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Journal of Applied Mathematics
Volume 2014, Article ID 642167, 7 pages
http://dx.doi.org/10.1155/2014/642167
Research Article

Convergence of Viscosity Iteration Process for a Finite Family of Generalized Asymptotically Quasi-Nonexpansive Mappings

College of Mathematics and Computer Science, Yangtze Normal University, Chongqing 408100, China

Received 22 August 2013; Revised 10 December 2013; Accepted 18 December 2013; Published 2 January 2014

Academic Editor: Fernando Simões

Copyright © 2014 Zhiming Cheng. 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 general iteration method for a finite family of generalized asymptotically quasi-nonexpansive mappings. The results presented in the paper extend and improve some recent results in the works by Shahzad and Udomene (2006); L. Qihou (2001); Khan et al. (2008).

1. Introduction and Preliminaries

Let be a nonempty subset of a real Banach space and a self-mapping of . The set of fixed points of is denoted by and we assume that . The mapping is said to be(i)contractive mapping if there exists a constant in such that , for all ;(ii)asymptotically nonexpansive mapping if there exists a sequence in with such that , for all and ;(iii)asymptotically quasi-nonexpansive if there exists a sequence in with such that , for all , and ;(iv)generalized asymptotically quasi-nonexpansive [1] if there exist two sequences , in with and such that where ;(v)uniformly -Lipschitzian if there exists a constant such that , for all and ;(vi) uniform -Lipschitz if there are constants and such that , for all and ;(vii)semicompact if for a sequence in with , there exists a subsequence of such that .

In (1), if for all , then becomes an asymptotically quasi-nonexpansive mapping; if and for all , then becomes a quasi-nonexpansive mapping. It is known that an asymptotically nonexpansive mapping is an asymptotically quasi-nonexpansive and a uniformly -Lipschitzian mapping is uniform -Lipschitz.

The mapping is said to be demiclosed at if for each sequence converging weakly to and converging strongly to , we have .

A Banach space is said to satisfy Opial’s property if for each and each sequence weakly convergent to , the following condition holds for all :

Let be a nonempty closed convex subset of a real Banach space and a finite family of asymptotically nonexpansive mappings of into itself. Suppose that , , and . Then we consider the following mapping of into itself: where (identity mapping). Such a mapping is called the modified -mapping generated by and (see [2, 3]).

In the sequel, we assume that .

In 2008, Khan et al. [4] introduced the following iteration process for a family of asymptotically quasi-nonexpansive mappings, for an arbitrary : where , , , and proved that the iterative sequence defined by (4) converges strongly to a common fixed point of the family of mappings if and only if , where . With the help of (3), we write (4) as

Recently, Chang et al. [5] introduced the following iteration process of asymptotically nonexpansive mappings in Banach space: where and is a fixed contractive mapping, and necessary and sufficient conditions are given for the iterative sequence to converge to the fixed points of .

For a family of mappings, it is quite significant to devise a general iteration scheme which extends the iteration processes (4) and (6), simultaneously. Thereby, to achieve this goal, we introduce a new iteration process for a family of mappings as follows.

Let be a nonempty closed convex subset of a real Banach space , a family of generalized asymptotically quasi-nonexpansive mappings, and a fixed contractive mapping with contractive coefficient . For a given , the iteration scheme is defined as follows: where and is the modified -mapping generated by , and for all positive integers .

The purpose of this paper is to study the convergence problem of the iterative sequences defined by (7). The obtained results extend the corresponding results in [48], and Lemma 11 partly improves the method of proof of Lemma  3.1 in [4].

In what follows, we need the following useful known lemmas.

Lemma 1 (see [9]). Let , , and be nonnegative real sequences satisfying the following condition: where and ; then exists.

Moreover, if in addition, , then .

Lemma 2 (see [4]). Let be a uniformly convex Banach space, for all , and let and be sequences in . Assume that , , and for some . Then .

2. Main Results

Lemma 3. Let be a nonempty closed convex subset of a real Banach space and an asymptotically quasi-nonexpansive self-mapping of with for all . Suppose . Then is a closed subset in .

Proof. Let be an arbitrary sequence of and as . Since is closed, we have . For any , there exists a natural number such that
Thus, we get
Since is arbitrary, it follows that ; that is, . Hence and is closed. This completes the proof.

Lemma 4. Let be a nonempty closed convex subset of a real Banach space . Let be generalized asymptotically quasi-nonexpansive self-mappings of with such that and for all . Suppose and for all . Let be the modified -mapping generated by and . Let the sequence be defined by (7) and assuming , then(1)there exist two sequences and in with such that (2)there exists a constant , such that where and .

Proof. Let , for all . Since for each , we can get . For all , it follows from (3) that
Assume that for some . Then
Thus, by induction, we have for all . Hence,
By (7) and (16), we obtain
Since , is bounded. Setting , we get that where and . This completes the proof of .
If , then and consequently, , . Thus, from part , we get for any positive integers , , where , . This completes the proof of .

Remark 5. Lemma 4 generalizes Lemma  2.1 in [4].

Theorem 6. Let be a nonempty closed convex subset of a real Banach space . Let be generalized asymptotically quasi-nonexpansive self-mappings of with such that and for all . Let for all and let be a modified -mapping generated by and . Suppose that is closed and . Starting from arbitrary , define the sequence by the recursion (7); then the sequence converges strongly to if and only if .

Proof. We will only prove the sufficiency; the necessity is obvious. From Lemma 4, we have for all and all . Therefore,
As , so . By Lemma 1 and , we get that . Next, we prove that is a Cauchy sequence. From Lemma 4, we have
Hence, for all integers and all ,
Taking infimum over in (23) gives
Now, since and , given , there exists an integer such that for all , and . So for all integers , , we obtain from (24) that
Hence, is a Cauchy sequence in . Since is complete, there exists such that . We now show that . Since and as , for each , there exists an integer such that, and for all . In particular, we have ; that is, there exists a such that ; hence
Since is a closed subset of , we obtain . This completes the proof.

Remark 7. Theorem 6 generalizes and extends Theorem 2.2 of Khan et al. [4], Theorem 3.1 of Ghosh and Debnath [8], Theorem 3.2 of Shahzad and Udomene [6], and Theorem of Qihou [7] together with its Corollaries and .

Asymptotically nonexpansive mappings and asymptotically quasi-nonexpansive mappings are all generalized asymptotically quasi-nonexpansive, by Theorem 6 and Lemma 3, so we have

Corollary 8. Let be a nonempty closed convex subset of a real Banach space . Let be asymptotically quasi-nonexpansive self-mappings of with such that for all . Let for all and let be a modified -mapping generated by and . Suppose and . Starting from arbitrary , define the sequence by the recursion (7). Then the sequence converges strongly to if and only if .

Corollary 9. Let be a nonempty closed convex subset of a real Banach space . Let be asymptotically nonexpansive self-mappings of with such that for all . Let for all and let be a modified -mapping generated by and . Suppose and . Starting from arbitrary , define the sequence by the recursion (7). Then the sequence converges strongly to if and only if .

Corollary 10. Let be a nonempty closed convex subset of a real Banach space . Let be generalized asymptotically quasi-nonexpansive self-mappings of with such that and for all . Let for all and let be a modified -mapping generated by and . Suppose that is closed and . Starting from arbitrary , define the sequence by the recursion (7). Then the sequence converges strongly to if and only if there exists a subsequence of which converges to .

3. Results in Uniformly Convex Banach Spaces

Lemma 11. Let be a nonempty closed convex subset of a uniformly convex Banach space . Let be uniform Lipschitz and generalized asymptotically quasi-nonexpansive self-mappings of with such that and for all . Let for some and let be a modified -mapping generated by and . Suppose and . Starting from arbitrary , define the sequence by the recursion (7). Then for each .

Proof. Let and , for all . By Lemma 1 and Lemma 4, it follows that exists for all . Assume that
From (15) and (27) we obtain that
From (7), we have therefore,
From (28) and (30) we can obtain that
Suppose that for some . Since so we obtain that
From (28) and (33), we have that
Thus, by induction, we have for each . That is, for each . From (28), we obtain for each . By Lemma 2, we get
If , from (38), we have
If , then we have
Hence,
Note that therefore, we have
Now, we observe that
By (41) and (43), we have for . This completes the proof.

Theorem 12. Let be a nonempty closed convex subset of a uniformly convex Banach space . Let be uniform Lipschitz and generalized asymptotically quasi-nonexpansive self-mappings of with such that and for all . Let for some and let be a modified -mapping generated by and . Suppose , and there exists one member in which is semicompact for some positive integer . Starting from arbitrary , define the sequence by the recursion (7). Then converges strongly to some common fixed point of the family .

Proof. By Lemma 11, we have for each . Without loss of generality, we may assume that is semicompact for some ; then we have
Since is semicompact, then there exists a subsequence of such that . Hence, we have for each . This implies that . By Corollary 10, converges strongly to some common fixed point of the family .

Theorem 13. Let be a nonempty closed convex subset of a uniformly convex Banach space . Let be uniform Lipschitz and generalized asymptotically quasi-nonexpansive self-mappings of with such that and for all . Let for some and let be a modified -mapping generated by and . Suppose , and each , , is demiclosed at . If satisfies Opial’s condition, then the sequence defined by (7) converges weakly to a common fixed point of the family .

Proof. From the proof of Lemma 11, we know that is a bounded sequence in . Since is uniformly convex, it must be reflexive. Therefore, there exists a subsequence in converging weakly to . By Lemma 11, and is demiclosed at for , so we obtain . That is, . Suppose that there exists another subsequence of converging weakly to . As above, we can prove . By (27) we know that and exist. Assume . Then by the Opial’s condition, we have which is a contradiction. Hence . This implies that converges weakly to a common fixed point of the family .

Remark 14. Lemma 11, Theorem 12, and Theorem 13 extend Lemma 3.1, Theorem 3.3, and Theorem 3.2 of Khan et al. [4], respectively.

Conflict of Interests

The author declares that there is no conflict of interests.

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