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Advances in Decision Sciences
Volume 2010 (2010), Article ID 250378, 8 pages
http://dx.doi.org/10.1155/2010/250378
Research Article

Parametric Set-Valued Vector Quasi-Equilibrium Problems

School of Mathematics Science, Liaocheng University, Liaocheng, 252059 Shandong, China

Received 9 June 2009; Revised 14 January 2010; Accepted 5 March 2010

Academic Editor: Graham Wood

Copyright © 2010 Liya Fan and Aiqin Li. 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.

Abstract

Two kinds of parametric set-valued vector quasi-equilibrium problems are introduced. The existence of solutions to these problems is studied. The upper and lower semicontinuities of their solution maps with respect to the parameters are investigated.

1. Introduction and Preliminaries

Equilibrium problems are a class of general problems that contains many other problems, such as optimization problems, variational inequality problems, saddle point problems, and complementarity problems, as special cases. Up to now, the main efforts for equilibrium problems have been made for the solution existence; see for example [16] and the references therein. A few results have been obtained for properties of solution sets, see [712].

Motivated and inspired by works in [1, 5, 812], in this paper, we will introduce two kinds of parametric set-valued vector quasi-equilibrium problems and study the solution existence of these problems. In addition, we will investigate the upper and lower semicontinuities of their solution maps with respect to the parameters.

Throughout this paper, let be real Hausdorff topological vector spaces, real topological vector spaces, and a nonempty compact convex subset of . We denote by , and the convex hull, interior, boundary, and closed hull of , respectively. Let , and be set-valued mappings such that for all and and be a closed convex pointed cone of with for each .

The mapping is said to be quasiconvex of type 2 with respect to (see [1]) if for any nonempty finite subset and any , there exist and such that . is said to be quasi convex-like of type 2 with respect to (see [1]) if for any nonempty finite subset and any , there exist and such that

Let be a nonempty subset of . A set-valued mapping is said to be upper semicontinuous (shortly, u.s.c) at if for any open set , there exists an open neighborhood of such that for each . is said to be u.s.c on if it is u.s.c at each point in .

The mapping is said to be lower semicontinuous (shortly, l.s.c) at if for each and any open neighborhood of there exists an open neighborhood of such that for each , or, equivalently, if for any net with and any , there exists a net with for each such that . is said to be l.s.c on if it is l.s.c at each point in .

The mapping is said to be closed at if for any net and for each , one has . is said to be a closed set-valued mapping if its graph, denoted by , is a closed set in , where . is said to have closed values if is a closed set for each .

A set-valued mapping is said to be a KKM mapping if for each nonempty finite subset of , one has .

Lemma 1.1 (Fan-KKM Theorem). Let a nonempty convex subset of and be a KKM mapping. If is a closed set for every and there exists such that is a compact set, then .

Lemma 1.2 (see [13]). If a set-valued mapping is u.s.c and has closed values, then it is a closed set-valued mapping.

Lemma 1.3 (see [14]). Let the set-valued mapping have a compact value at . Then is u.s.c at if and only if for any nets and for each there exist and a subnet of such that .

For any given parameters and , in this paper, we consider the following two parametric set-valued vector quasi-equilibrium problems.

PSVQEP 1. Find such that for each there exists satisfying

PSVQEP 2. Find such that for each there exists satisfying We denote their solution sets by and , respectively. Obviously, .

2. Solution Existence

In this section, we will study the existence of solutions for PSVQEP 1 and PSVQEP 2 without any monotonicity. Since parameters play no role in considering solution existence, for the sake of convenience, we state and prove existence results without parameters. We denote the above problems without parameters by SVQEP1 and SVQEP2, and their solution sets by and , respectively.

Theorem 2.1. Let(i) for all ,(ii) be an open set,(iii) be quasi convex of type 2 with respect to ,(iv) be a closed set for each . Then (SVQEP1) has at least a solution.

Proof. Put and define three set-valued mappings , and byFirstly, we show that is a KKM mapping.
Suppose to the contrary that is not a KKM mapping. Then there exist a nonempty finite subset and a point , where and , such that , which implies that .
If , then for all and , which contradicts (iii).
If , then , which indicates that and then . This is a contradiction.
Thus, is a KKM mapping.
Secondly, we show that .
For any given , we can deduce that By (ii) and (iv), we can conclude that is a closed set. Since is a Hausdorff topological vector space and is a compact set, we have that is compact for each . By Lemma 1.1, we get .
Finally, we prove that the assertion of the theorem holds.
Taking arbitrarily , we have for all , which indicates that . As for all , we know that and then . Consequently, for each , there exists such that , which shows that .

By a similar proof as for Theorem 2.1, we obtain the following result.

Theorem 2.2. Let hypotheses (i) and (ii) in Theorem 2.1 hold and let
(iii) be quasi convex-like of type 2 with respect to ,
(iv) be a closed set for each . Then (SVQEP2) has at least a solution.

3. Upper Semicontinuity of Solution Sets

In this section, we will study the upper semicontinuity of the solution sets and with respect to parameters . For this end, we assume that and are nonempty for any . Let and .

Theorem 3.1. Let(i) and be closed set-valued mappings, where for each ;(ii)for any nets for each and any , there exist nets for each , and such that and ;(iii) be l.s.c on . Then is both closed and u.s.c at .

Proof. We first show that is closed at .
Suppose to the contrary that is not closed at . Then there exist nets and and for each such that .
implies that for each . By the closedness of , we get , which together with indicates that there exists such that For , by (ii), there exists for each such that . Due to , for each , there exists such that . Again by (ii), there exist a subnet and a point such that andFor , by , there exists such that By the lower semicontinuity of , there exists for each such that , which together with the closedness of and implies that . This contradicts . Hence, is closed at .
Next, we show that is u.s.c at .
By the closedness of at , is closed and hence compact as is .
Suppose to the contrary that is not u.s.c at . By Lemma 1.3, there exist nets and for each such that for any and any subnet one has implies that for each and . By the compactness of , there exists a convergent subnet of such that . By the closedness of , we have . By , we get , that is, By using a similar argument as in part one, we can complete the proof.

Theorem 3.2. Let hypotheses (i) and (ii) in Theorem 3.1 hold and let
(iii) be u.s.c on .
Then is both u.s.c and closed at .

Proof. We first prove that is closed at .
Suppose to the contrary that is not closed at . Then there exist nets and and for each such that . By using a similar reasoning as in part one of the proof of Theorem 3.1, we can conclude that there exists a net such that andwhere with and with . By the upper semicontinuity of and , we know that there exists such that which contradicts . Hence, is closed at .
Next, we prove that is u.s.c at .
By the closedness of at , is closed and hence compact as is .
Suppose to the contrary that is not u.s.c at . By Lemma 1.3, there exist nets and for each such that holds for any and any subnet .
implies that for each and . By the compactness of and the closedness of , it follows that there exists a convergent subnet of such that . By , we get , that is, By using a similar argument as in part one, we can complete the proof.

4. Lower Semicontinuity of Solution Sets

In this section, we will consider the lower semicontinuity of the solution sets and with respect to parameters .

Theorem 4.1. Let(i) be l.s.c on and u.s.c at ;(ii)for any nets for each and any , there exist nets for each , and a point such that and ;(iii) be u.s.c and have compact values on ;(iv) for all , and . Then is l.s.c at .

Proof. Suppose to the contrary that is not l.s.c at . Then there exist a net and a point such that for any net for each one has implies that . By the lower semicontinuity of , there exists a net for each such that , which combining with shows that there exists a subnet of   such that for all . Consequently, for each , there exists satisfyingBy (ii), there exist a subnet and a point such that , which together with and (ii) indicates that there exist and such that for all and Take arbitrarily for each . By Lemma 1.3, there exist and a subset of such that .
Since for each , by the upper semicontinuity of and Lemma 1.2, we know that , which together with (iv) shows that . This contradicts . Hence, is l.s.c at .

Theorem 4.2. Let hypotheses (i) and (ii) in Theorem 4.1 hold and let
(iii) be l.s.c on ;
(iv) for all , and .
Then is l.s.c at .

Proof. By arguments similar to those for Theorem 4.1, we can conclude that there exists a net such that , and where for all and and .
For any given , by the lower semicontinuity of , there exists for each such that . By , we have for each . By the upper semicontinuity of and Lemma 1.2, it follows that , which together with (iv) implies that . This is a contradiction. Hence, is l.s.c at .

Acknowledgments

This research is supported by National Natural Science Foundation of China (10871226) and Natural Science Foundation of Shandong Province (ZR2009AL006).

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