- Research Article
- Open Access
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.
- Linear Operator
- Point Theorem
- Operator Space
- Open Neighborhood
- Existence Result
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:
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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