Weak Convergence Theorems for Nonself Mappings

LIU YoNG-QUAN AND GUo WEI-PING

(School of Mathematics and Physics,Suzhou University of Science and Technology, Suzhou,Jiangsu,215009)

Communicated by Ji You-qing

Weak Convergence Theorems for Nonself Mappings

LIU YoNG-QUAN AND GUo WEI-PING

(School of Mathematics and Physics,Suzhou University of Science and Technology, Suzhou,Jiangsu,215009)

Communicated by Ji You-qing

LetEbe a real uniformly convex and smooth Banach space,andKbe a nonempty closed convex subset ofEwithPas a sunny nonexpansive retraction.LetT1,T2:K→Ebe two weakly inward nonself asymptotically nonexpansive mappings with respect toPwith a sequence{k(i)n}⊂[1,∞)(i=1,2),andF:=F(T1)∩F(T2)/=∅.An iterative sequence for approximation common fxed points of the two nonself asymptotically nonexpansive mappings is discussed.IfEhas also a Fr´echet diferentiable norm or its dualE∗has Kadec-Klee property,then weak convergence theorems are obtained.

asymptotically nonexpansive nonself-mapping,weak convergence,uniformly convex Banach space,common fxed point,smooth Banach space

1 Introduction and Preliminaries

Throughout this work,we assume thatEis a real Banach space,E∗is the dual space ofEandJ:E→2E∗is the normalized duality mapping defned by

where〈·,·〉denotes the duality pairing betweenEandE∗.A single-valued normalized duality mapping is denoted byj.It is well known that ifEis a smooth Banach space,thenJis single-valued.

A subsetKofEis said to be retract ofEif there exists a continuous mappingP:E→Ksuch thatPx=xfor allx∈K.Every closed convex subset of a uniformly convex Banach space is retract.A mappingP:E→Eis said to be a retraction ifP2=P.It follows that if a mappingPis a retraction,thenPy=yfor allyin the range ofP.LetCandKbe subsets of a Banach spaceE.A mappingPfromCintoKis called sunny ifP(Px+t(x-Px))=Pxforx∈CwithPx+t(x-Px)∈Candt≥0.

For anyx∈K,the inward setIK(x)is defned as follows:

A mappingT:K→Eis said to satisfy the inward condition ifTx∈IK(x)for allx∈K.Tis said to be weakly inward ifTx∈clIK(x)for eachx∈K,where clIK(x)is the closure ofIK(x).

A Banach spaceEis said to have the Kadec-Klee property(see[2])if for every sequence{xn}inE,withxn→xweakly and‖xn‖→‖x‖,it follows thatxn→xstrongly.

We denote byF(T)the set of fxed points ofT,i.e.,F(T)={x∈K:Tx=x},and by∩the set of common fxed points of two mappingsT1andT2.

LetKbe a nonempty closed convex subset of a real uniformly convex Banach spaceE.Nonself asymptotically nonexpansive mappings have been studied by many authors(see [3-8]).Chidumeet al.[3]studied the following iteration scheme: {

where{αn}is a sequence in(0,1),and proved some strong and weak convergence theorems of the iteration scheme(1.2).

Wang[4]studied the following iteration scheme:

where{αn}and{βn}are two sequences in[0,1),T1,T2:K→Eare two asymptotically nonexpansive nonself mappings,and proved strong and weak convergence theorems of the iteration scheme(1.3).Guo and Guo[5]completed the weak convergence theorems of the iteration scheme(1.3).

Remark 1.1IfT:K→Eis an asymptotically nonexpansive mapping andP:E→Kis a nonexpansive retraction,thenPT:K→Kis asymptotically nonexpansive.Indeed,for allx,y∈Kandn≥1,it follows that

Therefore,Zhouet al.[7]introduced the following generalized defnition:

T is said to be uniformly L-Lipschitzian with respect to P if there exists a constant L＞0such that for all x,y∈K,

Furthermore,by studying the following iterative scheme:

where{αn}and{βn}and{γn}are three sequences in[a,1-a]for somea∈(0,1),satisfyingαn+βn+γn=1,Zhouet al.[7]obtained some strong and weak convergence theorems for common fxed points of nonself asymptotically nonexpansive mappings with respect toPin uniformly convex Banach spaces.As a consequence,the main results of Chidumeet al.[3]can be deduced.

Recently,Turkmenet al.[8]generalized the iteration process(1.5)as follows:

where{αn},{βn}are two sequences in[0,1),T1,T2:K→Eare two asymptotically nonexpansive nonself mappings,andPis as in Defnition 1.2,and obtained the following weak convergence theorem:

Only Theorem 1.1 has been obtained from the weak convergence problem for the sequence defned by(1.6).The purpose of this paper is to prove some new weak convergence theorems of the iteration scheme(1.6)for two asymptotically nonexpansive nonself-mappings in uniformly convex and smooth Banach spaces.

2 Some Lemmas

In order to prove the main results,we need the following lemmas:

Lemma 2.1[8]Let E be a real normed linear space,K be a nonempty closed convex subset of E,and T1,T2:K→E be two asymptotically nonexpansive mappings with respect to Pwith a sequencesatisfyingSuppose that{xn}is defned by(1.6)and F/=Ø.Thenexists for all p∈F.

Lemma 2.3[9]Let X be a uniformly convex Banach space and C be a convex subset of X. Then there exists a strictly increasing continuous convex function γ:[0,∞)→[0,∞)with γ(0)=0such that for each S:C→C with Lipschitz constant L,

Lemma 2.5[10]Let E be a uniformly convex Banach space,K be a nonempty closed convex subset of E,and T:K→K be an asymptotically nonexpansive mapping with F/=Ø.Then I-T is demiclosed at zero,i.e.,for each sequence{xn}in K,if{xn}converges weakly to q∈K and{(I-T)xn}converges strongly to0,then(I-T)q=0.

Lemma 2.6[8]Let E be a real smooth Banach space,K be a nonempty closed convex subset of E with P as a sunny nonexpansive retraction,and T:K→E be a mapping satisfying weakly inward condition.Then F(PT)=F(T).

3 Main Results

In this section,we prove weak convergence theorems for the iterative scheme(1.6)for two asymptotically nonexpansive nonself-mappings in uniformly convex and smooth Banach spaces.

Defne the mappingHn:K→Kby

Then,for allx,y∈K,by(1.4)we have

Set

From(3.1)and(3.2),we can obtain that

andfor eachq∈F.Let

Using(3.3),(3.4)and Lemma 2.3,we have

denotes the set of all weak subsequential limits of{xn},then〈x∗-y∗,j(q1-q2)〉=0for all q1,q2∈F and x∗,y∗∈Ww(xn).

Proof.This follows basically as in the proof of Lemma 4 in[1].For completeness,we sketch the details.Setx=q1-q2andh=t(xn-q1),0≤t≤1 in(1.1).Sincebis increasing,and‖xn-q1‖≤Mfor someM＞0 and alln≥1,by Lemma 2.1,we have

It follows from(3.7)and Lemma 3.1 that

So by(3.8)we have

From(3.9)and

This shows that〈x∗-y∗,j(q1-q2)〉=0 for allq1,q2∈Fandx∗,y∗∈Ww(xn).This completes the proof.

1)＜∞and F/=Ø.Let{xn}be defned by(1.6),where{αn}and{βn}are sequences in[a,1-a]for some a＞0.Then{xn}converges weakly to a common fxed point of T1and T2.

Proof.By Lemma 2.1,{xn}is bounded.SinceEis refexive,there exists a subsequence{xnk}of{xn}which converges weakly to someq∈K.By Lemma 2.2,we have

Now,we prove that{xn}converges weakly toq.Suppose that there exists some subsequence{xnj}of{xn}such that{xnj}converges weakly to someq1∈K.Then by the same method as given above,we can also prove thatq1∈F.Soq,q1∈F∩Ww(xn).It follows from Lemma 3.2 that

Therefore,q=q1and so{xn}converges weakly toq.This completes the proof.

Proof.Using the same method as given in Theorem 3.1,we can prove that there exists a subsequence{xnk}of{xn}which converges weakly to someq∈F.

Remark 3.1(1)It is well known(see,for example,[11])that some Banach spaces, such asLpspace withp/=2,do not satisfy Opial’s condition.Also,it is well known thatevery Banach space,which is both uniformly convex and uniformly smooth,has a Fr´echet diferentiable norm.In particular,Lpspace,1＜p＜∞,has a Fr´echet diferentiable norm. That shows that a Banach space which has a Fr´echet diferentiable norm is diferent from the Banach space satisfying Opial’s condition.

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A

1674-5647(2015)01-0015-08

10.13447/j.1674-5647.2015.01.02

Received date:Nov.21,2012.

Foundation item:The NSF(11271282)of China.

E-mail address:lyq60913016@hotmail.com(Liu Y Q).

2010 MR subject classifcation:47H09,47H10