Abstract and Applied Analysis

Volume 2012 (2012), Article ID 543040, 14 pages

doi:10.1155/2012/543040

Research Article

Some Results on Strictly Pseudocontractive Nonself-Mappings and Equilibrium Problems in Hilbert Spaces

Yan Hao¹ and Sun Young Cho²

¹School of Mathematics Physics and Information Science, Zhejiang Ocean University, Zhoushan 316004, China

²Department of Mathematics, Gyeongsang National University, Jinju 660-701, Republic of Korea

Received 12 September 2012; Accepted 6 October 2012

Academic Editor: Yonghong Yao

Copyright © 2012 Yan Hao and Sun Young Cho. 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.

An equilibrium problem and a strictly pseudocontractive nonself-mapping are investigated. Strong convergence theorems of common elements are established based on hybrid projection algorithms in the framework of real Hilbert spaces.

1. Introduction

Bifunction equilibrium problems which were considered by Blum and Oettli [1] have intensively been studied. It has been shown that the bifunction equilibrium problem covers fixed point problems, variational inequalities, inclusion problems, saddle problems, complementarity problem, minimization problem, and the Nash equilibrium problem; see [1–4] and the references therein. Iterative methods have emerged as an effective and powerful tool for studying a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, network, elasticity, and optimization; see [5–16] and the references therein. In this paper, we investigate an equilibrium problem and a strictly pseudocontractive nonself-mapping based on hybrid projection algorithms. Strong convergence theorems of common elements lie in the solution set of the equilibrium problem and the fixed point set of the strictly pseudocontractive nonself-mapping.

Throughout this paper, we always assume that is a real Hilbert space with the inner product and the norm . Let be a nonempty closed convex subset of and a bifunction of into , where denotes the set of real numbers. In this paper, we consider the following equilibrium problem:

The set of such an is denoted by , that is,

Given a mapping , let for all . Then if and only if for all , that is, is a solution of the variational inequality.

To study the equilibrium problem (1.1), we may assume that satisfies the following conditions:

(A1) for all ; (A2) is monotone, that is, for all ; (A3) for each ,

(A4) for each ,  is convex and lower semicontinuous.

If is an Euclidean space, then we see that is a simple example satisfying the above assumptions. See [2] for more details. Let , where and denote the domain and the range of the mapping . If , then the mapping is said to be a self-mapping. If , then the mapping is said to be a nonself-mapping. Let be a nonself-mapping. In this paper, we use to denote the fixed point set of . Recall the following definitions. is said to be nonexpansive if

is said to be strictly pseudocontractive if there exists a constant such that

For such a case, is also said to be -strict pseudocontraction. It is clear that (1.5) is equivalent to

is said to be pseudocontractive if

It is clear that (1.7) is equivalent to

The class of -strict pseudocontractions which was introduced by Browder and Petryshyn [17] in 1967 has been considered by many authors. It is easy to see that the class of strict pseudocontractions falls into the one between the class of nonexpansive mappings and the class of pseudocontractions. For studying the class of strict pseudocontractions, Zhou [18] proposed the following convex combination method: define a mapping by

He showed that is nonexpansive if ; see [18] for more details.

Recently, many authors considered the problem of finding a common element in the fixed point set of a nonexpansive mapping and in the solution set of the equilibrium problem (1.1) based on iterative methods; see, for instance, [19–27].

In 2007, Tada and Takahashi [23] considered an iterative method for the equilibrium problem (1.1) and a nonexpansive nonself-mapping. To be more precise, they obtained the following results.

Theorem TT. Let be a closed convex subset of a real Hilbert space , let be a bifunction satisfying (A1)–(A4), and let be a nonexpansive mapping of into such that . Let and be sequences generated by , and let

for every , where , for some and satisfies . Then the sequence converges strongly to .

We remark that the iterative process (1.10) is called the hybrid projection iterative process. Recently, the hybrid projection iterative process which was first considered by Haugazeau [28] in 1968 has been studied for fixed point problem of nonlinear mappings and equilibrium problems by many authors. Since the sequence generated in the hybrid projection iterative process depends on the sets and , the hybrid projection iterative process is also known as “CQ” iterative process; see [29] and the reference therein.

Recently, Takahashi et al. [30] considered the shrinking projection process for the fixed point problem of nonexpansive self-mapping. More precisely, they obtain the iterative sequence monontonely without the help of the set ; see [30] for more details.

In this paper, we reconsider the same shrinking projection process for the equilibrium problem (1.1) and a strictly pseudocontractive nonself-mapping. We show that the sequence generated in the proposed iterative process converges strongly to some common element in the fixed point set of a strictly pseudocontractive nonself-mapping and in the solution set of the equilibrium problem (1.1). The main results presented in this paper mainly improved the corresponding results in Tada and Takahashi [23].

2. Preliminaries

Let be a nonempty closed and convex subset of a real Hilbert space . Let be the metric projection from onto . That is, for , is the only point in such that . We know that the mapping is firmly nonexpansive, that is,

The following lemma can be found in [1, 2].

Lemma 2.1. Let be a nonempty closed convex subset of , and let be a bifunction satisfying (A1)–(A4). Then, for any and , there exists such that

Further, define

for all and . Then, the following hold:

(a) is single valued; (b) is firmly nonexpansive, that is,

(c); (d) is closed and convex.

Lemma 2.2 (see [18]). Let be a real Hilbert space, a nonempty closed convex subset of , and a strict pseudocontraction. Then the mapping is demiclosed at zero, that is, if is a sequence in such that and , then .

Lemma 2.3 (see [18]). Let be a nonempty closed convex subset of a real Hilbert space and a -strict pseudocontraction. Define a mapping for all . If , then the mapping is a nonexpansive mapping such that .

3. Main Results

Theorem 3.1. Let be a nonempty closed convex subset of a real Hilbert space . Let and be two bifunctions from to which satisfy (A1)–(A4), respectively. Let be a -strict pseudocontraction. Assume that . Let , , and be sequences in . Let be a sequence generated in the following manner:

where is chosen such that

and is chosen such that

Assume that the control sequences , , , , and satisfy the following restrictions:

(a); (b)

for some . Then the sequence generated in the converges strongly to some point , where .

Proof. First, we show that is closed and convex for each . It is easy to see that is closed for each . We only show that is convex for each . Note that is convex. Suppose that is convex for some positive integer . Next, we show that is convex for the same . Note that is equivalent to

Take and in , and put . It follows that ,,

Combining (3.4), we can obtain that , that is, . In view of the convexity of , we see that . This shows that . This concludes that is closed and convex for each .

Define a mapping by for all . It follows from Lemma 2.3 that is nonexpansive and for all . Next, we show that for all . It is easy to see that . Suppose that for some integer . We intend to claim that for the same . For any , we have

This shows that . This proves that for all .

Since and , we have that

It follows that

On the other hand, for any , we see that . In particular, we have

This shows that the sequence is bounded. In view of (3.7), we see that exists. It follows from (3.6) that

This yields that

Since , we see that

It follows that

From (3.10), we obtain that

On the other hand, we have

From the restriction (a), we obtain from (3.13) that

For any , we have that

This implies that

In a similar way, we get that

It follows from (3.17) and (3.18) that

This implies that

In view of the restriction (a), we obtain from (3.13) that

It also follows from (3.19) that

In view of the restriction (a), we obtain from (3.13) that

Since is bounded, there exists a subsequence of such that . From (3.21) and (3.23), we see that and , respectively. From (3.21) and the restriction (b), we see that

Now, we are in a position to show that . Note that

From (A2), we see that

Replacing by , we arrive at

In view of (3.24) and (A4), we get that

For any with and , let . Since and , we have and hence . It follows that

which yields that

Letting , we obtain from (A3) that

This means that . In the same way, we can obtain that . Next, we show that . Note that

It follows from (3.21) and (3.23) that

On the other hand, we have

It follows from (3.15) and (3.33) that

Note that

which yields that

This implies from the restriction (a) and (3.35) that

It follows from Lemma 2.2 that . This shows that . Since , we obtain that

which yields that

It follows that converges strongly to . Therefore, we can conclude that the sequence converges strongly to . This completes the proof.

From Theorem 3.1, we have the following results.

Corollary 3.2. Let be a nonempty closed convex subset of a real Hilbert space . Let be a bifunction from to which satisfies (A1)–(A4). Let be a -strict pseudocontraction. Assume that . Let and be sequences in . Let be a sequence generated in the following manner:

where is chosen such that

Assume that the control sequences , , and satisfy the following restrictions:

(a); (b)

for some . Then the sequence converges strongly to some point , where .

Proof. Putting and in Theorem 3.1, we see that . From the proof of Theorem 3.1, we can conclude the desired conclusion immediately.

Remark 3.3. Corollary 3.2 improves Theorem TT in the following aspects.

(1)From the viewpoint of mappings, the class of nonexpansive mappings is extended to the class of strict pseudocontractions. (2)From the viewpoint of computation, the set is removed.

Corollary 3.4. Let be a nonempty closed convex subset of a real Hilbert space . Let be a -strict pseudocontraction with fixed points. Let and be sequences in . Let be a sequence generated in the following manner:

Assume that the control sequences and satisfy the restrictions and for some . Then the sequence converges strongly to some point , where .

Proof. Putting and , we can obtain from Theorem 3.1 the desired conclusion easily.

If is nonexpansive and , then Corollary 3.4 is reduced to the following.

Corollary 3.5. Let be a nonempty closed convex subset of a real Hilbert space . Let be a nonexpansive mapping with fixed points. Let be a sequence in . Let be a sequence generated in the following manner:

Assume that the control sequence satisfies the restriction for some . Then the sequence converges strongly to some point , where .

Recently, many authors studied the following convex feasibility problem (CFP):

where is an integer and each is a nonempty closed and convex subset of a real Hilbert space . There is a considerable investigation on CFP in the setting of Hilbert spaces which captures applications in various disciplines such as image restoration, computer tomography, and radiation therapy treatment planning.

Next, we consider the case that each is the solution set of an equilibrium problem.

Theorem 3.6. Let be a nonempty closed convex subset of a real Hilbert space . Let be a bifunction from to which satisfies (A1)–(A4) for each . Assume that . Let , , and be sequences in . Let , and be sequences in . Let be a sequence generated in the following manner:

Assume that the control sequences satisfy the following restrictions:

(a),, ;(b),

where . Then the sequence generated in the above iterative process converges strongly to some point , where .

Proof. Let be the identity mapping and , then we can obtain from Theorem 3.1 the desired conclusion easily.

Acknowledgment

This work was supported by Natural Science Foundation of Zhejiang Province (Y6110270).

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