Research Article | Open Access

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

**Academic Editor:**Jinde Cao

#### 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).

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#### Copyright

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.