Journal of Applied Mathematics

Volume 2013 (2013), Article ID 602582, 6 pages

http://dx.doi.org/10.1155/2013/602582

Research Article

An Implicit Iteration Process for Common Fixed Points of Two Infinite Families of Asymptotically Nonexpansive Mappings in Banach Spaces

Wei-Qi Deng and Peng Bai

College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China

Received 30 October 2012; Accepted 18 January 2013

Academic Editor: Jinde Cao

Copyright © 2013 Wei-Qi Deng and Peng Bai. 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 nonempty, closed, and convex subset of a real uniformly convex Banach space . Let and be two infinite families of asymptotically nonexpansive mappings from to itself with . For an arbitrary initial point , is defined as follows: , , , where and with and satisfying the positive integer equation: , ; and are two countable subsets of and respectively; , , , , , and are sequences in for some , satisfying and . Under some suitable conditions, a strong convergence theorem for common fixed points of the mappings and is obtained. The results extend those of the authors whose related researches are restricted to the situation of finite families of asymptotically nonexpansive mappings.

1. Introduction

Let be a nonempty, closed, and convex subset of a real uniformly convex Banach space . A mapping is said to be nonexpansive if for all . is said to be asymptotically nonexpansive if there exists a sequence with () such that

It is obvious that a nonexpansive mapping is an asymptotically nonexpansive one, but the converse is not true. Denote by the set of fixed points of , that is, . Throughout this paper, we always assume that . As an important generalization of nonexpansive mappings, the class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] in 1972, who proved that if is a nonempty, closed, and convex subset of a real uniformly convex Banach space and is an asymptotically nonexpansive mapping, then has a fixed point.

Since then, iterative techniques for approximating fixed points of asymptotically nonexpansive mappings have been studied by various authors (see, e.g., [2–9]). However, these researches are all restricted to the situation of at most finite families of asymptotically nonexpansive mappings. For the extension of finite families to infinite ones, we develop an original method, namely, a specific way of choosing the indexes, for the iterative approximation of common fixed points of the involved mappings.

We now cite an announced result as the object of our extension. In 2010, Wang et al. [10] constructed the following iteration process for two asymptotically nonexpansive mappings and obtained some strong convergence theorems for common fixed points of the given mappings in Banach spaces. For an arbitrary initial point , are defined as follows:

where are two asymptotically nonexpansive mappings; , , , , , and are real sequences in satisfying and .

In this paper, a modified iteration scheme of (2) is used for approximating common fixed points of two infinite families of asymptotically nonexpansive mappings; a strong convergence theorem is established in the framework of uniformly convex Banach spaces. The results show the feasibility of the newly developed technique and extend those of the authors whose related researches are restricted to the situation of finite families of such mappings.

2. Preliminaries

Throughout this paper, we use to denote the set of common fixed points of two infinite families of asymptotically nonexpansive mappings and , that is, .

Let be a nonempty, closed, and convex subset of a real Banach space . Let and be two countable subsets of and , respectively. In order to approximate some member of , we define, from an arbitrary , the following implicit iteration scheme:

where , ,  , , , and are sequences in for some , satisfying and ; and with and being the solutions to the positive integer equation: , that is, for each , there exist unique and such that

For convenience, we restate the following concepts and results.

A Banach space is said to satisfy Opial’s condition if, for any sequence in , implies that

for all with , where denotes that converges weakly to .

A mapping with domain and range in is said to be demiclosed at if whenever is a sequence in such that converges weakly to and converges strongly to , then .

We now need the following lemmas for our main results.

Lemma 1 (see [11]). Let , and be sequences of nonnegative real numbers satisfying

If and , then exists.

Lemma 2 (see [6]). Let be a real uniformly convex Banach space, and let be two constants with . Suppose that is a real sequence and are two sequences in . Then, the conditions

imply that where is a constant.

Lemma 3 (see [2]). Let be a real uniformly convex Banach space, a nonempty, closed, convex subset of , and let be an asymptotically nonexpansive mapping with a sequence and as . Then, is demiclosed at zero.

Lemma 4. The unique solutions to the positive integer equation

are

where denotes the maximal integer that is not larger than x.

Proof. It follows from (8) that

and hence

that is,

which implies that

Thus,

that is,

while the difference of the two sides of the inequality above is

Then, it follows from (15) that (9) holds obviously.

3. Main Results

Lemma 5. Let be a nonempty, closed, and convex subset of a real uniformly convex Banach space , and let and be two countable subsets of the asymptotically nonexpansive mappings and from to itself, respectively, with corresponding sequences and such that and . Suppose that is generated by (3), where for all . If , then exists for each .

Proof. Set and for each positive integer , where and satisfy the positive integer equation: . For any , it follows from (3) that

Substituting (17) into (18) yields that

Note that . We have

which implies that , where

Note that and , which implies that . Then, for a given , there exists a positive such that

as . Then, it follows from (20) and (22) that

where , and so . Hence, it follows from (23) and Lemma 1 that exists for each . The proof is completed.

Remark 6. Because of the importance of the condition that is nonempty, we now give an example satisfying the lemma with the set of common fixed points of and being a non single point set. Let , and let . Define an infinite family of mappings by

and an infinite family of mappings by

Then, clearly, and are two infinite families of asymptotically nonexpansive mappings with .

Lemma 7. Let , , , , and be the same as those in Lemma 5. If , then for each , there exists a subsequence of such that

Proof. By Lemma 5, we may assume that for a given , that is,

where . It follows from (17) that

Taking on both sides in (27) yields that

Next, it follows from (26) that

where . It then follows from (26), (28), (29), and Lemma 2 that

which, in addition to (3), implies that

Now, we show that as . It follows from (18) that

Taking on both sides in the inequality above yields that

Combining (28) with (33), we have . Then, by ways similar to the preceding ones, it is easily shown that

which means that

Set . Since

then

where . It then follows from (36)–(38) and Lemma 2 that

which implies that

Then, it follows from (31), (35), and (40) that

Similarly, we have

Next, for any , we consider the corresponding subsequence of , where . For example, by the definition of , we have and . For simplicity, , , and are written as , and , respectively. Note that , that is, , and whenever . Then, we have

It hence follows from (31), (35), (40), and (41) that That is, for each , there exists a subsequence of such that Since , we have, for each ,

Similarly, it can be shown that, for each ,

This completes the proof.

Remark 8. The key point of the proof of Lemma 7 lies in the use of a specific way of choosing the indexes of the involved mappings, which makes the generalization of finite families of nonlinear mappings to infinite ones possible.

Theorem 9. Let , , , , and be the same as those in Lemma 5. If and there exist a or an and a nondecreasing function with and for all such that or for all , then converges strongly to some point of .

Proof. By Lemma 7, there exists a subsequence of such that Since

or

by taking as on both sides in the inequality above, we have

which implies by the definition of the function .

Now, we will show that is a Cauchy sequence. By Lemma 5, there exists a constant such that as . And for any , there exists a positive integer such that for all . Then, for any and , we have

Taking the infimum in the above inequalities for all yields that

which implies that is a Cauchy sequence. Therefore, there exists a such that as since is complete. Furthermore, shows that , which implies that since is closed. It follows from the existence of that as . This completes the proof.

Remark 10. The result of Theorem 9 extends that of Wang et al. [10] whose related research is restricted to the situation of two asymptotically nonexpansive mappings.

Acknowledgments

The authors are greatly grateful to the referees for their useful suggestions by which the contents of this paper are improved. This study is supported by the National Natural Science Foundation of China (Grant no. 11061037).

References

K. Goebel and W. A. Kirk, “A fixed point theorem for asymptotically nonexpansive mappings,” Proceedings of the American Mathematical Society, vol. 35, pp. 171–174, 1972.

S.-s. 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.

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.

J. Schu, “Iterative construction of fixed points of asymptotically nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 158, no. 2, pp. 407–413, 1991.

Y. Zhou and S.-S. Chang, “Convergence of implicit iteration process for a finite family of asymptotically nonexpansive mappings in Banach spaces,” Numerical Functional Analysis and Optimization, vol. 23, no. 7-8, pp. 911–921, 2002.

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.

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.

Z.-h. Sun, “Strong convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 286, no. 1, pp. 351–358, 2003.

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.

L. Wang, I. Yildirim, and M. Özdemir, “Strong convergence of an implicit iteration process for two asymptotically nonexpansive mappings in Banach spaces,” Analele Stiintifice Ale Universitatii Ovidius Constanta, vol. 18, no. 2, pp. 281–294, 2010.

M. O. Osilike, S. C. Aniagbosor, and B. G. Akuchu, “Fixed points of asymptotically demicontractive mappings in arbitrary Banach spaces,” Panamerican Mathematical Journal, vol. 12, no. 2, pp. 77–88, 2002.