Journal of Applied Mathematics

Volume 2012 (2012), Article ID 384108, 18 pages

doi:10.1155/2012/384108

Research Article

Weak Convergence Theorems for Strictly Pseudocontractive Mappings and Generalized Mixed Equilibrium Problems

Jong Soo Jung

Department of Mathematics, Dong-A University, Busan 604-714, Republic of Korea

Received 15 February 2012; Accepted 6 April 2012

Academic Editor: Yonghong Yao

Copyright © 2012 Jong Soo Jung. 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 iterative method for finding a common element of the set of fixed points of a strictly pseudocontractive mapping, the set of solutions of a generalized mixed equilibrium problem, and the set of solutions of a variational inequality problem for an inverse-strongly-monotone mapping in Hilbert spaces and then show that the sequence generated by the proposed iterative scheme converges weakly to a common element of the above three sets under suitable control conditions. The results in this paper substantially improve, develop, and complement the previous well-known results in this area.

1. Introduction

Let be a real Hilbert space with inner product and induced norm . Let be a nonempty closed convex subset of . Let be a nonlinear mapping and be a function, and be a bifunction of into , where is the set of real numbers.

Then, we consider the following generalized mixed equilibrium problem of finding such that

which was introduced by Peng and Yao [1] recently. The set of solutions of the problem (1.1) is denoted by . Here some special cases of the problem (1.1) are stated as follows.

If , then the problem (1.1) reduced the following generalized equilibrium problem of finding such that

which was studied by S. Takahashi and M. Takahashi [2]. The set of solutions of the problem (1.2) is denoted by .

If , then the problem (1.1) reduces the following mixed equilibrium problem of finding such that

which was studied by Ceng and Yao [3] (see also [4]). The set of solutions of the problem (1.3) is denoted by .

If and , then the problem (1.1) reduces the following equilibrium problem of finding such that

The set of solutions of the problem (1.4) is denoted by .

If and for all , the problem (1.1) reduces the following variational inequality problem of finding such that

The set of solutions of the problem (1.5) is denoted by .

The problem (1.1) is very general in the sense that it includes, as special cases, fixed point problems, optimization problems, variational inequality problems, minimax problems, Nash equilibrium problems in noncooperative games, and others; see, for example, [3, 5–7].

The class of pseudocontractive mappings is one of the most important classes of mappings among nonlinear mappings. We recall that a mapping is said to be -strictly pseudocontractive if there exists a constant such that

Note that the class of -strictly pseudocontractive mappings includes the class of nonexpansive mappings as a subclass. That is, is nonexpansive (i.e., ) if and only if is 0-strictly pseudocontractive. The mapping is also said to be pseudocontractive if and is said to be strongly pseudocontractive if there exists a constant such that is pseudocontractive. Clearly, the class of -strictly pseudocontractive mappings falls into the one between classes of nonexpansive mappings and pseudocontractive mappings. Also we remark that the class of strongly pseudocontractive mappings is independent of the class of -strictly pseudocontractive mappings (see [8, 9]). Recently, many authors have been devoting the studies on the problems of finding fixed points to the class of pseudocontractive mappings; see, for example, [10–15] and the references therein.

Recently, in order to study the problems (1.1)–(1.5) coupled with the fixed point problem, many authors have introduced some iterative schemes for finding a common element of the set of the solutions of the problem (1.1)–(1.5) and the set of fixed points of a countable family of nonexpansive mappings and have studied strong convergence of the sequences generated by the proposed schemes; see [1–4, 16–18] and the references therein. Also we refer to [19–21] for the problems (1.1), (1.3), and (1.5) combined to the fixed point problem for nonexpansive semigroups and strictly pseudocontractrive mappings.

In this paper, inspired and motivated by [18, 22–27], we introduce a new iterative method for finding a common element of the set of fixed points of a -strictly pseudocontractive mapping, the set of solutions of a generalized mixed equilibrium problem (1.1), and the set of solutions of the variational inequality problem (1.5) for an inverse-strongly monotone mapping in a Hilbert space. We show that, under suitable conditions, the sequence generated by the proposed iterative scheme converges weakly to a common element of the above three sets. The results in this paper can be viewed as an improvement and complement of the recent results in this direction.

2. Preliminaries and Lemmas

Let be a real Hilbert space and let be a nonempty closed convex subset of . In the following, we write to indicate that the sequence converges weakly to . implies that converges strongly to . We denote by the set of fixed points of the mapping .

In a real Hilbert space , we have

for all and . For every point , there exists a unique nearest point in , denoted by , such that

for all . is called the metric projection of onto . It is well known that is nonexpansive and satisfies

for every . Moreover, is characterized by the properties:

In the context of the variational inequality problem for a nonlinear mapping , this implies that

It is also well known that satisfies the Opial condition, that is, for any sequence with , the inequality

holds for every with .

A mapping of into is called -inverse-strongly monotone if there exists a constant such that

We know that if , where is a nonexpansive mapping of into itself and is the identity mapping of , then is -inverse-strongly monotone and . A mapping of into is called strongly monotone if there exists a positive real number such that

In such a case, we say is -strongly monotone. If is -strongly monotone and -Lipschitzian, continuous, that is, for all , then is -inverse-strongly monotone. If is an -inverse-strongly monotone mapping of into , then it is obvious that is -Lipschitzian. We also have that for all and ,

So, if , then is a nonexpansive mapping of into . The following result for the existence of solutions of the variational inequality problem for inverse-strongly monotone mappings was given in Takahashi and Toyoda [27].

Proposition 2.1. Let be a bounded closed convex subset of a real Hilbert space and let be an -inverse-strongly monotone mapping of into . Then, is nonempty.

A set-valued mapping is called monotone if for all , and imply . A monotone mapping is maximal if the graph of is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping is maximal if and only if for , for every implies . Let be an inverse-strongly monotone mapping of into and let be the normal cone to at , that is, , and define

Then is maximal monotone and if and only if ; see [28, 29].

For solving the equilibrium problem for a bifunction , let us assume that and satisfy the following conditions:

(A1) for all ,

(A2) is monotone, that is, for all ,

(A3) for each ,

(A4) for each is convex and lower semicontinuous,

(A5) for each , is weakly upper semicontiunuous,

(B1) for each and , there exist a bounded subset and such that for any ,

(B2) is a bounded set.

The following lemmas were given in [1, 5].

Lemma 2.2 (see [5]). Let be a nonempty closed convex subset of and a bifunction of into satisfying (A1)–(A4). Let and . Then, there exists such that

Lemma 2.3 (see [1]). Let be a nonempty closed convex subset of . Let be a bifunction form to which satisfies (A1)–(A5) and a proper lower semicontinuous and convex function. For and , define a mapping as follows:

for all . Assume that either (B1) or (B2) holds. Then, the following hold:

(1) for each ,

(2) is single-valued,

(3) is firmly nonexpansive, that is, for any ,

(4),

(5) is closed and convex.

We also need the following lemmas for the proof of our main results.

Lemma 2.4 (see [30]). Let be a real Hilbert space, let be a sequence of real numbers such that for all and let and be sequences in such that, for some

Then .

Lemma 2.5 (see [27]). Let be a nonempty closed convex subset of a real Hilbert spaces and let be a sequence in . If

then converges strongly to some , where stands for the metric projection of onto .

Lemma 2.6 (see [31]). Let be a Hilbert space, a closed convex subset of . If is a -strictly pseudocontractive mapping on , then the fixed point set is closed convex, so that the projection is well defined.

Lemma 2.7 (see [31]). Let be a Hilbert space, a closed convex subset of , and a -strictly pseudocontractive mapping. Define a mapping by for all . Then, as , is a nonexpansive mapping such that .

3. Main Results

In this section, we introduce a new iterative scheme for finding a common point of the set of fixed points of a -strictly pseudocontractive mapping, the set of solutions of the problem (1.1), and the set of solutions of the problem (1.5) for an inverse-strongly monotone mapping.

Theorem 3.1. Let be a nonempty closed convex subset of a real Hilbert space . Let be a bifunction from to satisfying (A1)–(A5) and a lower semicontinuous and convex function. Let be two , -inverse-strongly monotone mappings of into , respectively. Let be a -strictly pseudocontractive mapping of into itself for some such that . Assume that either (B1) or (B2) holds. Let and be sequences generated by and

where is a mapping defined by for and . Assume that for some and . Then and converge weakly to , where .

Proof. From now, we put .

We divide the proof into several steps.

Step 1. We show that is bounded. To this end, let and be a sequence of mappings defined as in Lemma 2.3. Then, since by Lemma 2.7, . Also, from (4) in Lemma 2.3 and (2.5), it follows that and . From and the fact that and are nonexpansive, it follows that

Also, by and the -inverse-strongly monotonicity of , we have with ,

that is, , and so

So, by using the convexity of , (3.2) and (3.3), we have

So, there exists such that

Therefore, is bounded, and so are and by (3.2) and (3.4). Moreover, from (3.5), it follows that

which implies that

Step 2. We show that . To this end, let . Since is firmly nonexpansive and , we have

and hence

On the other hand, by using the convexity of , (3.2) and (3.10), we obtain

and hence

where . Since and in (3.8), we obtain

and so is the limit of since is Lipschitz.

Step 3. We show that . Indeed, let and set . Being nonexpansive and , from (3.4) we can write

and hence . By (3.4), we also have

By Lemma 2.4, we obtain .

Step 4. We show that . Using , , we compute

So, we get

From conditions and , it follows that

By , we obtain

On the other hand, using and (2.3), we observe that

that is,

Thus, by (3.21), we have

which implies that

where . From in (3.19) and , we conclude that .

Step 5. We show that . Indeed, since

by Step 2, Step 3, and Step 4, we have

Step 6. We show that any of its weak cluster point of belongs in . In this case, there exists a subsequence which converges weakly to . By Step 2 and Step 4, without loss of generality, we may assume that converges weakly to . Since

from Step 2, Step 3, and Step 4, it follows that and .

We will show that . First we show that . Assume that . Since and , by the Opial condition, we obtain

which is a contradiction. Thus we have .

Next we prove that . Let

where is normal cone to at . We have already known that in this case the mapping is maximal monotone, and if and only if . Let . Since and , we have

On the other hand, from , we have

that is,

Thus, we obtain

Since in Step 4 and is -inverse-strongly monotone, it follows from (3.32) that

Since is maximal monotone, we have and hence .

Finally, we show that . By , we know that

It follows from (A2) that

Hence

For with and , let . Since and , we have and hence . So, from (3.36), we have

Since by Step 2, we have and , that is, . Also by in Step 4, we have . Moreover, from the inverse-strongly monotonicity of , we have . So, from (A4) and the weak lower semicontinuity of , if follows that

By (A1), (A4), and (3.38), we also obtain

and hence

Letting in (3.40), we have for each

This implies that . Therefore, we have .

Let be another subsequence of such that . Then, we have . If , from the Opial condition, we have

This is a contradiction. So, we have . This implies that

Also from Step 2, it follows that .

Let . Since , we have

Since for , by Lemma 2.5, we have that converges strongly to some . Since converges weakly to , we have

Therefore, we obtain

This completes the proof.

As direct consequences of Theorem 3.1, we also obtain the following new weak convergence theorems for the problems (1.2) and (1.3) and fixed point problem of a strict pseudocontractive mapping.

Corollary 3.2. Let , and be as in Theorem 3.1. Let be a -strictly pseudocontractive mapping of into itself for some such that . Assume that either (B1) or (B2) holds. Let and be sequences generated by and

where is a mapping defined by for and . Assume that for some and . Then and converge weakly to , where .

Proof. Putting in Theorem 3.1, we obtain the desired result.

Corollary 3.3. Let , and be as in Corollary 3.2. Let be a -strictly pseudocontractive mapping of into itself for some such that . Assume that either (B1) or (B2) holds. Let and be sequences generated by and

where is a mapping defined by for and . Assume that for some and . Then and converge weakly to , where .

Proof. Putting in Corollary 3.2, we obtain the desired result.

Corollary 3.4. Let , and be as in Theorem 3.1. Let be a -strictly pseudocontractive mapping of into itself for some such that . Assume that either (B1) or (B2) holds. Let and be sequences generated by and

where is a mapping defined by for and . Assume that for some and . Then and converge weakly to , where .

Proof. Putting in Theorem 3.1, we obtain the desired result.

Corollary 3.5. Let and be as in Theorem 3.1. Let be a -strictly pseudocontractive mapping of into itself for some such that . Assume that either (B1) or (B2) holds. Let and be sequences generated by and

where is a mapping defined by for and . Assume that for some and . Then and converge weakly to , where .

Proof. Putting in Corollary 3.4, we obtain the desired result.

Remark 3.6.

(1)As a new result for a new iterative scheme, Theorem 3.1 develops and complements the corresponding results, which were obtained recently by many authors in references and others; for example, see [22–24, 26]. In particular, even though in Theorem 3.1, Theorem 3.1 develops and complements Theorem 3.1 of Ceng et al. [22] in the following aspects:

(a)the iterative scheme (3.1) in Theorem 3.1 is a new one different from those in Theorem 3.1 of [22].(b)the equilibrium problem in Theorem 3.1 of [22] is extended to the case of generalized mixed equilibrium problem.

(2)We point out that our iterative schemes in Corollaries 3.2, 3.3, 3.4 and 3.5 are new ones different from those in the literature (see [22–24, 26] and others in references).

Acknowledgments

The author thanks the referees for their valuable comments and suggests, which improved the presentation of this paper, and for providing some recent related papers. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011-0003901).

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