Fixed Point Theory and Applications

, 2010:783178 | Cite as

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

Open Access
Research Article

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 .

A mapping Open image in new window is called nonexpansive provided

for all Open image in new window .

Open image in new window is called 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.

Open image in new window is called 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 .

Asymptotically nonexpansive self-mappings using Ishikawa iterative and the Mann iterative processes have been studied extensively by various authors to approximate fixed points of asymptotically nonexpansive mappings (see [2, 12]). Noor [3] introduced a three-step iterative scheme and studied the approximate solutions of variational inclusion in Hilbert spaces. Glowinski and Le Tallec [4] applied a three-step iterative process for finding the approximate solutions of liquid crystal theory, and eigenvalue computation. It has been shown in [1] that the three-step iterative scheme gives better numerical results than the two-step and one-step approximate iterations. Xu and Noor [5] introduced and studied a three-step scheme to approximate fixed point of asymptotically nonexpansive mappings in a Banach space. Very recently, Nilsrakoo and Saejung [6] and Suantai [7] defined new three-step iterations which are extensions of Noor iterations and gave some weak and strong convergence theorems of the modified Noor iterations for asymptotically nonexpansive mappings in Banach space. It is clear that the modified Noor iterations include Mann iterations [8], Ishikawa iterations [9], and original Noor iterations [3] as special cases. Consequently, results obtained in this paper can be considered as a refinement and improvement of the previously known results

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.

If Open image in new window , then (1.4) reduces to the modified Noor iterations defined by Suantai [7] as follows:

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.

If Open image in new window , then (1.4) reduces to Noor iterations defined by Xu and Noor [5] as follows:
If Open image in new window , then (1.4) reduces to modified Ishikawa iterations as follows:
If Open image in new window , then (1.4) reduces to Mann iterative process as follows:

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 .

Iterative techniques for converging fixed points of nonexpansive nonself-mappings have been studied by many authors (see, e.g., Khan and Hussain [10], Wang [11]). Evidently, we can obtain the corresponding nonself-versions of (1.5) Open image in new window (1.7). We will obtain the weak and strong convergence theorems using (1.12) for nonself asymptotically nonexpansive mappings in a uniformly convex Banach space. Very recently, Suantai [7] introduced iterative process and used it for the weak and strong convergence of fixed points of self-mappings in a uniformly convex Banach space. As remarked earlier, Suantai [7] has established weak and strong convergence criteria for asymptotically nonexpansive self-mappings, while Chidume et al. [12] studied the Mann iterative process for the case of nonself-mappings. Our results will thus improve and generalize corresponding results of Suantai [7] and others for nonself-mappings and those of Chidume et al. [12] in the sense that our iterative process contains the one used by them. The concept of nonself asymptotically nonexpansive mappings was introduced by Chidume et al. [12] as the generalization of asymptotically nonexpansive self-mappings and obtained some strong and weak convergence theorems for such mappings given (1.9) as follows: for Open image in new window

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

A nonself-mapping Open image in new window is called asymptotically nonexpansive 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 . Open image in new window is called 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

Throughout this paper, we assume that Open image in new window is a real Banach space, Open image in new window is a nonempty closed convex subset of Open image in new window , and Open image in new window is the set of fixed points of mapping Open image in new window . A Banach space Open image in new window is said to be 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

Recall that a Banach space Open image in new window is said to satisfy 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]).

Let Open image in new window be a uniformly convex Banach space and Open image in new window , then there exists a continuous strictly increasing convex function 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 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.

Thus, we have

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

Open image in new window ,

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 . By Lemma 3.2, we know that Open image in new window exits for any Open image in new window . Then the sequence Open image in new window is bounded. It follows that the sequences Open image in new window and Open image in new window are also bounded. Since Open image in new window is a nonself asymptotically nonexpansive mapping, then the sequences Open image in new window , Open image in new window , and Open image in new window are also bounded. Therefore, there exists Open image in new window such that 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 . By Lemma 2.4 and (1.12), we have

Let Open image in new window .

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

Thus, we have
From the last inequality, we have
By condition
there exists a positive integer Open image in new window and Open image in new window such that Open image in new window and Open image in new window for all Open image in new window then it follows from (3.7) that
From Open image in new window is continuous strictly increasing with Open image in new window and (1), then we have
By using a similar method for inequalities (3.8) and (3.10), we have
Next, to prove
By Lemma 3.1, 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 . This together with (3.18) implies that for Open image in new window ,
It follows from (3.15) and (3.16) that

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 ,

Open image in new window ,

then Open image in new window .

Proof.

We first consider
We note that every asymptotically nonexpansive mapping is uniformly Open image in new window -Lipschitzian. Also note that
In addition,
We denote as Open image in new window the identity maps from Open image in new window into itself. Thus, by above inequality, we write
which implies that

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 ,

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.

By Lemma 3.2, Open image in new window is bounded. It follows by our assumption that Open image in new window is completely continuous, there exists a subsequence Open image in new window of Open image in new window such that Open image in new window as Open image in new window . Therefore, by Lemma 3.4, we have Open image in new window which implies that Open image in new window as Open image in new window . Again by Lemma 3.4, we have

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 ,

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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Copyright information

© Seyit Temir. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Department of Mathematics, Art, and Science FacultyHarran UniversitySanliurfaTurkey

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