Abstract and Applied Analysis

Volume 2012 (2012), Article ID 207896, 9 pages

doi:10.1155/2012/207896

Research Article

New Iterative Manner Involving Sunny Nonexpansive Retractions for Pseudocontractive Mappings

Yaqin Zheng¹ and Jinwei Shi²

¹College of Science, Agricultural University of Hebei, Baoding 071001, China

²North China Electric Power University, Baoding 071003, China

Received 19 September 2012; Accepted 8 October 2012

Academic Editor: Yonghong Yao

Copyright © 2012 Yaqin Zheng and Jinwei Shi. 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.

Iterative methods for pseudocontractions have been studied by many authors in the literature. In the present paper, we firstly propose a new iterative method involving sunny nonexpansive retractions for pseudocontractions in Banach spaces. Consequently, we show that the suggested algorithm converges strongly to a fixed point of the pseudocontractive mapping which also solves some variational inequality.

1. Introduction

Let be a nonempty closed convex subset of a real Banach space . A mapping is said to be nonexpansive, if

for all .

Now we know that the involved operators in the many practical applications can be reduced to the nonexpansive mappings, that is, there are a large number of applied areas which are closely related to the nonexpansive mappings, for example, inverse problem, partial differential equations, image recovery, and signal processing. Based on these facts, recently, iterative methods for finding fixed points of nonexpansive mappings have received vast investigations. For related works, please see [1–26] and the references therein.

In the present paper, we focus on a class of strictly pseudocontractive mappings which strictly includes the class of nonexpansive mappings. Recall that a mapping is said to be strictly pseudocontractive if there exists a constant and such that

for all . We use to denote the set of fixed points of .

We know that the strict pseudocontractions have more powerful applications than nonexpansive mappings in solving inverse problems. There are some related references in the literature for strictly pseudocontractive mappings; see, for example, [27–30]. Motivated and inspired by the works in the literature, in the present paper, we firstly propose a new iterative method involving sunny nonexpansive retractions for pseudocontractions in Banach spaces. Consequently, we show that the suggested algorithm converges strongly to a fixed point of the pseudocontractive mapping which also solves some variational inequality.

2. Preliminaries

Let be the dual space of a Banach space . Let () be the generalized duality mapping from into given by

where denotes the generalized duality pairing. In particular, is called the normalized duality mapping and it is usually denoted by . It is well known that is a uniformly smooth Banach space if and only if is single valued and uniformly continuous on any bounded subset of .

Let be a nonempty closed convex subset of a Banach space , and let be a nonempty subset of . Recall that a mapping is called a retraction from onto provided for all . A retraction is sunny provided for all and whenever . A sunny nonexpansive retraction is a sunny retraction which is also nonexpansive. Sunny non-expansive retractions are characterized as follows.

Lemma 2.1. If is a smooth Banach space, then is a sunny nonexpansive retraction if and only if there holds the inequality

for all and .

Lemma 2.2 (see [31]). Let be a real -uniformly smooth Banach space, and let . Then, one has

for all .

Lemma 2.3 (see [16]). Let and be two bounded sequences in Banach spaces, and let be a sequence in with . Suppose that   for all and . Then .

Lemma 2.4 (see [14]). Let be a nonempty closed convex subset of a real -uniformly smooth and uniformly convex Banach space . Let be a strictly pseudocontractive mapping. Then is demiclosed.

Lemma 2.5 (see [32]). Assume is a sequence of nonnegative real numbers such that   for where is a sequence in and is a sequence in such that

(i); (ii) or .

Then .

3. Main Results

In this section, we will give our main results. In the sequel, we assume the following:

(C1) is a uniformly convex and 2-uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping from to ; (C2) is a nonempty closed convex subset of ; (C3) is a sunny nonexpansive retraction from onto ; (C4) is a -strict pseudocontraction; (C5) is a -contraction; (C6) is strongly positive (i.e., for some ) and linear bounded operator with for all and ; (C7).

First, we consider the following VI: finding such that

The set of solutions of (3.1) is denoted by . In the sequel, we assume that . Note that (3.1) has the unique solution.

Next, we propose our algorithm.

Algorithm 3.1. For the initial point , we generate a sequence via the following manner:

where is a sequence in and .

Theorem 3.2. If the sequence satisfies and , then the sequence generated by (3.2) converges strongly to the unique solution of VI (3.1).

Proof. First, by using Lemma 2.2, we know that is nonexpansive. is sunny nonexpansive. Thus, is nonexpansive. Let . From (3.2), we have

Thus,

This indicates that is bounded.

We write for all . So,

Hence,

It follows that

where is a constant satisfying . This implies that

By Lemma 2.3, we deduce

Therefore,

Note that . As a matter of fact, if , that is , then . Since is a self-mapping, it is clear that . So, . Therefore, . Conversely, if , that is , we also have . Thus, . Set . We observe that . Next, we estimate .

Since

we have

Next, we show

where .

First, we have

Since the sequence is bounded, hence is bounded. Thus, we can take a subsequence of such that weakly. Without loss of generality, we may assume that weakly. Note that is nonexpansive and . By using the demiclosed principle of nonexpansive mappings (see Lemma 2.4), we get . At the same time, is weakly sequentially continuous. Therefore,

Finally we show that . From (3.2), we have

It follows that

It can be checked easily that and . From Lemma 2.5, we deduce . This completes the proof.

Algorithm 3.3. For the initial point , we generate a sequence via the following manner:

where is a sequence in and .

Corollary 3.4. If the sequence satisfies and , then the sequence generated by (3.18) converges strongly to the unique solution of VI: finding such that

Algorithm 3.5. For the initial point and , we generate a sequence via the following manner:

where is a sequence in and .

Corollary 3.6. If the sequence satisfies and , then the sequence generated by (3.20) converges strongly to the unique solution of VI: finding such that

Algorithm 3.7. For the initial point , we generate a sequence via the following manner:

where is a sequence in and .

Corollary 3.8. If the sequence satisfies and , then the sequence generated by (3.22) converges strongly to .

Acknowledgments

J. Shi was supported in part by the scientific research fund of the Educational Commission of Hebei Province of China (no. 936101101) and the National Natural Science Foundation of China (no. 51077053).

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