- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Abstract and Applied Analysis

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

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

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

^{1}Department of Mathematics, Arts and Science Faculty, Harran University, 63200 Şanliurfa, Turkey^{2}Graduate School of Natural and Applied Sciences, Adıyaman University, 02040 Adıyaman, Turkey^{3}Department 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.

#### References

- K. Goebel and W. A. Kirk, “A fixed point theorem for asymptotically nonexpansive mappings,”
*Proceedings of the American Mathmatical Society*, vol. 35, pp. 171–174, 1972. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. C. Chang, Y. J. Cho, and H. Zhou, “Demi-closed principle and weak convergence problems for asymptotically nonexpansive mappings,”
*Journal of the Korean Mathematical Society*, vol. 38, no. 6, pp. 1245–1260, 2001. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - Y. J. Cho, H. Zhou, and G. Guo, “Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings,”
*Computers and Mathematics with Applications*, vol. 47, no. 4-5, pp. 707–717, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - M. A. Noor, “New approximation schemes for general variational inequalities,”
*Journal of Mathematical Analysis and Applications*, vol. 251, no. 1, pp. 217–229, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - M. O. Osilike and S. C. Aniagbosor, “Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings,”
*Mathematical and Computer Modelling*, vol. 32, no. 10, pp. 1181–1191, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - S. Plubtieng, R. Wangkeeree, and R. Punpaeng, “On the convergence of modified Noor iterations with errors for asymptotically nonexpansive mappings,”
*Journal of Mathematical Analysis and Applications*, vol. 322, no. 2, pp. 1018–1029, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - J. Schu, “Weak and strong convergence to fixed points of asymptotically nonexpansive mappings,”
*Bulletin of the Australian Mathematical Society*, vol. 43, no. 1, pp. 153–159, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. Xu and M. A. Noor, “Fixed-point iterations for asymptotically nonexpansive mappings in Banach spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 267, no. 2, pp. 444–453, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - C. E. Chidume and B. Ali, “Approximation of common fixed points for finite families of nonself asymptotically nonexpansive mappings in Banach spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 326, no. 2, pp. 960–973, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - I. Yıldırım and M. Özdemir, “Approximating common fixed points of asymptotically quasi-nonexpansive mappings by a new iterative process,”
*Arabian Journal for Science and Engineering*, vol. 36, no. 3, pp. 393–403, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - J. Quan, S.-S. Chang, and X. J. Long, “Approximation common fixed point of asymptotically quasi-nonexpansive-type mappings by the finite steps iterative sequences,”
*Fixed Point Theory and Applications*, vol. 2006, Article ID 70830, 8 pages, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - J. W. Peng, “On the convergence of finite steps iterative sequences with errors asymptotically nonexpansive mappings,”
*IAENG International Journal of Applied Mathematics*, vol. 37, no. 2, 5 pages, 2007. View at Google Scholar · View at Zentralblatt MATH - H. Kızıltunç and S. Temir, “Convergence theorems by a new iteration process for a finite family of nonself asymptotically nonexpansive mappings with errors in Banach spaces,”
*Computers and Mathematics with Applications*, vol. 61, no. 9, pp. 2480–2489, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - K. K. Tan and H. K. Xu, “Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process,”
*Journal of Mathematical Analysis and Applications*, vol. 178, no. 2, pp. 301–308, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - H.-K. Xu, “Inequalities in Banach spaces with applications,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 16, no. 12, pp. 1127–1138, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus