Approximation of the Nearest Common Fixed Point of Asymptotically Nonexpansive Mappings in Banach Spaces∗

WANG XIONG-RUI

(Department of Mathematics,Yibin University,Yibin,Sichuan,644007)

Approximation of the Nearest Common Fixed Point of Asymptotically Nonexpansive Mappings in Banach Spaces∗

WANG XIONG-RUI

(Department of Mathematics,Yibin University,Yibin,Sichuan,644007)

asymptotically nonexpansive mapping,sunny nonexpansive retraction, uniformly Gˆateaux differentiable,eeakly sequentially continuous duality

1 Introduction

In this paper,we are interested in the following iteration for a finite family of asymptotically nonexpansive mappings{T1,T2,···,TN}in the setting of uniformly convex Banach spaces:

where Tn=Tn(modN),n=(k−1)N+i,and i=i(n)∈{1,2,···,N},k=k(n)∈N,the set of natural numbers.

Especially,if N=1,then(1.1)is reduced to the following iteration

where T is an asymptotically nonexpansive map.Particularly,if T is a nonexpanive mapping and if Tnis seen as Tn,i.e.,if{T,T2,···,Tn,···}is replaced by an in finite family ofnonexpansive maps{T1,T2,···,Tn,···},then(1.2)is reduced to the following iteration for an in finite family of nonexpansive maps{T,T2,···,Tn,···}:

which is studied by Chang[1]in the setting of Hilbert spaces.

Throughout this paper,we assume that E is a uniformly convex Banach space whose norm is uniformly Gˆateaux differentiable,C is a nonempty closed convex subset of H,I is the identity mapping,and F(T)={x∈C:x=Tx}is the set of fixed points of mapping T.Denote by→and⇀the strong convergence and weak convergence,respectively.

A mapping T:C→ C is called an asymptotically nonexpansive mapping,if for any x,y∈C,there exists a real sequence{hn}such that

A mapping P:E→C is said to be

(1)sunny,if for each x∈C and t∈[0,1]we have

(2)a retraction of E onto C,if Px=x for all x∈C;

(3) a sunny nonexpansive retraction,if P is a sunny,nonexpansive mapping and a retraction of E onto C.

C is said to be a sunny nonexpansive retract of E,if there exists a sunny nonexpansive retraction of E onto C.

For the sake of the convenience,we may recall the following lemma firstly(see[2]and [3]):

Lemma 1.1LetEbe a uniformly convex Banach space,Cbe a nonempty closed convex subset ofEandT:C→ Cbe an asymptotically nonexpansive mapping.ThenI−Tis semi-closed at zero,i.e.,for each sequence{xn}inC,if{xn}converges weakly toq∈Cand{(I−T)xn}converges strongly to0,then(I−T)q=0.

Lemma 1.2[4]Let{an},{bn},{cn}be three nonnegative real sequences such that

Lemma 1.4([6],Theorem 1)LetCbe a closed,convex subset of a uniformly convex Banach space whose norm is uniformly Gˆateaux differentiable and letTbe an asymptotically nonexpansive mapping fromCinto itself such that the setF(T)of fixed points ofTis nonempty.ThenF(T)is a sunny,nonexpansive retract ofC.

Lemma 1.5([7],Proposition 1)LetCbe a nonempty subset ofE,and{Tn:C→C,n= 1,2,···,N}be a finite family of asymptotically nonexpansive mappings.Then

(1)there exists a real sequence{hn}⊂[1,∞)withhn→1such that

(2) {Tn:C → C,n=1,2,···,N}is uniformly Lipschitzian with a Lipschitzian constantL≥1,i.e.,there exists a constantL≥1such that

In this paper,we study the iteration(1.1).We assume that{T1,T2,···,TN}is a finite family of asymptotically nonexpansive mappings.By Lemma 1.5 we know that there exists a real sequence{hn}⊂[1,+∞)withsuch that for any x,y∈C,

We may write it in the following compact form:for any x,y∈C,

where n=(k(n)−1)N+i(n).

Especially,in the case of N=1,(1.4)is reduced into the following:

Proposition 1.1[8](1)If the norm of a Banach spaceEis Gˆateaux differentiable,then the normalized duality mappingJis single-valued and norm-to-weak∗continuous;

(2)If the norm ofEis uniformly Gˆateaux differentiable,then the normalized duality mappingJis single-valued and norm-to-weak∗uniformly continuous on each bounded subset ofE;

(3)Every uniformly convex Banach space is re fl exive.

Next,we also need the following two propositions(see[9]–[13]).

Proposition 1.2LetCbe a convex subset of a smooth Banach space,Kbe a nonempty subset ofCandPa retraction fromContoK.ThenPis sunny and nonexpansive if and only if

Proposition 1.3A Banach spaceEis smooth if and only if the normalized duality mappingJ:E → 2E∗is single-valued.In this case,the normalized duality mappingJis strong-weak∗continuous.

2 Main Results

In this section,we assume that E is a uniformly convex Banach space with a weakly sequentially continuous duality,whose norm is uniformly Gˆateaux differentiable.In fact,there exists a lot of Banach spaces satisfying the conditions above,such as all the sequence spaces lp(1＜p＜∞),all Hilbert spaces,and so on.Motivated by some results and methods of [14]–[22],we give and prove the following theorem:

Remark 2.1(1) In the frame work of Hilbert spaces,the fixed point set F(T)of a nonexpansive mapping T:C→C is closed and convex.In addition,a sunny nonexpansive mapping from H to F(T)is equivalent to the nearest point projection from H to F(T). Thus,Theorem 2.1 of[1]is a corollary of our Corollary 2.1.Furthermore,we know from[1] that the corresponding results of[14]and[15]are corollaries of our Corollary 2.1.

(2)Corollary 2.1 can be at least applied to the case that T is an aysmptotically nonexpansive and uniformly asymptotically regular mapping.

De fi nition 2.1A mappingT:C→Cis called uniformly asymptotically regular if for eachε＞0,there exists an integern0∈Nsuch that

Proof.We only need to know whether the conditions(i)and(ii)of Corollary 2.1 are satis fi ed.For this purpose,we take

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Communicated by Ji You-qing

47H09

A

1674-5647(2011)04-0369-09

date:Oct.30,2010.

The Found(2011Z05)of the Key Project of Yibin University.