- Research Article
- Open Access

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

- Zhi Lin
^{1}Email author

**2009**:654370

https://doi.org/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.

## Keywords

- Linear Operator
- Point Theorem
- Operator Space
- Open Neighborhood
- Existence Result

## 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 concave if is convex.

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:

- (2)
for each , is open.

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;

- (3)
for each .

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

Proof.

whenever , ,

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;

- (4)
for each and each , is convex or quasiconvex-like;

- (5)
for each and each , if , then , where is the component of .

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;

- (4)
for each and each , is convex or quasiconvex-like;

- (5)
for each and each , if , then .

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

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