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Journal of Applied Mathematics
Volume 2013, Article ID 973408, 11 pages
http://dx.doi.org/10.1155/2013/973408
Research Article

Existence and Iterative Approximation Methods for Generalized Mixed Vector Equilibrium Problems with Relaxed Monotone Mappings

1Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand
2Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

Received 22 March 2013; Revised 3 June 2013; Accepted 21 June 2013

Academic Editor: Filomena Cianciaruso

Copyright © 2013 Rabian Wangkeeree and Panu Yimmuang. 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 first consider an auxiliary problem for the generalized mixed vector equilibrium problem with a relaxed monotone mapping and prove the existence and uniqueness of the solution for the auxiliary problem. We then introduce a new iterative scheme for approximating a common element of the set of solutions of a generalized mixed vector equilibrium problem with a relaxed monotone mapping and the set of common fixed points of a countable family of nonexpansive mappings. The results presented in this paper can be considered as a generalization of some known results due to Wang et al. (2010).

1. Introduction

Let be a real Hilbert space with inner productand norm, respectively. Letbe a nonempty closed convex subset ofLetbe a bifunction. The equilibrium problem is to findsuch that As pointed out by Blum and Oettli [1], EPprovides a unified model of several problems, such as the optimization problem, fixed point problem, variational inequality, and complementarity problem.

A mappingis called nonexpansive, if We denote the set of all fixed points of by, that is,It is well known that if is bounded, closed, convex and is a nonexpansive mapping of onto itself, then is nonempty (see [2]). A mapping is said to be relaxed - monotone if there exist a mapping and a function positively homogeneous of degree , that is, for allandsuch that whereis a constant; see [3]. In the case offor all,   is said to be relaxed-monotone. In the case offor alland, whereand,   is said to be-monotone; see [46]. In fact, in this case, if , then is a -strongly monotone mapping. Moreover, every monotone mapping is relaxedmonotone withfor alland.

In 2000, Moudafi [7] introduced an iterative scheme of finding the solution of nonexpansive mappings and proved a strong convergence theorem. Recently, Huang et al. [8] introduced the approximate method for solving the equilibrium problem and proved the strong convergence theorem.

Letbe a bifunction andnonlinear mappings. In 2010, Wang et al. [9] introduced the following generalized mixed equilibrium problem with a relaxed monotone mapping. Problem (4) is very general setting, and it includes special cases of Nash equilibrium problems, complementarity problems, fixed point problems, optimization problems, and variational inequalities (see, e.g., [8, 1013] and the references therein). Moreover, Wang et al. [9] studied the existence of solutions for the proposed problem and introduced a new iterative scheme for finding a common element of the set of solutions of a generalized equilibrium problem with a relaxed monotone mapping and the set of common fixed points of a countable family of nonexpansive mappings in a Hilbert space.

It is well known that the vector equilibrium problem provides a unified model of several problems, for example, vector optimization, vector variational inequality, vector complementarity problem, and vector saddle point problem [1416]. In recent years, the vector equilibrium problem has been intensively studied by many authors (see, e.g., [12, 1419] and the references therein).

Recently, Li and Wang [18] first studied the viscosity approximation methods for strong vector equilibrium problems and fixed point problems. Very recently, Shan and Huang [20] studied the problem of finding a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of the generalized mixed vector equilibrium problem, and the solution set of a variational inequality problem with a monotone Lipschitz continuous mapping in Hilbert spaces. They first introduced an auxiliary problem for the generalized mixed vector equilibrium problem and proved the existence and uniqueness of the solution for the auxiliary problem. Furthermore, they introduced an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of the generalized mixed vector equilibrium problem, and the solution set of a variational inequality problem with a monotone Lipschitz continuous mapping.

Letbe a Hausdorff topological vector space, and, let be a closed, convex and pointed cone ofwith intLetbe a vector-valued bifunction. The strong vector equilibrium problem (for short, SVEP) is to findsuch that and the weak vector equilibrium problem (for short, WVEP()) is to findsuch that

In this paper, inspired and motivated by the works mentioned previously, we consider the following generalized mixed vector equilibrium problem with a relaxed monotone mapping (for short, GVEPR()): findsuch that whereint,, andare the mappings. The set of all solutions of the generalized mixed vector equilibrium problem with a relaxed monotone mapping is denoted by SGVEPR(), that is, If, we denote the set ASGVEPR() by Some special cases of the problem (7) are as follows.(1)If,  , and, then GVEPR() (7) reduces to the generalized mixed equilibrium problem with a relaxed monotone mapping (4). (2)Ifand, then GVEPR() (7) reduces to the classic vector equilibrium problem (5).

We consider the auxiliary problem of GVEPR() and prove the existence and uniqueness of the solutions of auxiliary problem of GVEPR() under some proper conditions. By using the result for the auxiliary problem, we introduce a new iterative scheme for finding a common element of the set of solutions of a generalized mixed vector equilibrium problem with a relaxed monotone mapping and the set of common fixed points of a countable family of nonexpansive mappings and then obtain a strong convergence theorem. The results presented in this paper improve and generalize some known results of Wang et al. [9].

2. Preliminaries

Letbe a-inverse-strongly monotone mapping ofFor alland, one has [21] Hence, if , then is a nonexpansive mapping of into.

For each point , there exists a unique nearest point of , denoted by , such that for allSuch ais called the metric projection fromontoThe well-known Browder’s characterization ofensures thatis a firmly nonexpansive mapping fromonto, that is, Further, we know that for anyand,  if and only if Letbe a nonexpansive mapping ofinto itself such thatThen we have which is obtained directly from the following: This inequality is a very useful characterization ofObserve what is more that it immediately yields that Fix() is a convex closed set.

Definition 1 (see [6, 22]). Let and be two Hausdorff topological vector spaces, a nonempty, convex, subset of and a closed, convex and pointed cone of with Let be the zero point of ,  the neighborhood set of ,   be the neighborhood set of , and a mapping. (1)If for any in, and there existssuch that then is called upper -continuous on . If is upper -continuous for all ,then is called upper -continuous on (2)If for anyin , and there exists such that thenis called lower -continuous onIfis lower-continuous for all, then is called lower-continuous on(3)is called-continuous ifis upper-continuous and lower-continuous. (4)If for anyand, and the mappingsatisfies thenis called proper -quasiconvex. (5)If for anyand, and the mappingsatisfies thenis called-convex.

Lemma 2 (see [19]). Letandbe two real Hausdorff topological vector spaces,is a nonempty, compact, convex subset of, andis a closed, convex, and pointed cone ofAssume thatandare two vector valued mappings. Suppose thatandsatisfy the following:(i), for all(ii)is upper-continuous on(iii)is lower-continuous for all(iv)is proper-quasiconvex for all
Then there exists a pointsatisfying where

Definition 3 (see [3]). Letbe a Banach space with the dual spaceand letbe a nonempty subset ofLet, andbe two mappings. The mappingis said to be-hemicontinuous, if for any fixed, the functiondefined by is continuous at

3. The Existence of Solutions for the Generalized Mixed Vector Equilibrium Problem with a Relaxed Monotone Mapping

For solving the generalized mixed vector equilibrium problem with a relaxed monotone mapping, we give the following assumptions. Letbe a real Hilbert space with inner and norm , respectively. Assume thatis nonempty, compact, convex subset, is real Hausdorff topological vector space, and is a closed, convex, and pointed cone. Let ,   be two mappings. For any , define a mapping as follows: where is a positive number in andLet,  , and satisfy the following conditions: for all,is monotone, that is,for all is -continuous for all is -convex, that is, () for all , is continuous, and for any ()  is proper -quasiconvex for all and

Remark 4. Let ,  , and For any , if is upper semicontinuous and is continuous, then is lower -continuous. In fact, since is upper semicontinuous and is continuous, for any , there exists a such that, for all , we have whereis a point inThis means that is lower-continuous.

Remark 5. Let,  andAssume thatis a convex mapping for all. Then for anyand, we have which implies thatis proper-quasiconvex.

Now we are in the position to state and prove the existence of solutions for the generalized mixed vector equilibrium problem with a relaxed monotone mapping.

Theorem 6. Let be a nonempty, compact, convex subset of a real Hilbert space . Let be a closed, convex, and pointed cone of a Hausdorff topological vector space . Let be an -hemicontinuous and relaxed --monotone mapping. Let be a vector-valued bifunction. Suppose that all the conditions ()–() are satisfied. Let and define a mapping as follows: for allAssume that(i), for all(ii)for any,  
Then, the following holds.(1)for all.(2)is single-value.(3)is a firmly nonexpansive mapping, that is, for all(4),(5)ASGVEPR() is closed and convex.

Proof. (1) In Lemma 2, let, and, let for allandThen it is easy to check thatandsatisfy all the conditions of Lemma 2. Thus, there exists a pointsuch that which gives that, for any , Therefor we conclude that for all .
(2) For and , let . Then Letting in (32) and in (33), adding (32) and (33), we have By the monotonicity of , we have Thus Since is relaxed --monotone andand the property of, one has In (36) exchanging the position ofand, we get that is, Now, adding the inequalities (37) and (39), By using (iv), we have If, then This implies that From (41) and (43), we have which is a contradiction. Thus so Hence Therefore is single value.
(3) For any , let and . Then Lettingin (46) andin (47), adding (46) and (47), we have Sinceis monotone andis closed convex cone, we get that is, In (50) exchanging the position of and , we get Adding the inequalities (50) and (51), we have It follows from (iv) that This implies that This shows thatis firmly nonexpansive.
(4) We claim thatIndeed, we have the following:
(5) Since every firmly nonexpansive mapping is nonexpansive, we see thatis nonexpansive. Since the set of fixed point of every nonexpansive mapping is closed and convex, we have that ASGVEPRis closed and convex. This completes the proof.

4. Convergence Analysis

In this section, we prove a strong convergence theorem which is one of our main results.

Theorem 7. Letbe a nonempty, compact, convex subset of a real Hilbert spaceLetbe a closed, convex cone of a real Hausdorff topological vector space and. Letsatisfy ()–(). Letbe an -hemicontinuous and relaxed --monotone mapping. Let be a-inverse-strongly monotone mapping, and letbe a countable family of nonexpansive mappings from onto itself such that Assume that the conditions (i)-(ii) of Theorem 6 are satisfied. Putand assume thatis a strictly decreasing sequence. Assume thatwith someandwith someThen, for any, the sequence , generated by converges strongly toIn particular, ifcontains the origin 0 and taking, then the sequencegenerated by (57) converges strongly to the minimum norm element in, that is.

Proof. We divide the proof into several steps.
Step 1. We will show that  is closed and convex, the sequence generated by (57) is well defined, and , for all.
First, we prove that is closed and convex. It suffices to prove that SGVEPR is closed and convex. Indeed, it is easy to prove the conclusion by the following fact: This implies that Sinceis a nonexpansive mapping forand the set of fixed points of a nonexpansive mapping is closed and convex, we have that SGVEPRis closed and convex.
Next, we prove that the sequencegenerated by (57) is well defined andfor allBy Definition of, for all, the inequality is equivalent to It is easy to see that is closed and convex for all. Hence is closed and convex for all . For any , and since and is nonexpansive, we have This implies thatfor all. HenceThat is Sinceis nonempty closed convex, we get that the sequenceis well defined. This completes the proof of Step 1.
Step 2. We shall show that asand there issuch that .
It easy to see that for all from the construction ofHence Since, we have for all. This implies that is increasing. Note that is bounded, we get that is bounded. This shows that exists.
Since and for all , we have It follows from (66) that By takingin (67), we have Since the limits ofexist, we get that This implies that Moreover, from (67), we also have This shows that the sequenceis a Cauchy sequence. Hence there issuch that
Step 3. We shall show that as.
Sinceand, we have and hence Note that can be rewritten as for all . We take thus we have . Since  is -inverse-strongly monotone, and , we know that, for all Using (57) and (75), we have and hence Note thatandare bounded,, andconverges to, we get that Using Theorem 6, we have This implies that From (80), we have and hence From (78) and , we have
Step 4. We show that, for all .
It follows from definition of scheme (57) that that is, Hence, for any, one has Since eachis nonexpansive and by (36) we get that Hence, combining this inequality with (86), we have that is, Since and , we have
Step 5. We show that.
First, we show thatSince we have Hence . Next, we show that SGVEPR. Noting that , one obtains which implies that Put, for allandThen, we haveSo, from (95), we have Since and the properties of , we have From the monotonicity of , we have Thus So, from (96)–(99) and-hemicontinuity of , we have Since is -convex, we have Since for any , and the mapping is convex, we have This implies that From (101) and (103), we get that which implies that From (100) and (105), we have This implies that It follows that As , we obtain that for each HenceSGVEPRFinally, we prove thatFromand, we have Note that ; we take the limit in (110), and then we have We see that by (33). This completes the proof.

Remark 8. If ,  , and , then Theorem 7 extends and improves Theorem 3.1 of Wang et al. [9].

Acknowledgments

The authors would like to thank the Thailand Research Fund and “Centre of Excellence in Mathematics” under the Commission on Higher Education, Ministry of Education, Thailandm for financial support.

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