- 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.

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)

- (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;

(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

## References

- Lee GM, Kim DS, Lee BS, Yen ND:
**Vector variational inequality as a tool for studying vector optimization problems.***Nonlinear Analysis: Theory, Methods & Applications*1998,**34**(5):745–765. 10.1016/S0362-546X(97)00578-6MATHMathSciNetView ArticleGoogle Scholar - Cheng Y:
**On the connectedness of the solution set for the weak vector variational inequality.***Journal of Mathematical Analysis and Applications*2001,**260**(1):1–5. 10.1006/jmaa.2000.7389MATHMathSciNetView ArticleGoogle Scholar - Gong X-H:
**Efficiency and Henig efficiency for vector equilibrium problems.***Journal of Optimization Theory and Applications*2001,**108**(1):139–154. 10.1023/A:1026418122905MATHMathSciNetView ArticleGoogle Scholar - Gong X-H, Fu WT, Liu W:
**Super efficiency for a vector equilibrium in locally convex topological vector spaces.**In*Vector Variational Inequalities and Vector Equilibria, Nonconvex Optimization and Its Applications*.*Volume 38*. Edited by: Giannessi F. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2000:233–252. 10.1007/978-1-4613-0299-5_13View ArticleGoogle Scholar - Gong X-H:
**Connectedness of the solution sets and scalarization for vector equilibrium problems.***Journal of Optimization Theory and Applications*2007,**133**(2):151–161. 10.1007/s10957-007-9196-yMATHMathSciNetView ArticleGoogle Scholar - Ansari QH, Oettli W, Schläger D:
**A generalization of vectorial equilibria.***Mathematical Methods of Operations Research*1997,**46**(2):147–152. 10.1007/BF01217687MATHMathSciNetView ArticleGoogle Scholar - Ansari QH, Yao J-C:
**An existence result for the generalized vector equilibrium problem.***Applied Mathematics Letters*1999,**12**(8):53–56. 10.1016/S0893-9659(99)00121-4MATHMathSciNetView ArticleGoogle Scholar - Konnov IV, Yao JC:
**Existence of solutions for generalized vector equilibrium problems.***Journal of Mathematical Analysis and Applications*1999,**233**(1):328–335. 10.1006/jmaa.1999.6312MATHMathSciNetView ArticleGoogle Scholar - Fu J-Y:
**Generalized vector quasi-equilibrium problems.***Mathematical Methods of Operations Research*2000,**52**(1):57–64. 10.1007/s001860000058MATHMathSciNetView ArticleGoogle Scholar - Hou SH, Yu H, Chen GY:
**On vector quasi-equilibrium problems with set-valued maps.***Journal of Optimization Theory and Applications*2003,**119**(3):485–498.MATHMathSciNetView ArticleGoogle Scholar - Tan NX:
**On the existence of solutions of quasivariational inclusion problems.***Journal of Optimization Theory and Applications*2004,**123**(3):619–638. 10.1007/s10957-004-5726-zMATHMathSciNetView ArticleGoogle Scholar - Peng J-W, Lee H-WJ, Yang X-M:
**On system of generalized vector quasi-equilibrium problems with set-valued maps.***Journal of Global Optimization*2006,**36**(1):139–158. 10.1007/s10898-006-9004-5MATHMathSciNetView ArticleGoogle Scholar - Ansari QH, Flores-Bazán F:
**Recession methods for generalized vector equilibrium problems.***Journal of Mathematical Analysis and Applications*2006,**321**(1):132–146. 10.1016/j.jmaa.2005.07.059MATHMathSciNetView ArticleGoogle Scholar - Lin L-J, Ansari QH, Huang Y-J:
**Some existence results for solutions of generalized vector quasi-equilibrium problems.***Mathematical Methods of Operations Research*2007,**65**(1):85–98. 10.1007/s00186-006-0102-4MATHMathSciNetView ArticleGoogle Scholar - Long X-J, Huang N-J, Teo K-L:
**Existence and stability of solutions for generalized strong vector quasi-equilibrium problem.***Mathematical and Computer Modelling*2008,**47**(3–4):445–451. 10.1016/j.mcm.2007.04.013MATHMathSciNetView ArticleGoogle Scholar - Aubin J-P, Ekeland I:
*Applied Nonlinear Analysis, Pure and Applied Mathematics*. John Wiley & Sons, New York, NY, USA; 1984:xi+518.MATHGoogle Scholar - Warburton AR:
**Quasiconcave vector maximization: connectedness of the sets of Pareto-optimal and weak Pareto-optimal alternatives.***Journal of Optimization Theory and Applications*1983,**40**(4):537–557. 10.1007/BF00933970MATHMathSciNetView ArticleGoogle Scholar - Robertson AP, Robertson W:
*Topological Vector Spaces, Cambridge Tracts in Mathematics and Mathematical Physics, no. 53*. Cambridge University Press, New York, NY, USA; 1964:viii+158.Google Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.