Existence of Solutions to the System of Generalized Implicit Vector Quasivariational Inequality Problems
© Zhi Lin. 2009
Received: 31 March 2009
Accepted: 3 September 2009
Published: 27 September 2009
We study the system of generalized implicit vector quasivariational inequality problems and prove a new existence result of its solutions by Kakutani-Fan-Glicksberg's fixed points theorem. As a special case, we also derive a new existence result of solutions to the generalized implicit vector quasivariational inequality problems.
The system of generalized implicit vector quasivariational inequality problems generalizes the generalized implicit vector quasivariational inequality problems, and the latter had been studied in [1–3]. In this paper, we study the system of generalized implicit vector quasivariational inequality problems and prove a new existence result of its solutions by Kakutani-Fan-Glicksberg's fixed points theorem. For other existence results with respect to the system of generalized implicit vector quasivariational inequality problems, we refer the reader to [4–6] and references therein.
Throughout this paper, unless otherwise specified, assume that for each , is a nonempty convex compact subset of a Banach space is a Hausdorff topological vector space, and is a closed, convex, and pointed cone of with , where denotes the interior of .
In this section, we introduce some useful notations and results.
Lemma 2.4 ([7, Theorem 1]).
Let be a nonempty paracompact subset of a Hausdorff topological space and, be a nonempty subset of a Hausdorff topological vector space . Suppose that be two set-valued maps with following conditions:
3. Existence of Solutions to the SGIVQIP
The proof is finished.
By Lemma 3.1, we obtain the following result.
If , then there exists such that for each . By Lemma 3.1, there exist open neighborhood of and open neighborhood of , such that whenever . By (1), there exist such that , which implies that there exists a positive integer such that whenever . Thus we have whenever , a contradiction. This shows that is closed,that is, is open.
We prove that the SGIVQIP has a solution.
The proof is finished.
The research was supported by the Natural Science Foundation of CQ CSTC.
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