- Research Article
- Open Access

# Upper Semicontinuity of Solution Maps for a Parametric Weak Vector Variational Inequality

- ZM Fang
^{1}Email author and - SJ Li
^{1}

**2010**:482726

https://doi.org/10.1155/2010/482726

© Z. M. Fang and S. J. Li. 2010

**Received:**7 March 2010**Accepted:**9 July 2010**Published:**26 July 2010

## Abstract

This paper investigates the upper semicontinuity of the solution map for a parametric weak vector variational inequality associated to a -hemicontinuous and weakly -pseudomonotone operator.

## Keywords

- Banach Space
- Linear Operator
- Euclidean Space
- Generalize Linearization
- Convex Cone

## 1. Introduction and Preliminaries

A vector variational inequality (VVI, in short) was first introduced by Giannessi [1] in the setting of finite-dimensional Euclidean space. Later on, it was studied and generalized to infinite-dimensional spaces. Existences of the solutions for VVI have been studied extensively in various versions; see [2–6] and references therein.

Stability of the solution map for VVI or vector equilibrium problems is an important topic in optimization theory. A great deal of papers have been denoted to study the semicontinuity and continuity of the solution maps; see [7–18] and references therein. All results of the stability of the solution map for VVI in the literature are obtained based on continuity of the operator. It is well known that -hemicontinuity is weaker than continuity. In this paper, our aim is to investigate the upper semicontinuity of the solution map for a parametric weak vector variational inequality associated to a -hemicontinuous and weakly -pseudomonotone operator.

where is a nonempty subset and is a vector-valued mapping.

where is a nonempty subset and is a vector-valued mapping.

Throughout the paper, we always assume that is nonempty for all in a neighborhood of . Now, we recall some basic definitions and their properties.

Let be a nonempty convex subset of and let be an operator. is said to be -hemicontinuous if and only if, for every , and , the mapping is continuous at .

Definition 1.2 (see [6]).

Let be a nonempty subset of and be an operator. is weakly -pseudomonotone on if, for every pair of points , , one has that implies that .

Proposition 1.3 (see [6, Generalized Linearization Lemma]).

Let be a nonempty convex subset of and let be an operator. Consider the following problems:

(I) such that for all ,

(II) such that for all .

Then the following are obtained.

(i)Problem (I) implies Problem (II) if is weakly -pseudomonotone.

(ii)Problem (II) implies Problem (I) if is -hemicontinuous.

Let be a set-valued mapping, given that .

Definition 1.4 (see [19, 20]).

(i) is called lower semicontinuous (l.s.c) at if, for any open set satisfying , there exists such that for every , .

(ii) is called upper semicontinuous (u.s.c) at if, for any open set satisfying , there exists such that, for every , .

We say is l.s.c (resp., u.s.c) on , if it is l.s.c (resp., u.s.c) at each . is said to be continuous on if it is both l.s.c and u.s.c on .

Proposition 1.5 (see [19, 21]).

- (ii)
If has compact values (i.e., is a compact set for each ), then is u.s.c at if and only if, for any sequence with and for any , there exist and a subsequence of such that .

## 2. Main Results

In this section, we mainly discuss the upper semicontinuity of the solution map for (PWVVI).

Lemma 2.1.

Let be a nonempty compact convex subset of . Suppose that, for any , is -hemicontinuous and weakly -pseudomonotone on . Then, has compact values on , that is, is a compact set for each .

Proof.

That is . Thus, is a closed set. Furthermore, it follows from and the compactness of that is a compact set. The proof is complete.

Theorem 2.2.

Let be a nonempty compact convex subset of . Suppose that the following conditions are satisfied.

(i)For any , is -hemicontinuous on ,

(ii)For any , is weakly -pseudomonotone on ,

(iii)For any , is continuous on .

Then, is u.s.c on .

Proof.

that is .

Thus, for any sequence with and for any , there exist and a subsequence of such that . By Proposition 1.5 and Lemma 2.1, we have is u.s.c at . From the arbitrariness of , we can get is u.s.c on . The proof is complete.

Remark 2.3.

In [7–10], the upper semicontinuity of the solution map for has been discussed based on the continuity of the operator. Note that -hemicontinuity is weaker than continuity. Moreover, together with the assumption of weakly -pseudomonotonicity, -hemicontinuity may not derive the continuity of the operator. Thus, it is necessary to investigate the upper semicontinuity of the solution map for associated to a -hemicontinuous and weakly -pseudomonotone operator. Now we give an example to illustrate our result.

Example 2.4.

Then, by the definition of upper semicontinuity, it follows readily that the solutions map is u.s.c on .

However, for with , we have , but . Thus, for any , is not continuous at . Therefore, the theorems concerning the upper semicontinuity in the literatures are not applicable.

## Declarations

### Acknowledgments

The authors would like to thank two anonymous referees for their valuable comments and suggestions, which helped to improve the paper. This research was partially supported by the National Natural Science Foundation of China (Grant no. 10871216) and Chongqing University Postgraduates Science and Innovation Fund (Project no. 201005B1A0010338).

## Authors’ Affiliations

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