# Convergence of Three-Step Iterations Scheme for Nonself Asymptotically Nonexpansive Mappings

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## Abstract

Weak and strong convergence theorems of three-step iterations are established for nonself asymptotically nonexpansive mappings in uniformly convex Banach space. The results obtained in this paper extend and improve the recent ones announced by Suantai (2005), Khan and Hussain (2008), Nilsrakoo and Saejung (2006), and many others.

### Keywords

Banach Space Nonexpansive Mapping Strong Convergence Nonempty Closed Convex Subset Convex Banach Space## 1. Introduction

Suppose that Open image in new window is a real uniformly convex Banach space, Open image in new window is a nonempty closed convex subset of Open image in new window . Let Open image in new window be a self-mapping of Open image in new window .

for all Open image in new window .

*asymptotically nonexpansive*mapping if there exists a sequence Open image in new window with Open image in new window such that

for all Open image in new window and Open image in new window .

The class of asymptotically nonexpansive maps which is an important generalization of the class nonexpansive maps was introduced by Goebel and Kirk [1]. They proved that every asymptotically nonexpansive self-mapping of a nonempty closed convex bounded subset of a uniformly convex Banach space has a fixed point.

*uniformly*Open image in new window

*-Lipschitzian*if there exists a constant Open image in new window such that Open image in new window , the following inequality holds:

for all Open image in new window .

where Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , and Open image in new window in Open image in new window satisfy certain conditions.

where Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window and Open image in new window in Open image in new window satisfy certain conditions.

Let Open image in new window be a real normed space and Open image in new window be a nonempty subset of Open image in new window . A subset Open image in new window of Open image in new window is called a *retract* of Open image in new window if there exists a continuous map Open image in new window such that Open image in new window for all Open image in new window . Every closed convex subset of a uniformly convex Banach space is a rectract. A map Open image in new window is called a retraction if Open image in new window . In particular, a subset Open image in new window is called a *nonexpansive retract* of Open image in new window if there exists a *nonexpansive retraction* Open image in new window such that Open image in new window for all Open image in new window .

where Open image in new window and Open image in new window for some Open image in new window .

*asymptotically nonexpansive*if there exists a sequence Open image in new window with Open image in new window such that

*uniformly*Open image in new window

*-Lipschitzian*if there exists constant Open image in new window such that

for all Open image in new window , and Open image in new window . From the above definition, it is obvious that nonself asymptotically nonexpansive mappings are uniformly Open image in new window -Lipschitzian.

Now, we give the following nonself-version of (1.4):

Open image in new window , where Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , and Open image in new window in Open image in new window satisfy certain conditions.

The aim of this paper is to prove the weak and strong convergence of the three-step iterative sequence for nonself asymptotically nonexpansive mappings in a real uniformly convex Banach space. The results presented in this paper improve and generalize some recent papers by Suantai [7], Khan and Hussain [10], Nilsrakoo and Saejung [6], and many others.

## 2. Preliminaries

*uniformly convex*if the modulus of convexity of Open image in new window is as follows:

for all Open image in new window (i.e., Open image in new window is a function Open image in new window

*Opial's condition*[13] if, for each sequence Open image in new window in Open image in new window , the condition Open image in new window weakly as Open image in new window and for all Open image in new window with Open image in new window implies that

Lemma 2.1 (see [12]).

Let Open image in new window be a uniformly convex Banach space, Open image in new window a nonempty closed convex subset of Open image in new window and Open image in new window a nonself asymptotically nonexpansive mapping with a sequence Open image in new window and Open image in new window , then Open image in new window is demiclosed at zero.

Lemma 2.2 (see [12]).

Let Open image in new window be a real uniformly convex Banach space, Open image in new window a nonempty closed subset of Open image in new window with Open image in new window as a sunny nonexpansive retraction and Open image in new window a mapping satisfying weakly inward condition, then Open image in new window .

Lemma 2.3 (see [14]).

Let Open image in new window , Open image in new window , and Open image in new window be sequences of nonnegative real sequences satisfying the following conditions: Open image in new window , Open image in new window , where Open image in new window and Open image in new window , then Open image in new window exists.

Lemma 2.4 (see [6]).

for all Open image in new window , and Open image in new window with Open image in new window .

Lemma 2.5 (See [7], Lemma Open image in new window ).

Let Open image in new window be a Banach space which satisfies Opial's condition and let Open image in new window be a sequence in Open image in new window . Let Open image in new window be such that Open image in new window and Open image in new window . If Open image in new window , Open image in new window are the subsequences of Open image in new window which converge weakly to Open image in new window , respectively, then Open image in new window .

## 3. Main Results

In this section, we prove theorems of weak and strong of the three-step iterative scheme given in (1.12) to a fixed point for nonself asymptotically nonexpansive mappings in a uniformly convex Banach space. In order to prove our main results the followings lemmas are needed.

Lemma 3.1.

If Open image in new window and Open image in new window are sequences in Open image in new window such that Open image in new window and Open image in new window is sequence of real numbers with Open image in new window for all Open image in new window and Open image in new window , then there exists a positive integer Open image in new window and Open image in new window such that Open image in new window for all Open image in new window .

Proof.

from which we have Open image in new window . Put Open image in new window , then we have Open image in new window for all Open image in new window .

Lemma 3.2.

Let Open image in new window be a real Banach space and Open image in new window a nonempty closed and convex subset of Open image in new window . Let Open image in new window be a nonself asymptotically nonexpansive mapping with the nonempty fixed-point set Open image in new window and a sequence Open image in new window of real numbers such that Open image in new window and Open image in new window . Let Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , and Open image in new window be real sequences in Open image in new window , such that Open image in new window and Open image in new window in Open image in new window for all Open image in new window . Let Open image in new window be a sequence in Open image in new window defined by (1.12), then we have, for any Open image in new window , Open image in new window exists.

Proof.

Since Open image in new window and from Lemma 2.3, it follows that Open image in new window exits.

Lemma 3.3.

Let Open image in new window be a real uniformly convex Banach space and Open image in new window a nonempty closed and convex subset of Open image in new window . Let Open image in new window be a nonself asymptotically nonexpansive mapping with the nonempty fixed-point set Open image in new window and a sequence Open image in new window of real numbers such that Open image in new window and Open image in new window . Let Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , and Open image in new window be real sequences in Open image in new window , such that Open image in new window and Open image in new window in Open image in new window for all Open image in new window . Let Open image in new window be a sequence in Open image in new window defined by (1.12), then one has the following conclusions.

If Open image in new window , then Open image in new window

If either Open image in new window or Open image in new window and Open image in new window , then Open image in new window

If the following conditions

either Open image in new window and Open image in new window or Open image in new window and Open image in new window are satisfied, then Open image in new window

Proof.

Let Open image in new window .

Therefore, the assumption Open image in new window implies that Open image in new window .

This completes the proof.

Next, we show that Open image in new window .

Lemma 3.4.

Let Open image in new window be a real uniformly convex Banach space and Open image in new window a nonempty closed convex subset of Open image in new window . Let Open image in new window be a nonself asymptotically nonexpansive mapping with the nonempty fixed-point set Open image in new window and a sequence Open image in new window of real numbers such that Open image in new window and Open image in new window . Let Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , and Open image in new window be real sequences in Open image in new window , such that Open image in new window and Open image in new window in Open image in new window for all Open image in new window . Let Open image in new window be a sequence in Open image in new window defined by (1.12) with the following restrictions:

Open image in new window and Open image in new window ,

then Open image in new window .

Proof.

In the next result, we prove our first strong convergence theorem as follows.

Theorem 3.5.

Let Open image in new window be a real uniformly convex Banach space and Open image in new window a nonempty closed convex subset of Open image in new window . Let Open image in new window be a nonself asymptotically nonexpansive mapping with the nonempty fixed-point set Open image in new window and a sequence Open image in new window of real numbers such that Open image in new window and Open image in new window . Let Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , and Open image in new window be real sequences in Open image in new window , such that Open image in new window and Open image in new window in Open image in new window for all Open image in new window . Let Open image in new window be a sequence in Open image in new window defined by (1.12) with the following restrictions:

Open image in new window and Open image in new window ,

If, in addition, Open image in new window is either completely continuous or demicompact, then Open image in new window converges strongly to a fixed point of Open image in new window .

Proof.

It folows that Open image in new window . Moreover, since Open image in new window exists, then Open image in new window , that is, Open image in new window converges strongly to a fixed point Open image in new window of Open image in new window .

We assume that Open image in new window is demicompact. Then, using the same ideas and argument, we also prove that Open image in new window converges strongly to a fixed point of Open image in new window .

Finally, we prove the weak convergence of the iterative scheme (1.12) for nonself asymptotically nonexpansive mappings in a uniformly convex Banach space satisfying Opial's condition.

Theorem 3.6.

Let Open image in new window be a real uniformly convex Banach space satisfying Opial's condition and Open image in new window a nonempty closed convex subset of Open image in new window . Let Open image in new window be a nonself asymptotically nonexpansive mapping with the nonempty fixed-point set Open image in new window and a sequence Open image in new window of real numbers such that Open image in new window and Open image in new window . Let Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , and Open image in new window be real sequences in Open image in new window , such that Open image in new window and Open image in new window in Open image in new window for all Open image in new window . Let Open image in new window be a sequence in Open image in new window defined by (1.12) with the following restrictions:

Open image in new window and Open image in new window ,

then Open image in new window converges weakly to a fixed point of Open image in new window .

Proof.

Let Open image in new window . Then as in Lemma 3.2, Open image in new window exists. We prove that Open image in new window has a unique weak subsequential limit in Open image in new window . We assume that Open image in new window and Open image in new window are weak limits of the subsequences Open image in new window , Open image in new window , or Open image in new window , respectively. By Lemma 3.4, Open image in new window and Open image in new window is demiclosed by Lemma 2.1, Open image in new window and in the same way, Open image in new window . Therefore, we have Open image in new window . It follows from Lemma 2.5 that Open image in new window . Thus, Open image in new window converges weakly to an element of Open image in new window This completes the proof.

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