- Somyot Plubtieng
^{1}Email author and - Kanokwan Sitthithakerngkiet
^{1}

**2011**:967515

https://doi.org/10.1155/2011/967515

© S. Plubtieng and K. Sitthithakerngkiet. 2011

**Received: **30 November 2010

**Accepted: **18 February 2011

**Published: **9 March 2011

## Abstract

This paper deals with the generalized strong vector quasiequilibrium problems without convexity in locally -convex spaces. Using the Kakutani-Fan-Glicksberg fixed point theorem for upper semicontinuous set-valued mapping with nonempty closed acyclic values, the existence theorems for them are established. Moreover, we also discuss the closedness of strong solution set for the generalized strong vector quasiequilibrium problems.

## 1. Introduction

Problem (1.1) was studied by Blum and Oettli [1]. The set of solution of (1.1) is denoted by . The equilibrium problem contains many important problems as special cases, including optimization, Nash equilibrium, complementarity, and fixed point problems (see [1–3] and the references therein). Recently, there has been an increasing interest in the study of vector equilibrium problems. Many results on the existence of solutions for vector variational inequalities and vector equilibrium problems have been established (see, e.g., [4–16]).

where is a multivalued map with nonempty values.

It is called generalized vector equilibrium problem (for short, GVEP), and it has been studied by many authors; see, for example, [20–22] and references therein. For other possible ways to generalize VEP, we refer to [23–25]. If is nonempty and satisfies (1.4), then we call a weak efficient solution for VEP, and if satisfies (1.5), then we call a strong solution for VEP. Moreover, they also proved an existence theorem for a strong vector equilibrium problem (1.5) (see [4]).

Moreover, they gave an existence theorem for a generalized strong vector quasiequilibrium problem without assuming that the dual of the ordering cone has a weak compact base.

Throughout this paper, motivated and inspired by Hou et al. [27], Long et al. [16], and Yuan [28], we will introduce and study the generalized vector quasiequilibrium problem on locally -convex Hausdorff topological vector spaces. Let , , and be real locally -convex Hausdorff topological vector spaces, and nonempty compact subsets, and a nonempty closed convex cone. We also suppose that , and are set-valued mappings.

We denote the set of all solution to the (GSVQEP I) and (GSVQEP II) by and , respectively. The main motivation of this paper is to prove the existence theorems of the generalized strong vector quasiequilibrium problems in locally -convex spaces, by using Kakutani-Fan-Glicksberg fixed point theorem for upper semicontinuous set-valued mapping with nonempty closed acyclic values, and the closedness of and . The results in this paper generalize, extend, and unify some well-known some existence theorems in the literature.

## 2. Preliminaries

Let be the standard -dimensional simplex in with vertices . For any nonempty subset of , we denote by the convex hull of the vertices . The following definition was essentially given by Park and Kim [29].

Definition 2.1.

A generalised convex space, or say, a -convex space consists of a topological space , a nonempty subset of and a function such that

(ii)for each with , there exists a continuous function such that for each , where and denotes the face of corresponding to the subindex of in .

A subset of the -convex space is said to be -convex if for each for all . For the convenience of our discussion, we also denote by or if there is no confusion for , where is the set of all indices for the set ; that is, . A space is said to have a -convex structure if and only if is a -convex space.

In order to cover general economic models without linear convex structures, Park and Kim [29] introduced another abstract convexity notion called a -convex space, which includes many abstract convexity notions such as -convex spaces as special cases. For the details on G-convex spaces, see [30–34], where basic theory was extensively developed.

Definition 2.2.

A -convex is said to be a locally -convex space if is a uniform topological space with uniformity , which has an open base of symmetric entourages such that for each , the set is a -convex set for each .

We recall that a nonempty space is said to be acyclic if all of its reduced Čech homology groups over the rationals vanish.

Definition 2.3 (see [35]).

Let be a topological space. A subset of is called contractible at , if there is a continuous mapping such that for all and for all .

In particular, each contractible space is acyclic and thus any nonempty convex or star-shaped set is acyclic. Moreover, by the definition of contractible set, we see that each convex space is contractible.

Definition 2.4.

Let and be two topological vector spaces and a nonempty subset of , and let be a set-valued mapping.

(i) is called upper -continuous at if, for any neighbourhood of the origin in , there is a neighbourhood of such that, for all ,

(ii) is called lower -continuous at if, for any neighbourhood of the origin in , there is a neighbourhood of such that for all ,

Definition 2.5.

Definition 2.6.

Let and be two topological vector spaces and a set-valued mapping.

(i) is said to be upper semicontinuous at if, for any open set containing , there exists an open set containing such that for all , ; is said to be upper semicontinuous on if it is upper semicontinuous at all .

(ii) is said to be lower semicontinuous at if, for any open set with , there exists an open set containing such that for all , ; is said to be lower semicontinuous on if it is lower semicontinuous at all .

(iii) is said to be continuous on if it is at the same time upper semicontinuous and lower semicontinuous on .

(iv) is said to be closed if the graph, , of , that is, , is a closed set in .

Lemma 2.7 (see [36]).

Let and be two Hausdorff topological vector spaces and a set-valued mapping. Then, the following properties hold:

(i)if is closed and is compact, then is upper semicontinuous, where and denotes the closure of the set ,

(ii)if is upper semicontinuous and for any is closed, then is closed,

(iii) is lower semicontinuous at if and only if for any and any net , there exists a net such that and .

We now have the following fixed point theorem in locally -convex spaces given by Yuan [28] which is a generalization of the Fan-Glickberg-type fixed point theorems for upper semicontinuous set-valued mapping with nonempty closed acyclic values given in several places (e.g., see Kirk and Shin [37], Park and Kim [29], and others in locally convex spaces).

Lemma 2.8 (see [28]).

Let be a compact locally -convex space and an upper semicontinuous set-valued mappings with nonempty closed acyclic values. Then, has a fixed point; that is, there exists an such that .

## 3. Main Results

In this section, we apply the Kakutani-Fan-Glicksberg fixed point theorem for upper semicontinuous set-valued mapping with nonempty closed acyclic values to establish two existence theorems of strong solutions and obtain the closedness of the strong solutions set for generalized strong vector quasiequilibrium problem.

Theorem 3.1.

Let , , and be real locally -convex topological vector spaces, and nonempty compact subsets, and a nonempty closed convex cone. Let be a continuous set-valued mapping such that for any , the set is a nonempty closed contractible subset of . Let be an upper semicontinuous set-valued mapping with nonempty closed acyclic values and a set-valued mapping satisfy the following conditions:

(ii)for all are properly -quasiconvex,

(iv)for all are lower -continuous.

Then, the solutions set is nonempty and closed subset of .

Proof.

Since for any is nonempty. So, by assumption (i), we have that is nonempty. Next, we divide the proof into five steps.

Step 1 (to show that is acyclic).

which contradicts . Therefore, , and hence is contractible.

Step 2 (to show that is a closed subset of ).

We claim that . Assume that there exists and . Then, we note that , and the set is closed. Thus, is open, and . Since is a locally -convex space, there exists a neighbourhood of the origin such that and . Thus, we note that , and hence , which contradicts to (3.7). Hence, , and therefore, . Then, is a closed subset of .

Step 3 (to show that is upper semicontinuous).

it is a contradiction to . Hence, , and therefore, is a closed mapping. Since is a compact set and is a closed subset of , is compact. This implies that is compact. Then, by Lemma 2.7(i), we have is upper semicontinuous.

Step 4 (to show that the solutions set is nonempty).

Step 5 (to show that the solutions set is closed).

This means that belongs to . Therefore, the solutions set is closed. This completes the proof.

Theorem 3.1 extends Theorem 3.1 of Long et al. [16] to locally -convex which includes locally convex Hausdorff topological vector spaces.

Corollary 3.2.

Let , and be real locally convex Hausdorff topological vector spaces, and two nonempty compact convex subsets, and a nonempty closed convex cone. Let be a continuous set-valued mapping such that for any , is a nonempty closed convex subset of . Let be an upper semicontinuous set-valued mapping such that for any , is a nonempty closed convex subset of . Let be a set-valued mapping satisfying the following conditions:

(ii)for all are properly -quasiconvex,

(iv)for all are lower -continuous.

Then, the solutions set is nonempty and closed subset of .

Theorem 3.3.

Let , and be real locally -convex topological vector spaces, and nonempty compact subsets, and a nonempty closed convex cone. Let be a continuous set-valued mapping such that for any , the set is a nonempty closed contractible subset of . Let be an upper semicontinuous set-valued mapping with nonempty closed acyclic values and a set-valued mapping satisfying the following conditions:

(ii)for all are properly -quasiconvex,

(iv)for all are lower -continuous.

Then, the solutions set is nonempty and closed subset of .

Proof.

Proceeding as in the proof of Theorem 3.1, we need to prove that is closed acyclic subset of for all . We divide the remainder of the proof into three steps.

Step 1 (to show that is a closed subset of ).

This means that and so is a closed subset of .

Step 2 (to show that is upper semicontinuous).

it is a contradiction to . Hence, , and therefore, is a closed mapping. Since is a compact set and is a closed subset of , is compact. This implies that is compact. Then, by Lemma 2.7(i), we have that is upper semicontinuous.

Step 3 (to show that the solutions set is nonempty and closed).

Hence, . Similarly, by the proof of Step 5 in Theorem 3.1, we have is closed. This completes the proof.

## 4. Stability

In this section, we discuss the stability of the solutions for the generalized strong vector quasiequilibrium problem (GSVQEP II).

Throughout this section, let , be Banach spaces, and let be a real locally -convex Hausdorff topological vector space. Let and be nonempty compact subsets, and let be a nonempty closed convex cone. Let is a continuous set-valued mapping with nonempty closed contractible values, and is an upper semicontinuous set-valued mapping with nonempty closed acyclic values}.

Thus, , which conclude that defines a set-valued mapping from into .

We also need the following lemma in the sequel.

Let be a metric space, and let be compact sets in . Suppose that for any open set , there exists such that for all . Then, any sequence satisfying has a convergent subsequence with limit in .

In the following theorem, we replaced the convex set by the contractible set and acyclic set in Theorem 4.1 in [16]. The following theorem can acquire the same result appearing on the Theorem 4.1 by utilized Lemma 4.1. Now, we need only to present stability theorem for the solution set mapping for (GSVQEP II).

Theorem 4.2.

is an upper semicontinuous mapping with compact values.

Proof.

By the same argument as in the proof of Theorem 4.1 in [16], we can show that and .

## Declarations

### Acknowledgments

S. Plubtieng would like to thank the Thailand Research Fund for financial support under Grant no. BRG5280016. Moreover, K. Sitthithakerngkiet would like to thank the Office of the Higher Education Commission, Thailand, for supporting by grant fund under Grant no. CHE-Ph.D-SW-RG/41/2550, Thailand.

## Authors’ Affiliations

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