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Discrete Dynamics in Nature and Society
Volume 2010 (2010), Article ID 307245, 19 pages
http://dx.doi.org/10.1155/2010/307245
Research Article

Convergence Theorems on a New Iteration Process for Two Asymptotically Nonexpansive Nonself-Mappings with Errors in Banach Spaces

Department of Mathematics, Faculty of Science, Ataturk University, Erzurum 25240, Turkey

Received 21 October 2009; Accepted 20 January 2010

Academic Editor: Guang Zhang

Copyright © 2010 Murat Ozdemir et al. 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

We introduce a new two-step iterative scheme for two asymptotically nonexpansive nonself-mappings in a uniformly convex Banach space. Weak and strong convergence theorems are established for this iterative scheme in a uniformly convex Banach space. The results presented extend and improve the corresponding results of Chidume et al. (2003), Wang (2006), Shahzad (2005), and Thianwan (2008).

1. Introduction

Let be a real normed space and be a nonempty subset of . A mapping is called nonexpansive if for all . A mapping is called asymptotically nonexpansive if there exists a sequence with such that for all and . is called uniformly -Lipschitzian if there exists a real number such that for all and . It is easy to see that if is an asymptotically nonexpansive, then it is uniformly -Lipschitzian with the uniform Lipschitz constant .

Iterative techniques for nonexpansive and asymptotically nonexpansive mappings in Banach spaces including Mann type and Ishikawa type iteration processes have been studied extensively by various authors; see [18]. However, if the domain of , , is a proper subset of (and this is the case in several applications), and maps into , then the iteration processes of Mann type and Ishikawa type studied by the authors mentioned above, and their modifications introduced may fail to be well defined.

A subset of is said to be a retract of if there exists a continuous map such that , for all . Every closed convex subset of a uniformly convex Banach space is a retract. A map is said to be a retraction if . It follows that if a map is a retraction, then for all the range of .

The concept of asymptotically nonexpansive nonself-mappings was firstly introduced by Chidume et al. [4] as the generalization of asymptotically nonexpansive self-mappings. The asymptotically nonexpansive nonself-mapping is defined as follows.

Definition (see [4]). Let be a nonempty subset of real normed linear space . Let be the nonexpansive retraction of onto . A nonself mapping is called asymptotically nonexpansive if there exists sequence , such that for all and . is said to be uniformly -Lipschitzian if there exists a constant such that for all and .

In [4], they study the following iterative sequence:

to approximate some fixed point of under suitable conditions. In [9], Wang generalized the iteration process (1.3) as follows:

where are asymptotically nonexpansive nonself-mappings and are sequences in . He studied the strong and weak convergence of the iterative scheme (1.4) under proper conditions. Meanwhile, the results of [9] generalized the results of [4].

In [10], Shahzad studied the following iterative sequence:

where is a nonexpansive nonself-mapping and is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space with nonexpansive retraction.

Recently, Thianwan [11] generalized the iteration process (1.5) as follows:

where , , ,, , are appropriate sequences in and , are bounded sequences in . He proved weak and strong convergence theorems for nonexpansive nonself-mappings in uniformly convex Banach spaces.

The purpose of this paper, motivated by the Wang [9], Thianwan [11] and some others, is to construct an iterative scheme for approximating a fixed point of asymptotically nonexpansive nonself-mappings (provided that such a fixed point exists) and to prove some strong and weak convergence theorems for such maps.

Let be a normed space, a nonempty convex subset of , the nonexpansive retraction of onto , and be two asymptotically nonexpansive nonself-mappings. Then, for given and , we define the sequence by the iterative scheme:

where , , , are appropriate sequences in satisfying and , are bounded sequences in . Clearly, the iterative scheme (1.7) is generalized by the iterative schemes (1.4) and (1.6).

Now, we recall the well-known concepts and results.

Let be a Banach space with dimension . The modulus of is the function defined by

A Banach space is uniformly convex if and only if for all .

A Banach space is said to satisfy Opial's condition [12] if for any sequence in , implies that

for all with , where denotes that converges weakly to .

The mapping with is said to satisfy condition [13] if there is a nondecreasing function with , for all such that

for all where ; (see [13, page 337]) for an example of nonexpansive mappings satisfying condition .

Two mappings are said to satisfy condition [14] if there is a nondecreasing function with , for all such that

for all where .

Note that condition reduces to condition ) when and hence is more general than the demicompactness of [13]. A mapping is called: (1) demicompact if any bounded sequence in such that converges has a convergent subsequence, (2) semicompact (or hemicompact) if any bounded sequence in such that as has a convergent subsequence. Every demicompact mapping is semicompact but the converse is not true in general.

Senter and Dotson [13] have approximated fixed points of a nonexpansive mapping by Mann iterates, whereas Maiti and Ghosh [14] and Tan and Xu [5] have approximated the fixed points using Ishikawa iterates under the condition of Senter and Dotson [13]. Tan and Xu [5] pointed out that condition is weaker than the compactness of . Khan and Takahashi [6] have studied the two mappings case for asymptotically nonexpansive mappings under the assumption that the domain of the mappings is compact. We shall use condition instead of compactness of to study the strong convergence of defined in (1.7).

In the sequel, we need the following usefull known lemmas to prove our main results.

Lemma 1.2 (see [5]). Let , and be sequences of nonnegative real numbers satisfying the inequality If and , then (i) exists;(ii)In particular, if has a subsequence which converges strongly to zero, then .

Lemma 1.3 (see [2]). Suppose that is a uniformly convex Banach space and for all . Suppose further that and are sequences of such that hold for some . Then .

Lemma 1.4 (see [4]). Let be a uniformly convex Banach space, a nonempty closed convex subset of , and be a nonexpansive mapping. Then, is demiclosed at zero, that is, if weakly and strongly, then , where is the set fixed point of .

2. Main Results

We shall make use of the following lemmas.

Lemma. Let be a normed space and let be a nonempty closed convex subset of which is also a nonexpansive retract of . Let be two asymptotically nonexpansive nonself-mappings of with sequences such that , , respectively and . Suppose that , are bounded sequences in such that , . Starting from an arbitrary , define the sequence by the recursion (1.7). Then, exists for all .

Proof. Let . Since and are bounded sequences in , we have Set = and = . Firstly, we note that From (1.7) and (2.3), we have Substituting (2.4) into (2.2), we obtain It follows from (1.7) and (2.5) that Note that and are equivalent to and , respectively. Since and , we have We obtained from (2.6) and Lemma 1.2 that exists for all . This completes the proof.

Lemma. Let be a normed space and let be a nonempty closed convex subset of which is also a nonexpansive retract of . Let be nonself uniformly -Lipschitzian, -Lipschitzian, respectively. Suppose that , are bounded sequences in such that , . Starting from an arbitrary , define the sequence by the recursion (1.7) and set for all . If , then

Proof. Since , are bounded, it follows from Lemma 2.1 that and are all bounded. We set Let and . Then, we have We find the following from (1.7) and (2.10): Substituting (2.11) into (2.9), we get It follows from (1.7) and (2.12) that Using (2.11) and (2.13), we obtain Combine (2.13) with (2.14) yields that from which it follows that It follows from that Similarly, we can show that . This completes the proof.

Lemma. Let be a real uniformly convex Banach space and let be a nonempty closed convex subset of which is also a nonexpansive retract of . Let be two asymptotically nonexpansive nonself-mappings of with sequences such that , , respectively, and . Suppose that , , , , , are appropriate sequences in satisfying and , are bounded sequences in such that , . Moreover, for all and some . Starting from an arbitrary , define the sequence by the recursion (1.7). Then,

Proof. Let and . By Lemma 2.1, we see that exists. Assume that . If , then by the continuity of and the conclusion follows. Now, suppose . Taking on both sides in the inequalities (2.2), (2.3), and (2.4), we have respectively. Next, we consider Taking on both sides in the above inequality and using (2.18), we get Observe that which implies that means that On the other hand, by using (2.23) and (2.5), we have Therefore, we have Combining (2.23) with (2.25), we obtain Hence, applying Lemma 1.3, we find Note that which yields that That is, . This implies that Similarly, we have Combining (2.30) with (2.32), we obtain On the other hand, we have Hence, using (2.32), (2.33), (2.35), and Lemma 1.3, we find Note that from (2.36), we have which yields that That is,
Again, means that
By using (2.39) and (2.3), we obtain Therefore, we have Combining (2.39) with (2.41), we obtain On the other hand, we have which implies that Notice that which implies that Using (2.42), (2.44), (2.46), and Lemma 1.3, we find Observe that which yields that That is, . This implies that Similarly, we have Combining (2.50) with (2.52), we obtain On the other hand, we have Hence, using (2.53), (2.54), (2.55), and Lemma 1.3, we find In addition, from and (2.47), we have Hence, from (2.36) and (2.57), we find That is, Since and are uniformly -Lipschitzian and uniformly -Lipschitzian, respectively, for some , it follows from (2.56), (2.59), and Lemma 2.2 that This completes the proof.

Theorem. Let be a real uniformly convex Banach space and let be a nonempty closed convex subset of which is also a nonexpansive retract of . Let be two asymptotically nonexpansive nonself -mappings of with sequences such that , , respectively, and . Suppose that , , , , , are appropriate sequences in satisfying and , are bounded sequences in such that , . Moreover, for all and some . If one of is completely continuous, then the sequence defined by the recursion (1.7) converges strongly to some common fixed point of .

Proof. By Lemma 2.1, is bounded. In addition, by Lemma 2.3;; then and are also bounded. If is completely continuous, there exists subsequence of such that as . It follows from Lemma 2.3 that . So by the continuity of and Lemma 1.4, we have and . Furthermore, by Lemma 2.1, we get that exists. Thus . The proof is completed.

The following result gives a strong convergence theorem for two asymptotically nonexpansive nonself-mappings in a uniformly convex Banach space satisfying condition .

Theorem. Let be a real uniformly convex Banach space and let be a nonempty closed convex subset of which is also a nonexpansive retract of . Let be two asymptotically nonexpansive nonself-mappings of with sequences such that , , respectively, and . Suppose that , , , , , are appropriate sequences in satisfying and , are bounded sequences in such that , . Moreover, for all and some . Suppose that satisfy condition (). Then, the sequence defined by the recursion (1.7) converges strongly to some common fixed point of .

Proof. By Lemma 2.1, we readily see that and so, exists for all . Also, by Lemma 2.3, . It follows from condition that That is, Since is a nondecreasing function satisfying , for all , therefore, we have Now we can take a subsequence of and sequence such that for all integers . Using the proof method of Tan and Xu [5], we have and hence We get that is a Cauchy sequence in and so it converges. Let . Since is closed, therefore, and then . As exists, . Thereby completing the proof.

Remark. If then the iterative scheme (1.7) reduces to the iterative scheme (1.4) of [9]. Moreover, the condition is weaker than both the compactness of and the semicompactness of the asymptotically nonexpansive nonself-mappings Also, the condition for all is weaker than the condition for all and some . Hence, Theorems 2.4 and 2.5 generalize Theorems 3.3 and 3.4 in [9], respectively.

In the next result, we prove the weak convergence of the iterative scheme (1.7) for two asymptotically nonexpansive nonself-mappings in a uniformly convex Banach space satisfying Opial's condition.

Theorem. Let be a real uniformly convex Banach space and let be a nonempty closed convex subset of which is also a nonexpansive retract of . Let be two asymptotically nonexpansive nonself-mappings of with sequences such that , , respectively, and . Suppose that , , , , are appropriate sequences in satisfying and , are bounded sequences in such that , . Moreover, for all and some . Suppose that satisfy Opial's condition. Then, the sequence defined by the recursion (1.7) converges weakly to some common fixed point of .

Proof. Let . By Lemma 2.1, we see that exists and bounded. Now we prove that has a unique weak subsequential limit in . Firstly, suppose that subsequences and of converge weakly to and , respectively. By Lemma 2.3, we have . And Lemma 1.4 guarantees that , that is., . Similarly, . Again in the same way, we can prove that .
Secondly, assume , then by Opial’s condition, we have
which is a contradiction, hence, . Then, converges weakly to a common fixed point of and . This completes the proof.

Remark. The above Theorem generalizes Theorem 3.5 of Wang [9].

3. Case of Two Nonself-Nonexpansive Mappings

Let be two nonexpansive nonself-mappings. Then, the iterative scheme (1.7) is written as follows:

Nothing prevents one from proving the results of the previous section for nonexpansive nonself-mappings case. Thus, one can easily prove the following.

Theorem. Let be a real uniformly convex Banach space and let be a nonempty closed convex subset of which is also a nonexpansive retract of . Let <