Weak and strong convergence theorems for a finite family of I-asymptotically nonexpansive mappings

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Abstract

In this paper, a new two-step iterative scheme for a finite family of Ii-asymptotically nonexpansive nonself-mappings {Ti}i=1r is constructed in a uniformly convex Banach space. Weak and strong convergence theorems of this iterative scheme to a common fixed point of {Ti}i=1r and {Ii}i=1r are proved in a uniformly convex Banach space. The results of this paper improve and extend the corresponding results of Temir [2].

Introduction

We assume that X is a real normed space and K is a nonempty subset of X. A mapping T:KK is said to be nonexpansive if Tx-Tyx-y holds for all x,yK. A mapping T:KK is said to be asymptotically nonexpansive [1] if there exists a sequence {kn}[1,),limnkn=1 such thatTnx-Tnyknx-yfor all x,yK and n1. A mapping T:KK is said to be uniformly Lipschitz with a Lipschitzian constant L0 if Tnx-TnyLx-y holds for all x,yK and n1. If F(T){xK:Tx=x} denotes the set of fixed points of a mapping T, and there exists a sequence {kn}[1,),limnkn=1 such that for all xK, the following inequality holds:Tnx-xknxn-x,xF(T),n1,then T is said to be asymptotically quasi-nonexpansive. The class of asymptotically nonexpansive mappings is a natural generalization of the important class of nonexpansive mappings. Every asymptotically nonexpansive mapping with a fixed point is asymptotically quasi-nonexpansive, but the converse may be not true. Goebel and Kirk [1] proved that if K is a nonempty closed and bounded subset of a uniformly convex Banach space, then every asymptotically nonexpansive self-mapping has a fixed point.

Let I,T:KK. A mapping T:KK is said to be I-Lipschitz [2] if there exists Γ0 such thatTx-TyΓIx-Iyfor all x,yK. If Γ<1 (respectively Γ=1), then T is said to be I-contraction (respectively I-nonexpansive) [3] on K.

T is said to be I-asymptotically nonexpansive [2], [3] if there exists a sequence {vn}[0,) with limnvn=0 such thatTnx-Tny1+vnInx-Inyfor all x,yK and n1.

T is said to be I-asymptotically quasi-nonexpansive [2] if there exists a sequence {vn}[0,) with limnvn=0 such thatTnx-q1+vnInx-qfor all xK and qF(T)F(I) and n1.

A subset K of X is said to be a retract of X if there exists a continuous map P:XK such that Px=x, for all xK. Every closed convex subset of a uniformly convex Banach space is a retraction. A mapping P:XK is said to be a retraction if P2=P. It follows that if a map P is a retraction, then Py=y for all y in the range of P.

For nonself nonexpansive mappings, some authors (see, e.g., [5], [6], [7], [8], [9] and the references therein) have studied the strong and weak convergence theorems in Hilbert space or uniformly convex Banach spaces. However, iterative algorithms for approximating fixed points of nonself asymptotically nonexpansive mappings have not been paid too much attention. The concept of nonself asymptotically nonexpansive mappings was introduced by Chidume et al. [10] in 2003 as the generalization of asymptotically nonexpansive self-mappings. The nonself asymptotically nonexpansive mapping is defined as follows:

Definition 1.1 [10]

Let K a nonempty subset of real normed linear space E. Let P:EK be the nonexpansive retraction of E onto K. A nonself-mapping T:KE is said to be asymptotically nonexpansive if there exists sequence {kn}[1,),kn1(n) such thatT(PT)n-1x-T(PT)n-1yknx-yfor all x,yK and n1.

T is said to be uniformly L-Lipschitz if there exists a constant L>0 such thatT(PT)n-1x-T(PT)n-1yLx-yfor all x,yK and n1.

If T is self-mapping, then P becomes the identity mapping, so that (1.4) reduces to (1.1).

By studying the following iterative sequence:xn+1=P((1-αn)xn+αnT(PT)n-1xn),x1K,n1

Chidume et al. [10] established demi-closed principle, strong and weak convergence theorems for nonself asymptotically nonexpansive mapping in a uniformly convex Banach space. Recently concerning the convergence problem of an explicit iterative process to a common fixed point for some nonself asymptotically nonexpansive mappings in uniformly convex Banach spaces have been considered by several authors (see, for example, Wang [11], Yang [12], Thainwan [13] and the references therein).

The purpose in this paper first introduce the class of I-asymptotically nonexpansive nonself-maps. Then, an iteration scheme for approximating common fixed points of a finite family of Ii-asymptotically nonexpansive nonself-mappings belonging to this class (when such common fixed points exist) is constructed; and strong and weak convergence theorems are proved. Our theorems improve and generalize important related results of the previously known results in this area.

Section snippets

Preliminaries

Let X be a Banach space with dimension X2. The modulus of X is the function δE:(0,2][0,1] defined byδX(ε)=inf1-12x+y:x=y=1,x-y=ε.

A Banach space X is uniformly convex if and only if δX(ε)>0 for all ε(0,2].

Let S(X)={xX:x=1}. The space X is said to be smooth iflimt0x+ty-xtexists for all x,yS(X). For any x,yX(x0), we denote this limit by (x,y). The norm · of X is said to be Fréchet differentiable if for all xS(X), the limit (x,y) exists uniformly for all yS(X).

Now, we give

Main results

In this section, we shall state the weak and strong convergence of the iteration scheme (2.3) to a common fixed point for a finite family of Ii-asymptotically nonexpansive nonself-mappings in a real uniformly convex Banach spaces. We first prove the following lemmas.

Lemma 3.1

Let X be a real uniformly convex space, K be a nonempty closed convex subset of X which is also a nonexpansive retract with retraction P. Let Ti:KX(iI0) be Ii-asymptotically nonexpansive nonself-mappings with sequences {vin}[0,)

Acknowledgments

The second author is supported by the National Natural Science Foundation of China (Grant No. 70871028).

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