Upper Semicontinuity of Solution Maps for a Parametric Weak Vector Variational Inequality
© Z. M. Fang and S. J. Li. 2010
Received: 7 March 2010
Accepted: 9 July 2010
Published: 26 July 2010
1. Introduction and Preliminaries
A vector variational inequality (VVI, in short) was first introduced by Giannessi  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.
Definition 1.2 (see ).
Proposition 1.3 (see [6, Generalized Linearization Lemma]).
Then the following are obtained.
2. Main Results
In this section, we mainly discuss the upper semicontinuity of the solution map for (PWVVI).
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.
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.
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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