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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 319241, 7 pages
http://dx.doi.org/10.1155/2013/319241
Research Article

Convergence of a New Modified Ishikawa Type Iteration for Common Fixed Points of Total Asymptotically Strict Pseudocontractive Semigroups

Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China

Received 28 April 2013; Revised 3 June 2013; Accepted 3 June 2013

Academic Editor: Abdellah Bnouhachem

Copyright © 2013 Yuanheng Wang and Chunjie Wang. 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

The purpose of this paper is to give a new modified Ishikawa type iteration algorithm for common fixed points of total asymptotically strict pseudocontractive semigroups. Under the reduction of some conditions, both strong convergence and weak convergence of the iteration algorithm are proved in Banach spaces with new methods of proofs, respectively. The main results presented in this paper extend and improve the corresponding recent results of many others.

1. Introduction

Throughout this paper, we assume that is a real Banach space with the norm , the dual space of , the duality between and , and a nonempty closed convex subset of . denotes the set of nonnegative real numbers and the natural number set. The mapping with is called the normalized duality mapping.

Let be a nonlinear mapping. denotes the set of the fixed points of .

As we know, a mapping is said to be pseudocontractive, if, for all , there exists , such that

Variational inequalities introduced by Stampacchia in the early sixties have had a great impact and influence on the development of almost all branches of pure and applied sciences and have witnessed an explosive growth in theoretical advances, algorithmic development, and so forth. Recently, some authors also studied the problem of finding the solution set of variational inequalities and the common element of the fixed point set for generalized nonexpansive mappings in the framework of real Hilbert spaces and Banach spaces. As is known to all, the variational inequality problem, nonlinear optimization problem, and fixed point problem are equivalent to each other under certain conditions.

In 2012, Chang et al. [1] introduced a more general class of pseudocontractive mappings and studied the methods for approximation of the split common fixed points.

Definition 1 (see [1]). (I) A mapping is said to be -totally asymptotically strictly pseudocontractive, if there exist a constant and sequences with and , such that, for all , where is continuous and a strict increasing function with .
(II) A mapping is said to be -asymptotically strictly pseudocontractive, if there exist a constant and a sequence with , such that

Definition 2. (I) One-parameter family is said to be a pseudocontractive semigroup on , if the following conditions are satisfied: (a) for each ; (b) for any and ; (c)the mapping is continuous for any given ; (d)for any , is pseudocontractive; that is, for any , there exists , such that
(II) One-parameter family is said to be strict pseudocontractive semigroup on , if the conditions (a)–(c) and the following condition (e) are satisfied. (e) For any , there exist and a bounded function , such that, for any ,
(III) is said to be asymptotically strict pseudocontractive semigroup, if the conditions (a)–(c) and the following condition (f) are satisfied. (f) There exist a bounded function and a sequence with as . For any given , there exists such that for, any ,

Osilike and Akuchu [2] established an iterative scheme for approximation of common fixed points of a finite family of asymptotically pseudocontractive mappings. Miao et al. [3] introduced an implicit iteration process for a finite family of total asymptotically pseudocontractive maps. And in recent years, many researchers focused on the convergence of pseudocontractive and asymptotically strict pseudocontractive semigroups; see [48] and their references. In [9, 10] especially, the authors gave the modified Mann type iteration algorithm and studied its convergence.

Inspired and motivated by the above works, in this paper, we give a new modified Ishikawa type iteration algorithm for total asymptotically strict pseudocontractive semigroups. Under the reducation of some conditions, we prove both strong convergence and weak convergence of the iteration algorithm by using the method of the subsequence of a subsequence of the sequence in Banach spaces, respectively. The results presented in this paper extend and improve the corresponding recent results of many authors, such as [1, 710].

2. Preliminaries

This section contains some definitions, notations, and lemmas, which will be used in the proofs of our main results in the next section.

A Banach space is said to be smooth if the limit exists for each . It is well known that if is reflexive and smooth, then the duality mapping is single valued.

A Banach space is said to have Opial condition if, for any sequence weakly convergent to , holds for any .

A mapping is said to be demiclosed, if, for any sequence ,   and imply that .

Definition 3 (see [9]). One-parameter is said to be a -total asymptotically strict pseudocontractive semigroup on , if the conditions (a)–(c) in Definition 2 and the following condition (g) are satisfied.(g) There exist a bounded function and sequences and with , , as . For any given ,  there exists , such that for any ,  for all  , where is continuous and strictly increasing function with .

A -total asymptotically strict pseudocontractive semigroup is said to be uniformly  Lipschitzian, if there exists a bounded measurable function , such that

Remark 4. According to the definitions, it is obvious that a pseudocontractive semigroup is a strict pseudocontractive semigroup with , and a strict pseudocontractive semigroup is an asymptotically strict pseudocontractive semigroup with . An asymptotically strict pseudocontractive semigroup is a -total asymptotically strict pseudocontractive semigroup with ,  ,  and  .

Definition 5 (see [11]). The normalized duality mapping of a Banach space is said to be weakly sequential continuous; if for all , , then there exist ,   such that , where weak convergence and weak star convergence are denoted by and , respectively.

In order to prove the main results of this paper, the following lemmas should be used.

Lemma 6 (see [4]). For any , one has

Lemma 7 (see [12]). Let ,  , and be the sequences of , which satisfy If ,  , then the limit exists.

3. Main Results

Theorem 8. Let be a nonempty closed convex subset of a real Banach space , and let be a uniformly Lipschitzian and -total asymptotically strict pseudocontractive semigroup defined in Definition 3. Suppose that and there exists a compact subset of such that . We assume that there exist positive constants and , such that for all . Let be the sequence defined by the modified Ishikawa type iteration algorithm: Then converges strongly to a common fixed point in , if the following conditions are satisfied:(i),  ,  ,  and ;(ii)  as  ,  ;(iii),  .

Proof. We divide the proof into four steps.
Step  1.  Firstly, we prove that exists for any .
By the definitions of and , we have This follows from that
Since is total asymptotically strict pseudocontractive semigroup, for any point and , by (9), we have
Since is an increasing function, it results in that , if ; , if . In either case, we can obtain that
Hence, by Lemma 6, we have where ,  .
By the conditions (i) and (ii), we have , . Thus, by Lemma 7, we can obtain that exists.
Step  2.  Now we prove that .
From (18), we know that As ,  , we can have Then,
Since , then (21) implies that
Otherwise, if , then there exists an , such that , when . So, we have This is in contradiction with (21).
Because and , we have
Step  3.  Now we prove that .
Consider Since and (25), we have Thus, there exists a subsequence such that
Step  4.  Finally, we prove the sequence converges strongly to a common fixed point of the semigroup .
Since is a compact subset of and , just as the proof in [9, 10], there exists a subsequence , such that . From (28), we have , and Hence we have that That is, .
Since, for any , exists, , and , so we have ; that is, converges strongly to an element of .

Remark 9. (a) If we take in the modified Ishikawa type iteration algorithm (13), then (13) is called the modified Mann type iteration algorithm in many articles, such as in [9, 10]. (b) In Theorem 8, because there is no limit to of , so our result is stronger and the conditions here are less than in [9, 10]. For example, the conditions “for any bounded subset , in [9, 10] can be removed in Theorem 8. (c) The condition “there exists a compact subset of such that ” does not look natural. But it easy to see that this condition is established naturally when we assume is a compact subset of . So, the result in Theorem 8 is still true if this condition is replaced by the condition “let be a compact subset of .” If there is no compactness assumption, we can get the following weak convergence theorem.

Theorem 10. Let be a reflexive Banach space satisfying the opial condition and be a nonempty bounded closed convex subset of . Let be a uniformly Lipschitzian and -total asymptotically strict pseudocontractive semigroup defined by Definition 3. Suppose that there exist positive constants and , such that , for all , and . Let be the sequence defined by (13). Then   converges weakly to a common fixed point in , if the following conditions are satisfied. (i),  ,  ,  .(ii) as ,  and .(iii),  .

Proof. It can be proved just like the proof in Theorem 8 that, for each , exists, and, for all ,   is bounded, . Thus, there exists a subsequence such that .
Now we prove that is demiclosed at zero (see [11]).
Since is a closed and convex subset of a reflexive Banach space , there exists a subsequence , such that . Without loss of generality, we can assume that replaces now.
In the following, we prove that .
Firstly, we choose and for . Since is uniformly Lipschitzian, we have
Because is totally asymptotically strictly pseudocontractive, we have Since , (note the of here instead of ), and is weakly sequential continuous duality mapping, we have Hence This implies that Let ; then we have , as , for ,  . Hence, , as , and . By the continuity of , we have .
Now, for the sequence generated by (13), we prove that .
Suppose the contrary; if there exists another subsequence , such that with , then we have that and exist. Since satisfies the Opial condition, we have This is a contraction, which shows . Therefore, . This completes the proof.

Remark 11. (a) Our results extend many other results that have been proved for this important class of general pseudocontractive mappings. For example, we extend the total asymptotically strict pseudocontractive mapping in [1] and Lipschitzian pseudo-contraction semigroup in [8] to the total asymptotically strict pseudocontractive semigroup. (b) In addition, we study the weak convergence of the total asymptotically strict pseudocontractive semigroup by using the demiclosedness of which, in some way, extends the result in [11] in Banach spaces. (c) And the method by using the subsequence of a subsequence of the sequence in this paper is different from the previous references.

Acknowledgments

The authors would like to thank editors and referees for many useful comments and suggestions for the improvement of the paper. This work was partially supported by the Natural Science Foundation of Zhejiang Province (Y6100696) and the National Natural Science Foundation of China (11271330).

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