

Thus, limra^oo \\xn — x*\\ exists and so the sequences {xn}, {yn}, {Txn}, {Tyn} are bounded.

Furthermore, for some positive constant M0 we then have

q\q-laq-lbn\\yn - Tyn\\q < \\xn - x*\\q - \\xn+l - x*\\q + M0(bn^ + bnc',° + cn)

Iterating downwards, we have

q\q~l Y, <~lbn\\yn - Tyn\\q < \\x0 - x*\\q + Mo ^ ( W l s + bj° + cn) < oo

ra>0 ra>0

so that from the hypothesis, we deduce that

liminf \\yn -Tyn\\ = 0

and so there exists a subsequence {ynj} C {yn} such that

ynj -^ V and ynj - Tynj -^ o as j - • oo

Since (I — T) is demiclosed at o e i f and K is weakly closed, we have that p G F(T), the fixed

point set of T. Thus, for any /* e i£* we have that

fiVnj -p) ->0 as j ^ oo

Now,

f*(xnj -P) = f*(xnj - ynj) + f*(yn. -p) = b'nJ*{xnj - Txnj) + dnj*(xn] - unj) + f*(yn. -p)

so that

I™ \f*{xni -p)\ < lim {b'n]\f{xn]-Txnj)\+d \f{xn]-unj)\ + \f*(yn. -p)\\ = 0j ^ o o J 3—>oo L J J J j J J j j

and hence xnj -^ p as j —>• . Observe that for some constant M0 > 0,

a'Jxn-Tyn\\ < \\yn-Tyn\\+b'n\\Txn-Tyn\\+c'n\\un-Tyn\\

< \\yn-Tyn\\+M0(b'n + c'n)

Hence,lim \\xnj -Tyn\\ = 0

We now claim thatf*(xnj+k-p)^0 as j ^oo ; Vfc>0

Suppose that the claim is true for some k = m. Then, from

f*(ynj+m -p) = b'n.+mf*(xnj+m - Txnj+m)

+c'nj+mf*(Unj+m ~ xnj+m) + f*(xnj+m-p)

we see that

Vrij+m -^P as j -^ oo

so that, additionally, from

.TOcrv+m+i ~p) = bnj+mf*(Tynj+m-xnj+m)

+Cnj+mf*(vnj+m - xnj+m) + f*(xnj+m - p)

we have that

\f*(xnj+m+i-p)\ < Mcnj+m + \f(xnj+m-p)\

+ \\f*\\-\\xnj+m-Tynj+m\\ -^ 0 as j -• oo

Since, the claim is trivially true for k = 0, it follows by the inductive hypothesis that the claimholds for all k > 0. Hence, {xn} converges weakly to a fixed point of T. This completes theproof. •

Theorem 2 Let E and the sequences be as in Theorem 1. Suppose that K is compact convexand that T : K i-> K is demicontractive. Then, the sequence {xn} converges strongly to a fixedpoint ofT.

Proof. Proceeding as in the proof of Theorem 1, we obtain that liminf \\yn — Tyn\\ = 0. Thisimmediately implies that some subsequence {ynj} of {yn} converges strongly to a fixed point,say p, of T. From

\\xn P\\ 2i \\xn yn\\ i^ \\yn p\\

<" yj ll/v. T1'?1 II 4 - r1 \\T ii— "rall"t/ra 1 -^nW \ l-n\vhn "-

II 4 - Wii-rill \ \\yn

it follows that {xnj} also converges strongly top. This implies that liminf \\xn— p\\ = 0. Observe

that from hypotheses of the theorem, σn = M0(bnb'n + bnc^ + cn) is summable and that with

Φn = \\xn -p\\q we have

Now, given any ε > 0 there exist an integer j0 sufficiently large such that

and another integer N 1 sufficiently large such that

since the tail of a summable series is arbitrarily small. Choose j * sufficiently large such that

njt > max{n j 0, N1}

Then, for any k > 0 we have that

k+1

r=0

< ε

+

ε = ε

- 4 4 2

Since, ε > 0 is arbitrary, it follows that xn —x a;* strongly as n —>• . This completes the proof.

n

Theorem 3 Let T,K,E be as in Theorem 1 and let {un} be a bounded sequence in K. Let

{an}, {bn}, {cn} ^ [0,1] be real sequences satisfying the following conditions:

(i) an + bn + cn = 1; n>0 (ii) J ^ bn = oo (iii) J ^ cn < oora>0 ra>0

Then, the sequence {xn} generated from an arbitrary X0 G K by

xn+1 = anxn + bnTxn + cnun; n > 0 (1 1)

converges weakly to a fixed point ofT.

Proof. Let

<Tn •= [q^ + CgL^bt^llXn-xT^+C^llUn-xT1] \K ~ X*

δn := q\q-laq

n-lbn

Then, proceeding as in the proof of Theorem 1, we easily obtain

A I I T _ T V I I * < I I T — T * I I * — I I T M _ / J - * | | 9 - | _ / TO r a l l - t r a -t • t r i l l ^ \\Xn X \\ W-Ln+l X \\ -\- (Tn cn

so that

5 3 Sn\\Xn ~ Txn\\q < \\XO - X*\\q + ^ σncn < OOra>0 ra>0

Since J2n>oδn = oc then liminfra^oo \\xn — Txn\\q = 0. So, there exists a subsequence {xnj} of

{xn} such that

xnj — Txnj —> o and xnj -^ p as j —> oo

Since K is weakly closed and (I — T) is demiclosed at o G K then p £ K and (I — T)p = o.

We now claim that

lim \f*{x nj+k-p)\=0; Vfc>0 and V f* e E*

Suppose that the claim is true for k = m. Then,

lim \f(xnj+m+i-p)\ < lim \f*(xnj+m - p)| + lim 6 r a .+ m |/*(a; r a .+ m - Txnj+m)|J —>OO J—>OO J—>OO

+ lim cnj+m\f*(xnj+m-unj+m)\

= 0

Since the claim is trivially true for k = 0 the claim follows by the inductive hypothesis. Thus,

the assertion of the theorem holds and the proof is complete. •

Theorem 4 Let E and the sequences be as in Theorem 3 and let K be compact and convex.

Suppose that T : K i—> K is demicontractive. Then, the sequence {xn} converges strongly to a

fixed point of T.

Proof. Follows as in the proof of Theorem 2 with minimal changes. •

Corollary 1 In Theorem 1, let T : K i—> K be demicontractive and continuous at a point p, a

cluster point of {xn}. Let

(i) 0 < α < bn < 1, n > 0 (ii) J ^ cn < oo, (iii) J ^ b^ < oo, (iv) ^ cn < s<n>0 n>0 n>0

where s = min{1, q — 1}. Then, p e F(T) and {xn} converges strongly to p.

Proof. Let δn = q\q~1aq~1bn. Then, for some positive constant d we have that δn > d > 0, so

that from

5n\\Vn ~ T y n \ \ q < \\Xn - X*\\q - \\Xn+1 - X*\\q + M i ( & ; s + 4 S + Cn)

and, consequently,

d E WVn ~ TVnW < \\XO ~ X*\\q + M i J^ibn + Cn + Cra) < OOn>0 n>0

we conclude that

and also

lim \\xn -Tyn\\ = 0n—>oo

Since p is a cluster point of {xn}, there is a subsequence {xn j} of {xn} which converges strongly

to p. Then,

'Jnj -p\\ < an.\\xnj -p\\ + b'n.\\Txnj -p\\ + c'n.\\unj -p\\ -»• 0 as j oo

Hence, ynj —>• p as j —>• oo and so Tynj —>• Tp as j —>• . Since l i m ^ o o \\ynj — Tynj\\ =

\\p -Tp\\ = 0, it follows that p G F(T). As in the proof of Theorem 1, we obtain

So that since M1 J2n>o(cn + &« + cn) < °°> it follows that lim \\xn — p\\ exists. Hence, since

Hindoo \\xnj — p\\ = 0, we have that lim^^oo \\xn —p\\ = 0. This completes the proof. •

Corollary 2 In Theorem 3, let T : K i—> K be demicontr active and continuous at a point p, a

cluster point of {xn}. Let

(i) an + bn + cn = 1; n > 0 (II) 0 < α < bn < 1; (iii) J ^ c n < oora>0

Then, {xn} converges strongly to p.

Proof. Follows easily from Corollary 1 on setting b'n = 0 = c'n. •

Corollary 3 In Corollary 1, letT : K i-> K be a demicontractive map which is also demicom-

pact at o G K. Then, {xn} converges strongly to a fixed point ofT.

Proof. Since {yn} is bounded and the sequence {yn — Tyn} converges strongly to o, then by

the demicompactness of T there is a subsequence {ynj} of {yn} which converges strongly to

some point p G F(T). From

\xnj - p\\ < \\ynj - p\\ + b'n. \\xnj - Txnj || + c'n. \\xnj - unj |nj \\ \\

< \\ynj -P\\ + M*Q/n. + c'n.) ^ 0 as j - . oo

we have that {xnj} converges strongly to p. The rest now follows as in the proof of Corollary

1. •

Corollary 4 In Corolary 2, letT : K i—> K be a demicontractive map which is also demicompact

at o G K. Then, {xn} converges strongly to a fixed point of T.

Proof. Follows easily from Corollary 3. •

4 Remarks

1. By setting cn = 0 = dn in our theorems, we obtain that the usual Mann and Ishikawa

iteration processes converge (strongly and weakly, according to conditions) to a fixed point

of T.

2. The convergence theorems in this paper do not depend on the Opial condition and as such

our theorems hold, in particualr, in the Lebesgue Lp (p = 2) spaces.

3. Suppose that A : E i-> E is a continuous linear operator, with zero as an eigenvalue,

satisfying the condition that for all z G D(A) and some λ a positive constant,

10

Suppose further that f G R(A). Then we may apply, say, Corollary 1 to prove that the

Mann and Ishikawa iteration methods (with errors) converge strongly to a solution of the

equation Ax = f. The details are routine.

Acknowledgments

This research was carried out while the author was visiting the Abdus Salam International

Centre for Theoretical Physics, Trieste, Italy as an Associate; a generous grant from the Swedish

International Development Cooperation Agency (SIDA) made the visit possible. The author is

most grateful to both.

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