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Upper Semicontinuity of Solution Maps for a Parametric Weak Vector Variational Inequality
Journal of Inequalities and Applicationsvolume 2010, Article number: 482726 (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.
1. Introduction and Preliminaries
A vector variational inequality (VVI, in short) was first introduced by Giannessi [1] in the setting of finitedimensional Euclidean space. Later on, it was studied and generalized to infinitedimensional 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.
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:
where is a nonempty subset and is a vectorvalued 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
where is a nonempty subset and is a vectorvalued mapping.
For each , we denote the solution map of (PWVVI) by , that is,
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 setvalued 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 .

(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
By Proposition 1.3 and the weakly pseudomonotonicity of , we get
From , we have as . It follows from the closedness of and (2.2) that
Moreover, by Proposition 1.3 and the hemicontinuity of , we have
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
Since is a compact set, there are an and a subsequence such that . Particularly, from (2.5), we get
By Proposition 1.3 and (iii), we can obtain that
Since is continuous and
we get , as . It follows from the closedness of and (2.7) that
Moreover, by Proposition 1.3 and (ii), we have
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.
Let , , , and
Then,
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,
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.
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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).
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Keywords
 Banach Space
 Linear Operator
 Euclidean Space
 Generalize Linearization
 Convex Cone