Abstract and Applied Analysis

Volume 2012 (2012), Article ID 127851, 8 pages

doi:10.1155/2012/127851

Research Article

A Strong Convergence Theorem for Total Asymptotically Pseudocontractive Mappings in Hilbert Spaces

Chuan Ding¹ and Jing Quan²

¹School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 610074, China

²Department of Mathematics, Yibin University, Yibin, Sichuan 644007, China

Received 19 May 2012; Accepted 27 August 2012

Academic Editor: Irena Lasiecka

Copyright © 2012 Chuan Ding and Jing Quan. 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.

Demiclosedness principle for total asymptotically pseudocontractive mappings in Hilbert spaces is established. The strong convergence to a fixed point of total asymptotically pseudocontraction in Hilbert spaces is obtained based on demiclosedness principle, metric projective operator, and hybrid iterative method. The main results presented in this paper extend and improve the corresponding results of Zhou (2009), Qin, Cho, and Kang (2011) and of many other authors.

1. Introduction

Throughout this article we assume that is a real Hilbert space, whose inner product and norm are denoted by and , respectively; is a nonempty closed convex subset of ; and denote the natural number set and the set of nonnegative real numbers, respectively. Let be a nonlinear mapping; denotes the set of fixed points of mapping . We use “” to stand for strong convergence and “” for weak convergence.

Recently, the iterative approximation of fixed points for asymptotically pseudocontractive mappings, total asymptotically pseudocontractive mappings in Hilbert, or Banach spaces has been studied extensively by many authors, see for example [1–5].

The asymptotically pseudocontractions and total asymptotically pseudo-contractions are defined as follows.

Definition 1.1 (see [3]). is said to be asymptotically pseudocontraction if there exists a sequence of positive real numbers with , , such that

holds; in [4], is said to be total asymptotically pseudo-contraction if there exists sequences with as and strictly increasing continuous functions with , such that

holds; Zhou [3, page 3144] have proved the following Theorem.

Theorem Zhou

Let be a bounded and closed convex subset of a real Hilbert space . Let be a uniformly -Lipschitzian asymptotically pseudo-contraction with a fixed point. Suppose the control sequence is chosen so that , for some . Let be a sequence generated in the following manner:

Then the iterative sequence converges strongly to .

Qin et al. [5] introduced the class of total asymptotically pseudocontractive mappings in Hilbert spaces and established the following weak convergence theorem of fixed points.

Theorem Qin

Let be a bounded and closed convex subset of a real Hilbert space . Let be a uniformly -Lipschitzian total asymptotically pseudo-contractive mapping. Assume that is nonempty and there exist positive constants and such that for all . Let be a sequence generated in the following manner:

where and are sequences in . Assume that the following restrictions are satisfied:

(a) and .(b) for some and some .

Then the iterative sequence converges weakly to fixed point of .

The purpose of this article is to prove the strong convergence for total asymptotically pseudo-contraction in Hilbert spaces. The results presented in the article improve and extend the corresponding results of Zhou [3], Qin et al. [5], and many other authors.

2. Preliminaries

A mapping is said to be uniformly -Lipschitzian if there exists some such that

holds for all and for all . Let be a nonempty closed convex subset of a real Hilbert space . For every point there exists a unique nearest point in , denoted by , such that holds for all , where is called the metric projection of onto .

In order to prove the results of this article, we will need the following lemmas.

Lemma 2.1 (see [6]). Let be a nonempty closed convex subset of a real Hilbert space . Given and , then if and only if there holds the relation

Lemma 2.2 (see [6]). Let C be a nonempty closed convex subset of a real Hilbert space H and the metric projection from H onto C. Then the following inequality holds:

3. Main Results

Theorem 3.1 (demiclosedness principle). Let C be a nonempty bounded and closed convex subset of a real Hilbert space H. Let be a uniformly -Lipschitzian and total asymptotically pseudo-contraction. Suppose there exists , such that , then is demiclosed at zero, where is the identical mapping.

Proof. Assume that , with and as . We want to prove and . Since is a closed convex subset of , so . In the following we prove .

Now we choose and let for arbitrary fixed . Because is uniformly -Lipschitzian, we have

Since is total asymptotically pseudo-contraction, we have

By assumption , and as , we have

By the -Lipschitz of and the definition of , we have

Thus we have

which implies that

When , so we have , that is, , so . By the continuity of , we have .

Theorem 3.2. Let C be a nonempty bounded and closed convex subset of a real Hilbert space H. Let be a uniformly -Lipschitzian and total asymptotically pseudo-contraction. Suppose there exists , such that , then is a closed convex subset of C.

Proof. Since is uniformly -Lipschitzian continuous, is closed. We need to show that is convex. We let , and for . We take and let , . Then for any , we have

This implies that

Now we take , multiplying and on both sides of above inequality, respectively, and adding up, and we can get

By , we get . Since is continuous, we have as , so that .

Theorem 3.3. Let be a bounded and closed convex subset of a real Hilbert space . Let be a uniformly -Lipschitzian and total asymptotically pseudo-contraction. Suppose there exists , such that , , is a sequence in , where . Let be a sequence generated by

where , then the iterative sequence converges strongly to in .

Proof. We divide the proof into seven steps.

(I) is well defined for every .

By Theorem 3.2, we know is closed and convex subset of . Moreover, by our assumption that is nonempty, therefore, is well defined for every .

(II) and are closed and convex for all .

From the definitions of and , it is obvious that and are closed and convex for each . We omit the details.

(III) We prove for each .

We first show . Let , by (3.10), and the uniform -Lipschitz continuity of and the total asymptotical pseudo-contractiveness of , we have

This implies that

This shows that for all . So for all . Next we prove for all . By induction, for , we have . Assume that . Since is the projection of onto , by Lemma 2.1, we have

for any , by the definition of , this shows that . So , for all .

(IV) We prove that exists.

From (3.10) and Lemma 2.1, we have , this with show , for all . As , we also have , for all . Consequently, exists and is bounded.

(V) We prove that as .

By Lemma 2.2, we have

as .

(VI) Now we prove as .

It follows from as , , is bounded, is bounded, and that

So as . Additional

So as .

(VII) Finally, we prove as .

Let be a subsequence of such that , then by Theorem 3.1, we have . We let . For any , and , so we get .

On the other hand, from the weak lower semicontinuity of the norm, we have

From the definition of , we obtain and hence . So we have . Thus we obtain that converges strongly to . Since is an arbitrary weakly convergent sequence of , we can conclude that converges strongly to . This completes the proof of Theorem 3.3.

Remark 3.4. Theorem 3.3 extends the main results of Zhou [3] and improves the main results of Qin et al. [5] and of many other authors.

Acknowledgments

The authors are grateful to the referees for their helpful and useful comments. This research is partially supported by the Fundamental Research Funds for the Central Universities (JBK120117), by the National Research Foundation of Yibin University (No. 2011B07) and by Scientific Research Fund Project of SiChuan Provincial Education Department (No. 12ZB345).

References

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