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 G?ateaux 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 G?ateaux 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 G?ateaux 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 G?ateaux differentiable,then the normalized duality mappingJis single-valued and norm-to-weak?continuous;

(2)If the norm ofEis uniformly G?ateaux 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 G?ateaux 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

[1]Zhang,S.,Lee,H.W.J.and Chan,C.K.,Approximation of nearest common fixed point of nonexpansive mappings in Hilbert spaces,Acta Math.Sinica(English Series),23(2007), 1889–1896.

[2]Chang,S.,Cho,Y.and Zhou,H.,Demi-closed principle and weak convergence problems for asymptotically nonexpansive mappings,J.Korean Math.Soc.,38(2001),1245–1260.

[3]Gomicki,J.,Weak convergence theorems for asymptotically nonexpansive mappings in uniformly convex Banach spaces,Comment.Math.Univ.Carolin.,301(1989),249–252.

[4]Liu,L.,Ishikawa and Mann iterative processes with errors for nonlinear strongly accretive mappings in Banach space,J.Math.Anal.Appl.,194(1995),114–125.

[5]Chang,S.,On Chidume’s open questions and approximation solutions of multivalued strongly accretive mappings equations in Banach spaces,J.Math.Anal.Appl.,216(1997),94–111.

[6]Shioji,N.and Takahashi,W.,Strong convergence of averaged approximants for asymptotically nonexpansive mappings in Banach spaces,J.Approx.Theory,97(1999),53–64.

[7]Chang,S.,Tan,K.,Lee,H.and Chan,C.,On the convergence of implicit iterative process with error for a finite family of asymptotically nonexpansive mappings,J.Math.Anal.Appl., 313(2006),273–283.

[8]Barbu,V.,Nonlinear Semigroups and differential Equations in Banach Spaces,Noordho ff, Leyden,the Nethedands,1976.

[9]Bruck,R.,Nonexpansive retracts of Banach spaces,Bull.Amer.Math.Soc.,76(1970),348–386.

[10]Reich,S.,Asymptotic behavior of contractions in Banach spaces,J.Math.Anal.Appl., 44(1973),57–70.

[11]Cioranescu,I.,Geometry of Banach Spaces,Duality Mappings and Nonlinear Problems,Math. Appl.Vol.62,Kluwer Acad.Publ.,Dordrecht,The Netherlands,1990.

[12]Chang,S.,Cho,Y.and Zhou,H.,Iterative Methods for Nonlinear Operator Equations in Banach SApaces,Nova Sci.Publ.,New York,2002.

[13]Chang,S.,Yao,J.,Kim,J.and Yang,L.,Iterative approximation to convex feasibility problems in Banach space,Fixed Point Theory Appl.,doi:10.1155/2007/46797.

[14]Shimizu,T.and Takahashi,W.,Strong convergence to common fixed points of families of nonexpansive mappings,J.Math.Anal.Appl.,211(1997),71–83.

[15]O’Hara,J.,O’Hara P.and Xu,H.,Iterative approaches to finding nearest common fixed points of nonexpansive mappings in Hilbert spaces,Nonlinear Anal.,54(2003),1417–1426.

[16]Rao,R.and Zhang,S.,Viscosity approximation methods for equilibrium problems in Hilbert spaces involving the new iterative process with error,J.Math.Res.Exposition,29(2009), 535–543.

[17]Rao,R.,Iteration xn+1=αn+1f(xn)+(1?αn+1)Tn+1xnfor an in finite family of nonexpansive maps{Tn}∞n=1,J.Math.Res.Exposition,29(2009),639–648.

[18]Rao,R.,Iterative algorithms of common solutions for quasi-variational inclusion and fixed point problems,J.Math.Res.Exposition,30(2010),701–715.

[19]Rao,R.,Strong convergence by the shrinking projection method for a generalized equilibrium problems and hemi-relatively nonexpansive mappings,J.Math.Res.Exposition,30(2010), 1099–1107.

[20]Rao,R.,New composite implicit iterative scheme with errors,Acta Math.Sci.,A29(2009), 823–831.

[21]Rao,R.,Iterative approximation to common fixed points of an in finite family of non-self nonexpansive maps and strong convergence of cesaro mean iterations,Acta Math.Sci.,A30(2010), 1666–1676.

[22]Rao,R.,Weak and strong convergence theorem of composite implicit iterative sequence with errors for asymptotically nonexpansive mappings,Chinese Ann.Math.Ser.A,29(2008),461–470.

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.