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International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 549364, 18 pages
http://dx.doi.org/10.1155/2011/549364
Research Article

Strong Convergence Theorems of Modified Ishikawa Iterative Method for an Infinite Family of Strict Pseudocontractions in Banach Spaces

1Department of Mathematics and Statistics, Faculty of Science and Agricultural Technology, Rajamangala University of Technology Lanna Tak, Tak 63000, Thailand
2Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand

Received 15 December 2010; Accepted 14 March 2011

Academic Editor: Vittorio Colao

Copyright © 2011 Phayap Katchang 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 modified Ishikawa iterative process and a new W-mapping for comput- ing fixed points of an infinite family of strict pseudocontractions mapping in the framework of q-uniformly smooth Banach spaces. Then, we establish the strong convergence theorem of the proposed iterative scheme under some mild conditions. The results obtained in this paper extend and improve the recent results of Cai and Hu 2010, Dong et al. 2010, Katchang and Kumam 2011 and many others in the literature.

1. Introduction

Let be a real Banach space with norm and a nonempty closed convex subset of . Let be the dual space of , and let denotes the generalized duality pairing between and . For , the generalized duality mapping is defined by for all . In particular, if , the mapping is called the normalized duality mapping and for . It is well known that if is smooth, then is single-valued, which is denoted by .

A mapping is called if We use to denote the set of fixed points of ; that is, .

is said to be a -strict pseudocontraction in the terminology of Browder and Petryshyn [1] if there exists a constant and for some such that

is said to be a strong pseudocontraction if there exists such that

Remark 1.1 (see [2]). Let be a -strict pseudocontraction in a Banach space. Let and . Then,

Recall that a self mapping is contraction on if there exists a constant and such that We use to denote the collection of all contractions on . That is, . Note that each has a unique fixed point in .

Very recently, Cai and Hu [3] also proved the strong convergence theorem in Banach spaces. They considered the following iterative algorithm: where is a non-self--strictly pseudocontraction, is a contraction, and is a strongly positive linear bounded operator.

Dong et al. [2] proved the sequence converges strongly in Banach spaces under certain appropriate assumptions and used the mapping defined by (1.11). Let the sequences be generated by

On the other hand, Katchang and Kumam [4, 5] introduced the following new modified Ishikawa iterative process for computing fixed points of an infinite family nonexpansive mapping in the framework of Banach spaces; let the sequences be generated by where is a contraction, is a strongly positive linear bounded self-adjoint operator, and mapping (see [6, 7]) is defined by where is an infinite family of nonexpansive mappings of into itself and is real numbers such that for every . In 2010, Cho [8] considered and proved the strong convergence of the implicit iterative process for an infinite family of strict pseudocontractions in an arbitrary real Banach space.

In this paper, motivated and inspired by Cai and Hu [9], we consider the mapping defined by where are real numbers such that , where is a -strict pseudocontraction of into itself and , where for all . By Lemma 2.3, we know that is a nonexpansive mapping, and therefore, is a nonexpansive mapping. We note that the -mapping (1.10) is a special case of a -mapping (1.11) when is constant for all .

Throughout this paper, we will assume that , for all and satisfies(H1) for all ,(H2) for all and , where satisfies .

The hypothesis (H2) secures the existence of for all . Set for all . Furthermore, we assume that(H3) for all .

It is obvious that satisfy (H1). Using condition (H3), from , we define mappings for all .

Our results improve and extend the recent ones announced by Cai and Hu [3], Dong et al. [2], Katchang and Kumam [4, 5], and many others.

2. Preliminaries

Recall that . A Banach space is said to be uniformly convex if, for any , there exists such that for any , implies . It is known that a uniformly convex Banach space is reflexive and strictly convex (see also [10]). A Banach space is said to be smooth if the limit exists for all . It is also said to be uniformly smooth if the limit is attained uniformly for .

In a smooth Banach space, we define an operator as strongly positive if there exists a constant with the property where I is the identity mapping and J is the normalized duality mapping.

If and are nonempty subsets of a Banach space such that is a nonempty closed convex and , then a mapping is sunny [11, 12] provided that for all and whenever . A mapping is called a retraction if . If a mapping is a retraction, then for all in the range of . A subset of is said to be a sunny nonexpansive retract of if there exists a sunny nonexpansive retraction of onto . A sunny nonexpansive retraction is a sunny retraction, which is also nonexpansive. Sunny nonexpansive retractions play an important role in our argument. They are characterized as follows [11, 12]: if is a smooth Banach space, then is a sunny nonexpansive retraction if and only if there holds the inequality

We need the following lemmas for proving our main results.

Lemma 2.1 (see [13]). In a Banach space E, the following holds: where .

Lemma 2.2 (see [14]). Let be a real -uniformly smooth Banach space, then there exists a constant such that In particular, if be a real 2-uniformly smooth Banach space with the best smooth constant , then the following inequality holds:

The relation between the -strict pseudocontraction and the nonexpansive mapping can be obtained from the following lemma.

Lemma 2.3 (see [15]). Let be a nonempty convex subset of a real -uniformly smooth Banach space and a -strict pseudocontraction. For , one defines . Then, as , is nonexpansive such that , where is the constant in Lemma 2.2.

Concerning , we have the following lemmas, which are important to prove the main results.

Lemma 2.4 (see [2]). Let be a nonempty closed convex subset of a -uniformly smooth and strictly convex Banach space E. Let , be a -strict pseudocontraction from into itself such that , and let . Let , be real numbers such that for any . Assume that the sequence satisfies (H1) and (H2). Then, for every and , the limit exists.

Using Lemma 2.4, we define the mappings and as follows: for all . Such is called the -mapping generated by and and .

Lemma 2.5 (see [2]). Let be a bounded sequence in a -uniformly smooth and strictly convex Banach space . Under the assumptions of Lemma 2.4, it holds

Lemma 2.6 (see [2]). Let be a nonempty closed convex subset of a -uniformly smooth and strictly convex Banach space E. Let , be a -strict pseudocontraction from into itself such that , and let . Let , be real numbers such that for any . Assume that the sequence satisfies (H1)–(H3). Then, .

Lemma 2.7 (see [16]). Assume that is a sequence of nonnegative real numbers such that where is a sequence in and is a sequence in such that (i), (ii) or . Then, .

Lemma 2.8 (see [17]). Let and be bounded sequences in a Banach space , and let be a sequence in with . Suppose that for all integers and . Then, .

Lemma 2.9 (see [3]). Assume that is a strong positive linear bounded operator on a smooth Banach space with coefficient and . Then, .

3. Main Results

In this section, we prove a strong convergence theorem.

Theorem 3.1. Let be a real -uniformly smooth and strictly convex Banach space which admits a weakly sequentially continuous duality mapping from to . Let be a nonempty closed and convex subset of which is also a sunny nonexpansive retraction of such that . Let be a strongly positive linear bounded operator on with coefficient such that , and let be a contraction of into itself with coefficient . Let , be -strict pseudocontractions from into itself such that and . Assume that the sequences , , and in satisfy the following conditions:(i); and ,(ii),(iii),(iv),(v) for some ,and the sequence satisfies (H1)–(H3). Then, the sequence generated by converges strongly to , which solves the following variational inequality:

Proof. By (i), we may assume, without loss of generality, that for all . Since is a strongly positive bounded linear operator on and by (2.1), we have Observe that This shows that is positive. It follows that
First, we show that is bounded. Let . By the definition of , and , we have and from this, we have
It follows that By induction on , we obtain for every and , then is bounded. So, , , and are also bounded.
Next, we claim that as . Let and . Fix for any with , and since and are nonexpansive, we have and , respectively. From (1.5), it follows that . We can set
From (1.11), we have for all . Similarly, we also have for all . We compute that and where . Observe that we put , then Now, we have Therefore, we have From the conditions (i)–(iv), (H2), and the boundedness of , , , , and , we obtain It follows from Lemma 2.8 that . Noting (3.13), we see that as . Therefore, we have We also have and as . Observing that it follows that By the conditions (i), (ii), (3.18), and the boundedness of , , and , we obtain Consider It follows that This implies that From the condition (v) and (3.21), we get On the other hand, From the boundedness of and using (2.7), we have as . It follows that
Next, we prove that where with being the fixed point of contraction . Noticing that solves the fixed point equation , it follows that It follows from Lemma 2.1 that where Since is linearly strong and positive and using (2.1), we have Substituting (3.32) in (3.30), we have Letting in (3.33) and noting (3.31) yield that where is a constant such that for all and . Taking from (3.34), we have On the other hand, we have which implies that
Noticing that is norm-to-norm uniformly continuous on bounded subsets of , it follows from (3.35) that
Therefore, we obtain that (3.28) holds.
Finally, we prove that as . Now, from Lemma 2.1, we have and consequently, where is an appropriate constant such that . Setting and , then we have By (3.28), (i) and applying Lemma 2.7 to (3.41), we have as . This completes the proof.

Corollary 3.2. Let be a real q-uniformly smooth and strictly convex Banach space which admits a weakly sequentially continuous duality mapping from to . Let be a nonempty closed and convex subset of which is also a sunny nonexpansive retraction of such that . Let be a strongly positive linear bounded operator on with coefficient such that , and let be a contraction of into itself with coefficient . Let , be -strict pseudocontractions from into itself such that and . Assume that the sequences , , and in satisfy the following conditions:(i); and ,(ii),(iii),(iv),(v) for some ,and the sequence satisfies (H1). Then, the sequence generated by converges strongly to , which solves the following variational inequality:

Corollary 3.3. Let be a real q-uniformly smooth and strictly convex Banach space which admits a weakly sequentially continuous duality mapping from to . Let be a nonempty closed and convex subset of which is also a sunny nonexpansive retraction of such that . Let be a strongly positive linear bounded operator on with coefficient such that , and let be a contraction of into itself with coefficient . Let , be a nonexpansive mapping from C into itself such that and . Assume that the sequences , , and in satisfy the following conditions:(i); and ,(ii), (iii),(iv)