Journal of Applied Mathematics

Volume 2012 (2012), Article ID 759718, 10 pages

doi:10.1155/2012/759718

Research Article

Strong Convergence Theorems for a Finite Family of -Strict Pseudocontractions in 2-Uniformly Smooth Banach Spaces by Metric Projections

Xin-dong Liu,¹ Shih-sen Chang,² and Xiong-rui Wang¹

¹School of Mathematics, Yibin University, Sichuan,Yibin 644007, China

²College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China

Received 16 July 2012; Accepted 27 August 2012

Academic Editor: Nan-Jing Huang

Copyright © 2012 Xin-dong Liu 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.

A new hybrid projection algorithm is considered for a finite family of -strict pseudocontractions. Using the metric projection, some strong convergence theorems of common elements are obtained in a uniformly convex and 2-uniformly smooth Banach space. The results presented in this paper improve and extend the corresponding results of Matsushita and Takahshi, 2008, Kang and Wang, 2011, and many others.

1. Introduction

Let be a real Banach space and let be the dual spaces of . Assume that is the normalized duality mapping from into defined by

where is the generalized duality pairing between and .

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

for all . Also a mapping is called a -strict pseudocontraction if there exists a constant such that for every and for some , the following holds:

From (1.3) we can prove that if is -strict pseudo-contractive, then is Lipschitz continuous with the Lipschitz constant .

It is well-known that the classes of nonexpansive mappings and pseudocontractions are two kinds important nonlinear mappings, which have been studied extensively by many authors (see [1–8]).

In [9] Reich considered the Mann iterative scheme

for nonexpansive mappings, where is a sequence in . Under suitable conditions, the author proved that converges weakly to a fixed point of . In 2005, Kim and Xu [10] proved a strong convergence theorem for nonexpansive mappings by modified Mann iteration. In 2008, Zhou [11] extended and improved the main results of Kim and Xu to the more broad 2-uniformly smooth Banach spaces for -strict pseudocontractive mappings.

On the other hand, by using metric projection, Nakajo and Takahashi [12] introduced the following iterative algorithms for the nonexpansive mapping in the framework of Hilbert spaces:

where , and is the metric projection from a Hilbert space onto . They proved that generated by (1.5) converges strongly to a fixed point of .

In 2006, Xu [13] extended Nakajo and Takahashi's theorem to Banach spaces by using the generalized projection.

In 2008, Matsushita and Takahashi [14] presented the following iterative algorithms for the nonexpansive mapping in the framework of Banach spaces:

where denotes the convex closure of the set , is normalized duality mapping, is a sequence in (0, 1) with , and is the metric projection from onto . Then, they proved that generated by (1.6) converges strongly to a fixed point of nonexpansive mapping .

Recently, Kang and Wang [15] introduced the following hybrid projection algorithm for a pair of nonexpansive mapping in the framework of Banach spaces:

where denotes the convex closure of the set , is a sequence in [0, 1], is a sequence in (0,1) with , and is the metric projection from onto . Then, they proved that generated by (1.7) converges strongly to a fixed point of two nonexpansive mappings and .

In this paper, motivated by the research work going on in this direction, we introduce the following iterative for finding fixed points of a finite family of -strict pseudocontractions in a uniformly convex and 2-uniformly smooth Banach space:

where denotes the convex closure of the set , is sequences in [0,1] and for each , is a sequence in (0,1) with , and is the metric projection from onto . we prove defined by (1.8) converges strongly to a common fixed point of a finite family of -strictly pseudocontractions, the main results of Kang and Wang is extended and improved to strictly pseudocontractions.

2. Preliminaries

In this section, we recall the well-known concepts and results which will be needed to prove our main results. Throughout this paper, we assume that is a real Banach space and is a nonempty subset of . When is a sequence in , we denote strong convergence of to by and weak convergence by . We also assume that is the dual space of , and is the normalized duality mapping. Some properties of duality mapping have been given in [16].

A Banach space is said to be strictly convex if for all with . is said to be uniformly convex if for each there is a such that for with and holds. The modulus of convexity of is defined by

is said to be smooth if the limit

exists for all . The modulus of smoothness of is defined by

A Banach space is said to be uniformly smooth if as . A Banach space is said to be -uniformly smooth, if there exists a fixed constant such that .

If is a reflexive, strictly convex, and smooth Banach space, then for any , there exists a unique point such that

The mapping defined by is called the metric projection from onto . Let and . Then it is known that if and only if

For the details on the metric projection, refer to [17–20].

In the sequel, we make use the following lemmas for our main results.

Lemma 2.1 (see [21]). Let be a real 2-uniformly smooth Banach space with the best smooth constant . Then the following inequality holds

for any .

Lemma 2.2 (see [11]). Let be a nonempty subset of a real 2-uniformly smooth Banach space with the best smooth constant and let be a -strict pseudocontraction. For , we define . Then is nonexpansive such that .

Lemma 2.3 (demiclosed principle, see [22]). Let be a real uniformly convex Banach space, let be a nonempty closed convex subset of , and let be a continuous pseudocontractive mapping. Then, is demiclosed at zero.

Lemma 2.4 (see [23]). Let be a closed convex subset of a uniformly convex Banach space. Then for each , there exists a strictly increasing convex continuous function such that and

for all , , , and , where =  and , , and is the set of all nonexpansive mappings from into .

3. Main Results

Now we are ready to give our main results in this paper.

Lemma 3.1. Let be a closed convex subset of a uniformly convex and 2-uniformly smooth Banach space with the best smooth constant , and be a -strict pseudocontraction. Then for each , there exists a strictly increasing convex continuous function such that and

for all , ,, where ,   and , .

Proof. Define the mapping as , for all . Then is nonexpansive. From Lemma 2.4, there exists a strictly increasing convex continuous function with and such that

Hence

This completes the proof.

Theorem 3.2. Let be a nonempty closed subset of a uniformly convex and 2-uniformly smooth Banach space with the best smooth constant , assume that for each , is a -strict pseudocontraction for some such that . Let be sequences in [0,1] with for each and be a sequence in (0,1) with . Let be a sequence generated by (1.8), where denotes the convex closure of the set and is the metric projection from onto . Then converges strongly to .

Proof. (I) First we prove that is well defined and bounded.

It is easy to check that is closed and convex and for all . Therefore is well defined.

Put . Since and , we have that

for all . Hence is bounded.

(II) Now we prove that as for all .

Since , there exist some positive integer ( denotes the set of all positive integers), and such that

for all . Put and . Take . It follows from Lemma 2.2 and (3.5) that

for all . Moreover, (3.7) implies

It follows from Lemma 3.1, (3.5)–(3.9) that

This shows that

for all .

(III) Finally, we prove that .

It follows from the boundedness of that there exists such that as . Since for each , is a -strict pseudocontraction, then is demiclosed. one has .

From the weakly lower semicontinuity of the norm and (3.4), we have

This shows and hence as . Therefore, we obtain . Further, we have that

Since is uniformly convex, we have . This shows that . This completes the proof.

Corollary 3.3. Let be a nonempty closed subset of a uniformly convex and 2-uniformly smooth Banach space with the best smooth constant , assume that is a -strict pseudocontraction for some such that . Let be a sequence generated by

where is a sequence in (0,1) with . denotes the convex closure of the set and is the metric projection from onto . Then converges strongly to .

Proof. Set for all , and , for all in Theorem 3.2. The desired result can be obtained directly from Theorem 3.2.

Remark 3.4. At the end of the paper, we would like to point out that concerning the convergence problem of iterative sequences for strictly pseudocontractive mappings has been considered and studied by many authors. It can be consulted the references [24–37].

Acknowledgment

The authors would like to express their thanks to the referees for their valuable suggestions and comments. This work is supported by the Scientific Research Fund of Sichuan Provincial Education Department (11ZA221) and the Scientific Research Fund of Science Technology Department of Sichuan Province 2011JYZ010.

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