Abstract and Applied Analysis

Volume 2013 (2013), Article ID 974317, 8 pages

http://dx.doi.org/10.1155/2013/974317

Research Article

Approximating Common Fixed Points for a Finite Family of Asymptotically Nonexpansive Mappings Using Iteration Process with Errors Terms

Seyit Temir1,2 and Adem Kiliçman³

¹Department of Mathematics, Arts and Science Faculty, Harran University, 63200 Şanliurfa, Turkey

²Graduate School of Natural and Applied Sciences, Adıyaman University, 02040 Adıyaman, Turkey

³Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia, 43400 Serdang, Selangor, Malaysia

Received 6 October 2013; Accepted 25 November 2013

Academic Editor: Mohammad Mursaleen

Copyright © 2013 Seyit Temir and Adem Kiliçman. 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.

Abstract.

Let be a real Banach space and a nonempty closed convex subset of . Let be asymptotically nonexpansive mappings with sequence , , and , where is the set of fixed points of . Suppose that ,  ,   are appropriate sequences in and ,   are bounded sequences in such that for . We give defined by The purpose of this paper is to study the above iteration scheme for approximating common fixed points of a finite family of asymptotically nonexpansive mappings and to prove weak and some strong convergence theorems for such mappings in real Banach spaces. The results obtained in this paper extend and improve some results in the existing literature.

1. Introduction

Let be a nonempty subset of a real Banach space and let be a mapping. Let be the set of fixed points of .

A mapping is called nonexpansive if

for all . Similarly, is called asymptotically nonexpansive if there exists a sequence with such that

for all and . The mapping is called uniformly L-Lipschitzian if there exists a positive constant such that

for all and .

It is easy to see that if is asymptotically nonexpansive, then it is uniformly -Lipschitzian with the uniform Lipschitz constant .

The class of asymptotically nonexpansive mappings 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.

The main tool for approximation of fixed points of generalizations of nonexpansive mappings remains iterative technique. Iterative techniques for nonexpansive self-mappings in Banach spaces including Mann type (one-step), Ishikawa type (two-step), and three-step iteration processes have been studied extensively by various authors; see, for example, ([2–8]).

Recently, Chidume and Ali [9] defined (4) and constructed the sequence for the approximation of common fixed points of finite families of asymptotically nonexpansive mappings. Yıldırım and Özdemir [10] introduced an iteration scheme for approximating common fixed points of a finite family of asymptotically quasi-nonexpansive self-mappings and proved some strong and weak convergence theorems for such mappings in uniformly convex Banach spaces. Quan et al. [11] studied sufficient and necessary conditions for finite step iterative schemes with mean errors for a finite family of asymptotically quasi-nonexpansive mappings in Banach spaces to converge to a common fixed point. Peng [12] proved the convergence of finite step iterative schemes with mean errors for asymptotically nonexpansive mappings in Banach spaces. More recently Kızıltunç and Temir [13] introduced and studied a new iteration process for a finite family of nonself asymptotically nonexpansive mappings with errors in Banach spaces.

In [9], the authors introduced an iterative process for a finite family of asymptotically nonexpansive mappings as follows:

where are asymptotically nonexpansive mappings and for .

Inspired and motivated by these facts, it is our purpose in this paper to construct an iteration scheme for approximating common fixed points of finite family of asymptotically nonexpansive mappings and study weak and some strong convergence theorems for such mappings in real Banach spaces.

Let be a real Banach space and a nonempty closed convex subset of . Let be asymptotically nonexpansive mappings with sequence , , and . Suppose that ,  , are appropriate sequences in and , are bounded sequences in such that for . Let be defined by

2. Preliminaries

Let be a real Banach space, a nonempty closed convex subset of , and the set of fixed points of . A Banach space is said to be uniformly convex if the modulus of convexity of

for all (i.e., . Recall that a Banach space is said to satisfy Opial’s condition if, for each sequence in , the condition implies that

for all with . It is well known that all spaces for have this property. However, the spaces do not have unless .

A mapping is said to be semicompact if, for any bounded sequence in such that as , there exists a subsequence say of such that converges strongly to some in . is said to be completely continuous if for every bounded sequence in , there exists a subsequence say of such that the sequence converges strongly to some element of the range of .

The following lemmas were given in [14, 15], respectively, and we need them to prove our main results.

Lemma 1. Let , , and be sequences of nonnegative real numbers satisfying the following conditions: for all , , where and . Then

(i) exists;(ii)in particular, if has a subsequence converging to 0, then .

Lemma 2. Let and be two fixed numbers. Then a Banach space is uniformly convex if and only if there exists a continuous, strictly increasing, convex function with such that

for all , and , where .

The following lemmas were proved in [3].

Lemma 3. Let be a uniformly convex Banach space and . Then there exists a continuous, strictly increasing, convex function with such that

for all and with .

Lemma 4. Let be a uniformly convex Banach space, a nonempty closed convex subset of , and an asymptotically nonexpansive mapping. Then ( is identity mapping) is demiclosed at zero; that is, if weakly and strongly, then , where is the set of fixed points of .

Definition 5. A family of asymptotically nonexpansive mappings on with is said to satisfy condition (A) on if there exists a nondecreasing function with , for all such that for all .

3. Main Results

In this section, we prove weak and strong convergence of the iterative sequence generated by iterative scheme (5) to a common element of the sets of fixed points of a finite family of asymptotically nonexpansive mappings in a real Banach space.

Lemma 6. Let be a real Banach space and a nonempty closed convex subset of . Let be asymptotically nonexpansive mappings with sequence , , and . Suppose that , , are appropriate sequences in and , are bounded sequences in such that for . Let be given by (5). Then is bounded and exists for .

Proof. For any given , since , are bounded sequences in , let

For each , using (5), we have

Then we have

which leads to

where

Since for all , the only assumption is enough for the boundedness for , then , for all , and for some . Hence holds for all . Therefore and also . Equation (13) and Lemma 1 guarantee that the sequence is bounded and exists.

Theorem 7. Let be a real uniformly convex Banach space and a nonempty closed convex subset of . Let be asymptotically nonexpansive mappings with sequence , and . Suppose that , , are appropriate sequences in and , are bounded sequences in such that for . Let be given by (5). Suppose that

for . Then

Proof. Let . Then by Lemma 6, exists. Since , are bounded sequences in , let ; moreover, it follows that is also bounded for each , and hence is also bounded for . By using (5), we obtain

Note that for all , the assumption implies that . Since is bounded, there exists such that . Then holds for all . Therefore, the assumption implies that . Then

It follows from (18) and (19) that

We first obtain that

Now if and , there exist a positive integer and such that ,   for all . This implies by (21) that

It follows from (22) that for ,

Then , and therefore , and by property of , we have . By a similar method, together with (20) and by property of , we have

for . Thus, we conclude that

for . From (5) and for

This together with (25) implies that for each

It follows from (5) that

Equations (24) and (28) imply that

It follows from (5) that

Thus, (24) and (30) guarantee that

Continuing in this fashion, for each we get,

Taking the limit on both sides inequality from (33), we have

Since is an asymptotically nonexpansive mapping with , we have

Taking the limit on both sides inequality (35), and by using (24), we get

Since is an asymptotically nonexpansive mapping with , we have

Also, taking the limit on both sides inequality (37), and by using (24), we get

In a similar way, one can prove that for each

Next, we consider

It follows from (34), (36), and the above inequality (40) that

It follows from (34), (38) and (42) that

Continuing similar process, for each we get

The proof is completed.

Theorem 8. Let be a real uniformly convex Banach space and a nonempty closed convex subset of . Let be asymptotically nonexpansive mappings with sequence , and . Suppose that , , are appropriate sequences in and , are bounded sequences in such that for . Suppose that

for . If one of is either completely continuous or semicompact, for some , then the sequence generated by (5) converges strongly to an element of .

Proof. Assume that there exists such that is semi-compact. Since is bounded and by Theorem 7, as , there exists a subsequence of such that converges strongly to . Since , it follows from Lemma 4 that . Also, from Theorem 7  , . Therefore, from Lemma 4 we obtain that . So converges strongly to .

If one of ’s is completely continuous, say , since is bounded, there exists a subsequence of such that converges strongly to . By Theorem 7, . It follows from continuity of that

Using as , , and Lemma 4, we obtain that . Also using as and Lemma 6, we obtain that . This completes the proof.

Next, we prove a strong convergence theorem for asymptotically nonexpansive mappings in a uniformly convex Banach space satisfying condition (A).

Theorem 9. Let be a real uniformly convex Banach space and a nonempty closed convex subset of . Let be m asymptotically nonexpansive mappings with sequence , and satisfying the condition (A). Suppose that , , are appropriate sequences in and , are bounded sequences in such that for . Suppose that and for . Then the sequence generated by (5) converges strongly to an element of .

Proof. Since exists for all by Lemma 6, then, for any such that

we have that exists. It follows from (47) that exists. From condition (A)

where is . From Theorem 7  . It then follows (48) that . By property of , . It also follows from (47) that . Therefore .

Now, we prove the weak convergence of iteration (5) for a family of asymptotically nonexpansive mappings in a uniformly convex Banach space.

Theorem 10. Let be a uniformly convex Banach space satisfying Opial’s condition, and let be a nonempty closed convex subset of . Let be asymptotically nonexpansive mappings with sequence , and let the sequences , , and , be the same as in Theorem 7. Then the sequence defined by (5) converges weakly to a common fixed point of .

Proof. It follows from Lemma 6 that exists. Therefore, is a bounded sequence in . Then by the reflexivity of and the boundedness of , there exists a subsequence of such that weakly. By Theorem 7, , and is demiclosed at 0 for . So we obtain for . Finally, we prove that converges to . Suppose , where denotes the weak limit set of . Let and be two subsequences of which converge weakly to and , respectively. Opial’s condition ensures that is a singleton set. It follows that . Thus converges weakly to an element of . This completes the proof.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors are grateful for the very useful comments regarding detailed remarks which improved the presentation and the contents of the paper. The first author acknowledged that this paper was partially supported by Turkish Scientific and Research Council (Tübitak) Program 2224. This joint work was done when the first author visited University Putra Malaysia as a visiting scientist during 5th February–15th February, 2013. Thus, he is very grateful to the administration of INSPEM for providing him local hospitalities.

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