- Research Article
- Open Access

# Connectedness and Compactness of Weak Efficient Solutions for Set-Valued Vector Equilibrium Problems

- Bin Chen
^{1}, - Xun-Hua Gong
^{1}Email author and - Shu-Min Yuan
^{1}

**2008**:581849

https://doi.org/10.1155/2008/581849

© Bin Chen et al. 2008

**Received:**1 November 2007**Accepted:**5 September 2008**Published:**21 September 2008

## Abstract

We study the set-valued vector equilibrium problems and the set-valued vector Hartman-Stampacchia variational inequalities. We prove the existence of solutions of the two problems. In addition, we prove the connectedness and the compactness of solutions of the two problems in normed linear space.

## Keywords

- Variational Inequality
- Lower Semicontinuous
- Normed Linear Space
- Norm Topology
- Vector Variational Inequality

## 1. Introduction

We know that one of the important problems of vector variational inequalities and vector equilibrium problems is to study the topological properties of the set of solutions. Among its topological properties, the connectedness and the compactness are of interest. Recently, Lee et al. [1] and Cheng [2] have studied the connectedness of weak efficient solutions set for single-valued vector variational inequalities in finite dimensional Euclidean space. Gong [3–5] has studied the connectedness of the various solutions set for single-valued vector equilibrium problem in infinite dimension space. The set-valued vector equilibrium problem was introduced by Ansari et al. [6]. Since then, Ansari and Yao [7], Konnov and Yao [8], Fu [9], Hou et al. [10], Tan [11], Peng et al. [12], Ansari and Flores-Bazán [13], Lin et al. [14] and Long et al. [15] have studied the existence of solutions for set-valued vector equilibrium and set-valued vector variational inequalities problems. However, the connectedness and the compactness of the set of solutions to the set-valued vector equilibrium problem remained unstudied. In this paper, we study the existence, connectedness, and the compactness of the weak efficient solutions set for set-valued vector equilibrium problems and the set-valued vector Hartman-Stampacchia variational inequalities in normed linear space.

## 2. Preliminaries

Throughout this paper, let , be two normed linear spaces, let be a nonempty subset of , let be a set-valued map, and let be a closed convex pointed cone in .

Definition 2.1.

is called a weak efficient solution to the SVEP. Denote by the set of all weak efficient solutions to the SVEP.

be the dual cone of .

Definition 2.2.

where means that , for all . Denote by the set of all -efficient solutions to the SVEP.

Definition 2.3.

Definition 2.4.

Definition 2.5.

Let be a nonempty subset of . Let be a set-valued map, where is the space of all bounded linear operators from into (let be equipped with operator norm topology). Set

is lower semicontinuous at 0.

(ii)Let . is said to be -pseudomonotone on if, for every pair of points , , for all , then , for all .

The definition of -hemicontinuity was introduced by Lin et al. [14].

Definition 2.6.

holds, where denoted the convex hull of .

For the definition of the upper semicontinuity and lower semicontinuity, see [16].

The following FKKM theorem plays a crucial role in this paper.

Lemma 2.7.

Let be a Hausdorff topological vector space. Let be a nonempty convex subset of , and let be a KKM map. If for each , is closed in , and if there exists a point such that is compact, then .

By definition, we can get the following lemma.

Lemma 2.8.

Let be a nonempty convex subset of . Let be a set-valued map, and let be a closed convex pointed cone. Moreover, suppose that is -convex in its second variable. Then, for each , is convex.

## 3. Scalarization

In this section, we extend a result in [3] to set-valued map.

Theorem 3.1.

Proof.

## 4. Existence of The Weak Efficient Solutions

Theorem 4.1.

Let be a nonempty closed convex subset of and let be a closed convex pointed cone with . Let be a set-valued map with for all . Suppose that for each , is lower semicontinuous on , and that is -convex in its second variable. If there exists a nonempty compact subset of , and , such that , for all , then, for any , , , , and .

Proof.

Therefore, . Next we show that . If , then . It follows from that , and by Theorem 3.1, we have .

Theorem 4.2.

Let be a nonempty closed convex subset of and let be a closed convex pointed cone with . Let . Assume that is a -hemicontinuous, -pseudomonotone mapping. Moreover, assume that the set-valued map defined by is -convex in its second variable. If there exists a nonempty compact subset of , and , such that , for all , then and .

Proof.

- (I)

- (II)

- (IV)

Since , . Hence since is continuous and . Therefore, for any and for each , we have . Hence . Thus . This means that there exists , for each , we have , for all . It follows that , thus . By the proof of Theorem 4.1, we know . Since , we have . The proof of the theorem is completed.

## 5. Connectedness and Compactness of The Solutions Set

In this section, we discuss the connectedness and the compactness of the weak efficient solutions set for set-valued vector equilibrium problems and the set-valued vector Hartman- Stampacchia variational inequalities in normed linear space.

Theorem 5.1.

Let be a nonempty closed convex subset of , let be a closed convex pointed cone with , and let be a set-valued map. Assume that the following conditions are satisfied:

(i)for each , is lower semicontinuous on ;

(ii) is -concave in its first variable and -convex in its second variable;

(iii) , for all ;

(iv) is a bounded subset in ;

(v)there exists a nonempty compact convex subset of , and , such that , for all .

Then is a nonempty connected compact set.

Proof.

that is . So is convex, therefore it is a connected set.

Thus by [17, Theorem 3.1] is a connected set.

This contradicts . Thus This means that is a closed set. Since is compact and , is compact.

Theorem 5.2.

Let be a nonempty closed convex subset of , and let be a closed convex pointed cone with . Assume that for each , is a -hemicontinuous, -pseudomonotone mapping. Moreover, assume that the set-valued map defined by is -convex in its second variable, and the set is a bounded set in . If there exists a nonempty compact convex subset of , and , such that , for all , then is a nonempty connected set.

Proof.

Then, by [17, Theorem 3.1], we know that is a connected set. The proof of the theorem is completed.

Lemma 5.3.

Let be a nonempty convex subset of , and let If is lower semicontinuous on , then is -hemicontinuous on .

Proof.

This means that is lower semicontinuous at 0. By definition, is -hemicontinuous on .

Theorem 5.4.

Let be a nonempty closed bounded convex subset of , and let be a closed convex pointed cone with . Assume that for each , is a -pseudomonotone, lower semicontinuous mapping. Moreover, assume that the set-valued map defined by is -convex in its second variable, and the set is a bounded set in . If there exists a nonempty compact convex subset of , and , such that , for all , then is a nonempty connected compact set.

Proof.

This contradicts (5.48), because Thus . This means that is a closed subset of .

## Declarations

### Acknowledgments

This research was partially supported by the National Natural Science Foundation of China and the Natural Science Foundation of Jiangxi Province, China.

## Authors’ Affiliations

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