# Existence of Solutions to the System of Generalized Implicit Vector Quasivariational Inequality Problems

- Zhi Lin
^{1}Email author

**2009**:654370

**DOI: **10.1155/2009/654370

© Zhi Lin. 2009

**Received: **31 March 2009

**Accepted: **3 September 2009

**Published: **27 September 2009

## Abstract

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.

## 1. Introduction

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.

If is a singleton, then the SGIVQIP coincides with the generalized implicit vector quasivariational inequality problems (briefly, GIVQIP). A GIVQIP is usually denoted by .

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 .

## 2. Preliminaries

In this section, we introduce some useful notations and results.

Definition 2.1.

Let and be two topological spaces and a nonempty convex subset of . is a set-valued map.

and upper semicontinuous on if it is upper semicontinuous at every point of

and lower semicontinuous on if it is lower semicontinuous at every point of

(3) is called continuous at if, it is both upper semicontinuous and lower semicontinuous at ; and continuous on if it is continuous at every point of .

Definition 2.2.

Let and be two topological vector spaces and a nonempty convex subset of .Also is a set-valued map.

and upper semicontinuous on if it is upper semicontinuous at every point of .

and lower semicontinuous on if it is lower semicontinuous at every point of .

(3) is called continuous at if it is upper semicontinuous and lower semicontinuous at ; and continuous on if it is continuous at every point of .

Definition 2.3.

Let and be two topological vector spaces and a nonempty convex subset of . Let be a set-valued map.

and quasiconcave-like if is quasiconvex-like.

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:

Then has a continuous selection, that is, there is a continuous map such that for each .

## 3. Existence of Solutions to the SGIVQIP

Lemma 3.1.

Let be three Hausdorff topological spaces, a topological vector space, and a closed, convex, and pointed cone of . Let and be two set-valued maps. Assume that and

(1) is upper semicontinuous on with nonempty and compact values;

Then there exist open neighborhood of and open neighborhood of , and open neighborhood of such that whenever , ,

Proof.

The proof is finished.

By Lemma 3.1, we obtain the following result.

Theorem 3.2.

Consider an SGIVQIP . For each , assume that

(1) is continuous on with convex compact values and for each ;

(2) is upper semicontinuous on with nonempty and compact values;

Proof.

Step 1.

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.

Without loss of generality, assume that .

Step 2.

We prove that for each is nonempty and convex.

For each , we have . By Lemma 3.1, there exists an open neighborhood of such that whenever , which implies that , that is, is open. By (4), it is easy to verify that is convex.

Since is convex and , then for each is nonempty and convex.

Step 3.

We prove that has a continuous selection .

whenever , which implies that , that is, is open. Hence, for each , the set is open.

By Lemma 2.4, has a continuous selection .

Step 4.

We prove that the SGIVQIP has a solution.

The proof is finished.

If is a singleton, we obtain the following existence result of solutions to the GIVQIP by Theorem 3.2.

Corollary 3.3.

Consider a GIVQIP . Assume that

(1) is continuous on with convex compact values and for each ;

(2) is upper semicontinuous on with nonempty and compact values;

Remark 3.4.

Theorem 3.2, Corollary 3.3, and each corresponding result in literatures [1–6] do not include each other as a special case.

## Declarations

### Acknowledgments

The research was supported by the Natural Science Foundation of CQ CSTC.

## Authors’ Affiliations

## References

- Chiang Y, Chadli O, Yao JC:
**Existence of solutions to implicit vector variational inequalities.***Journal of Optimization Theory and Applications*2003,**116**(2):251–264. 10.1023/A:1022472103162MathSciNetView ArticleMATHGoogle Scholar - Cubiotti P, Yao J:
**Discontinuous implicit generalized quasi-variational inequalities in Banach spaces.***Journal of Global Optimization*2007,**37**(2):263–274. 10.1007/s10898-006-9048-6MathSciNetView ArticleMATHGoogle Scholar - Lin L-J, Chen H-L:
**The study of KKM theorems with applications to vector equilibrium problems with implicit vector variational inequalities problems.***Journal of Global Optimization*2005,**32**(1):135–157. 10.1007/s10898-004-2119-7MathSciNetView ArticleMATHGoogle Scholar - Peng J-W, Lee H-WJ, Yang X-M:
**On system of generalized vector quasi-equilibrium problems with set-valued maps.***Journal of Global Optimization*2006,**36**(1):139–158. 10.1007/s10898-006-9004-5MathSciNetView ArticleMATHGoogle Scholar - Ansari QH:
**Existence of solutions of systems of generalized implicit vector quasi-equilibrium problems.***Journal of Mathematical Analysis and Applications*2008,**341**(2):1271–1283. 10.1016/j.jmaa.2007.11.033MathSciNetView ArticleMATHGoogle Scholar - Al-Homidan S, Ansari QH, Schaible S:
**Existence of solutions of systems of generalized implicit vector variational inequalities.***Journal of Optimization Theory and Applications*2007,**134**(3):515–531. 10.1007/s10957-007-9236-7MathSciNetView ArticleMATHGoogle Scholar - Wu X, Shen S:
**A further generalization of Yannelis-Prabhakar's continuous selection theorem and its applications.***Journal of Mathematical Analysis and Applications*1996,**197**(1):61–74. 10.1006/jmaa.1996.0007MathSciNetView ArticleMATHGoogle Scholar - Klein E, Thompson AC:
*Theory of Correspondences, Canadian Mathematical Society Series of Monographs and Advanced Texts*. John Wiley & Sons, New York, NY, USA; 1984:xi+256.Google Scholar - Aliprantis CD, Border KC:
*Infinite-Dimensional Analysis*. 2nd edition. Springer, Berlin, Germany; 1999:xx+672.View ArticleMATHGoogle Scholar

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