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  • Research Article
  • Open Access

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

Journal of Inequalities and Applications20102010:482726

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

  • Received: 7 March 2010
  • Accepted: 9 July 2010
  • Published:

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 [26] 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 [718] 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.

Let , , and (the spaces of parameters) be Banach spaces and let be a pointed closed and convex cone with nonempty interior . Let be the space of all linear continuous operators from to . The value of a linear operator at is denoted by . Consider the following weak vector variational inequality problem:
(WVVI)

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

When is perturbed by a parameter , which varies over a nonempty set , for a given , we can define the parametric weak vector variational inequality problem
(PWVVI)

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

For each , we denote the solution map of (PWVVI) by , that is,
(1.1)

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

Definition 1.1 (see [4, 6]).

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]).

(i) is l.s.c at if and only if, for any sequence with and any , there exists such that .
  1. (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.

For any , take any sequence with ; we have
(2.1)
By Proposition 1.3 and the weakly -pseudomonotonicity of , we get
(2.2)
From , we have as . It follows from the closedness of and (2.2) that
(2.3)
Moreover, by Proposition 1.3 and the -hemicontinuity of , we have
(2.4)

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.

For any , any sequences with , and , we have and
(2.5)
Since is a compact set, there are an and a subsequence such that . Particularly, from (2.5), we get
(2.6)
By Proposition 1.3 and (iii), we can obtain that
(2.7)
Since is continuous and
(2.8)
we get , as . It follows from the closedness of and (2.7) that
(2.9)
Moreover, by Proposition 1.3 and (ii), we have
(2.10)

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 [710], 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.

Let , , , and
(2.11)
Then,
(2.12)
It is clear that conditions (ii) and (iii) of Theorem 2.2 are satisfied. For any ray , is continuous. Thus, is -hemicontinuous on and condition (i) of Theorem 2.2 is satisfied. By Theorem 2.2, we conclude that is u.s.c on . In fact,
(2.13)

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

(1)
College of Mathematics and Science, Chongqing University, Chongqing, 400030, China

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Copyright

© Z. M. Fang and S. J. Li. 2010

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.

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