WANG XIoNG-RUI AND QUAN JING

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

Communicated by Ji You-qing

Strong Convergence for a Countable Family of Total Quasi-?-asymptotically Nonexpansive Nonself Mappings in Banach Space

WANG XIoNG-RUI AND QUAN JING

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

Communicated by Ji You-qing

The purpose of this article is to introduce a class of total quasi-?asymptotically nonexpansive nonself mappings.Strong convergence theorems for common fxed points of a countable family of total quasi-?-asymptotically nonexpansive mappings are established in the framework of Banach spaces based on modifed Halpern and Mann-type iteration algorithm.The main results presented in this article extend and improve the corresponding results of many authors.

strong convergence,total quasi-?-asymptotically nonexpansive nonself, generalized projection

1 Introduction and Preliminaries

Throughout this article we assume thatEis a real Banach space with norm‖·‖,E?is the dual space ofE,〈·,·〉is the duality pairing betweenEandE?,Cis a nonempty closed convex subset ofE,NandR+denote the set of natural numbers and the set of nonnegative real numbers,respectively.The mappingJ:E→2E?defned by

is called the normalized duality mapping.LetT:C→Cbe a nonlinear mapping,andF(T)denotes the set of fxed points of mappingT.

A subsetCofEis said to be retract if there exists a continuous mappingP:E→Csuch thatPx=xfor allx∈C.Every closed convex subset of a uniformly convex Banachspace is a retraction.A mappingP:E→Eis said to be a retraction ifP2=P.Note that if a mappingPis a retraction,thenPz=zfor allz∈R(P),the range ofP.A mappingP:E→Cis said to be a nonexpansive retraction,if it is nonexpansive and it is a retraction fromEtoC.

In this paper,we assume thatEis a smooth,strictly convex and refexive Banach space andCis a nonempty closed convex subset ofE.We use?:E×E→R+to denote the Lyapunov function,which is defned by

It is obvious that

and

Following Alber[1],the generalized projectionΠCx:E→Cis defned by

Lemma 1.1[1]Let E be a smooth,strictly convex,and refexive Banach space,and C be a nonempty closed convex subset of E.Then the following conclusions hold:

(i)?(x,ΠCy)+?(ΠCy,y)≤?(x,y)for all x∈C,y∈E;

(ii)If x∈E and z∈C,then z=ΠCx if and only if〈z-y,Jx-Jz〉≥0for all y∈C;

(iii)For any x,y∈E,?(x,y)=0if and only if x=y.

Lemma 1.2[2]Let E be a uniformly convex and smooth Banach space,and{xn}and {yn}be two sequences of E.If ?(xn,yn)→0and either{xn}or{yn}is bounded,then‖xn-yn‖→0.

Recently,many researchers have focused on studying the convergence of iterative scheme for quasi-?-asymptotically nonexspansive mappings and total quasi-?-asymptotically nonexspansive mappings.Related works can be found in[3-10].The quasi-?-nonexspansive,quasi?-asymptotically nonexspansive and total quasi-?-asymptotically nonexspansive mappings are defned as:

Defnition 1.1A mapping T:C→C is said to be quasi-?-nonexpansive,if F(T)/=? and ?(u,Tx)≤?(u,x)holds for all x∈C,u∈F(T).

A mapping T:C→C is said to be quasi-?-asymptotically nonexpansive,if F(T)/=?, and there exists a sequence{kn}?[1,+∞]with kn→1as n→∞such that ?(p,Tnx)≤kn?(p,x)holds for all x∈C,p∈F(T)and all n∈N.

A mapping T:C→C is said to be total quasi-?-asymptotically nonexpansive,if F(T)/=?,and there exist sequences{μn},{νn}withμn,νn→0as n→∞and a strictly increasing continuous function ψ:R+→R+with ψ(0)=0such that

holds for all x∈C,p∈F(T)and all n∈N.

Recently,the strong and weak convergence of nonself mappings has been considered extensively by several authors in the setting of Hilbert or Banach spaces(see,for example, [2,11-17]).Especially,Changet al.[3]studied the convergence theorems for a countable family of quasi-?-asymptotically nonexpansive nonself mappings in the framework of Banach spaces based on modifed Halpern and Mann-type iteration algorithm.Now we recall the following nonself mappings.

Defnition 1.2Let P:E→C be the nonexpansive retraction.

A mapping T:C→E is said to be quasi-?-nonexpansive nonself mapping,if F(T)/=? and ?(u,T(PT)n-1x)≤?(u,x)holds for all x∈C,u∈F(T)and all n∈N.

A mapping T:C→E is said to be quasi-?-asymptotically nonexpansive nonself mapping, if F(T)/=?,and there exists a sequence{kn}?[1,+∞]with kn→1as n→∞such that ?(u,T(PT)n-1x)≤kn?(u,x)holds for all x∈C,u∈F(T)and all n∈N.

A mapping T:C→E is said to be total quasi-?-asymptotically nonexpansive nonself mapping,if F(T)/=?,and there exist sequences{μn},{νn}withμn,νn→0as n→∞and a strictly increasing continuous function ψ:R+→R+with ψ(0)=0such that

holds for all x∈C,u∈F(T)and all n∈N.

Lemma 1.3Let E be a real uniformly smooth,strictly convex and refexive Banach space, and C be a nonempty closed convex subset of E.Let T:C→E be a total quasi-?asymptotically nonexpansive nonself mapping with respect to P defned by Defnition1.2. If ν1=0,then the fxed point set F(T)is a closed and convex set of C.

Proof.Letunbe any sequence inF(T)such thatun→u.Now we prove thatu∈F(T). In fact,sinceT:C→Eis a total quasi-?-asymptotically nonexpansive nonself mapping, we have

By Lemma 1.1(iii),we haveu=Tu.

We now prove thatF(T)is convex.Letu1,u2∈F(T)andu=tu1+(1-t)u2,wheret∈(0,1).By the defnition ofT,we have

and

In view of(1.2),we obtain

So we have

Multiply both sides of the above two inequalities bytand 1-t,respectively,and yield that

It follows that

In light of(1.1),we arrive at

SinceE?is refexive,without loss of generality,we assume thatJ(T(PT)n-1u)?e?∈E?.In view of the refexivity ofE,we haveJE=E?.So there exists an elemente∈Esuch thatJe=e?.It follows that

This implies thatu=e,that is,Ju=e?.SoJ(T(PT)n-1u)?Ju∈E?.By Kadec-Klee property ofE?,from

we obtain that

SinceJ-1:E?→Eis demicontinuous,we see thatT(PT)n-1u?u.By virtue of Kadec-Klee property ofE,from

we see that

Hence

i.e.,

In view of the closedness ofT,we can obtain thatTPu=u.Sinceu∈C,Pu=u,it shows thatTu=u.This proves thatF(T)is convex.The conclusion of Lemma 1.3 is proved.

Defnition 1.3A countable family of nonself mappings{Ti}:C→E is said to be uniformly total quasi-?-asymptotically nonexpansive nonself mapping if

there exist sequences{μn},{νn}withμn,νn→0as n→∞and a strictly increasing continuous function ψ:R+→R+with ψ(0)=0such that

A nonself mapping T:C→E is said to be uniformly L-Lipschitz continuous if there exists a constant L＞0such that

holds for all x,y∈C,n∈N.

Next,we prove the strong convergence theorems for common fxed points of a countable family of total quasi-?-asymptotically nonexpansive mappings in the framework of Banach spaces based on modifed Halpern and Mann-type iteration algorithm.The results improve and extend the corresponding results of many others.

2 Main Results

Theorem 2.1Let E be a a real uniformly convex and uniformly smooth Banach space, and C be a nonempty closed convex subset of E.Let Ti:C→E,i∈Nbe a family of uniformly total quasi-?-asymptotically nonexpansive nonself mappings defned by Defnition1.3.Suppose that Tiis uniformly Li-Lipschitz and

Suppose that there exists an M?＞0such that ψ(ηn)≤M?ηn.Let αnbe a sequence in[0,1],and βnbe a sequence in(0,1)satisfying the following conditions:

Let xnbe a sequence generated by

where

If ν1=0and F(T)is bounded in C,then the iterative sequence{xn}converges strongly to ΠF(T)x1in C.

Proof.(I)We prove thatF(T)andCn(n∈N)are all closed and convex subsets inC.

It follows from Lemma 1.3 that for eachi,F(Ti)is a closed and convex subset ofC.SoF(T)is closed and convex inC.By the assumption we know thatC1=Cis closed andconvex.We suppose thatCnis closed and convex for somen≥2.By the defnition of?, we have

This shows thatCn+1is closed and convex.

(II)We prove thatF(T)?Cnfor alln∈N.

In fact,F(T)?C1=C.Suppose thatF(T)?Cn,n≥2.Let

It follows from(1.2)that for anyu∈F(T)?Cn,we have

and

Therefore,

where

This shows that

So

(III)We prove that{xn}is a Cauchy sequence inC.

Sincexn=ΠCnx1,from Lemma 1.1(ii)we have

Again,sinceF(T)?Cn,n≥1,we have

It follows from Lemma 1.1(i)that for eachu∈F(T),n≥1,

Therefore,{?(xn,x1)}is bounded.By virtue of(1.1),xnis also bounded.Since

This shows that

It follows from Lemma 1.2 that

Hencexnis a Cauchy sequence inC.SinceCis complete,there isp?∈Csuch thatxn→p?. By the assumption,we have that

(IV)Now we prove thatp?∈F(T).

Sincexn+1∈Cn+1andαn→0,it follows from(2.1)and(2.2)that

Sincexn→p?,by Lemma 1.2,for eachi≥1 we have

This implies that{Ti(PTi)n-1xn}is uniformly bounded.For eachi≥1,we have

This implies that{ωn,i},t≥0 is also uniformly bounded.Sinceαn→0,from(2.1)we have

SinceEis uniformly smooth andJ-1is uniformly continuous on each bounded subset ofE?,it follows from(2.3)and(2.4)that

Sincexn→p?andJis uniformly continuous on each bounded subset ofE,we have thatJxn→Jp?,and for eachi≥1,

By the condition

we have

SinceJis uniformly continuous,this shows that

By the assumptions thatTi:i≥1 is closed and uniformlyLi-Lipschitz,we have

By

and(2.5),we have

So we get

By virtue of the continuity ofTiP,we haveTiPp?=p?.Sincep?∈CandPp?=p?,we getTip?=p?.By the arbitrariness ofi≥1,we havep?∈F(T).

(V)Finally,we prove thatxn→p?=ΠF(T)x1.

Letω=ΠF(T)x1.Sinceω∈F(T)?Cnandxn=ΠCnx1,we get

This implies that

By the defnition ofΠF(T)x1and from(2.6)we havep?=ω.Therefore,

This completes the proof of Theorem 2.1.

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A

1674-5647(2015)01-0031-09

10.13447/j.1674-5647.2015.01.04

Received date:Dec.16,2012.

Foundation item:Scientifc Research Fund(2011JYZ010)of Science Technology Department of Sichuan Province and Scientifc Research Fund(11ZA172 and 12ZB345)of Sichuan Provincial Education Department.

E-mail address:wxr888x@163.com(Wang X R).

2010 MR subject classifcation:47H05,47H09,49M05