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

## 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 [13]. 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 [46] and references therein.

Let be an index set (finite or infinite). For each , let and be two Hausdorff topological vector spaces, a nonempty subset of , and a closed, convex and pointed cone of with , where denotes the interior of . Denote that , . For each , we can write . For each , let be a nonempty subset of the continuous linear operators space from into and let , , be three set-valued maps, where and denote the family of all nonempty subsets of and , respectively. The system of generalized implicit vector quasivariational inequality problems (briefly, SGIVQIP) is as follows: find such that for each and

(1.1)

is said to be a solution of the SGIVQIP. An SGIVQIP is usually denoted by .

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.

(1) is called upper semicontinuous at if, for any open set , there exists an open neighborhood of in such that for all ,

(2.1)

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

(2) is called lower semicontinuous at if, for any open set , there exists an open neighborhood of in such that for all ,

(2.2)

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.

(1) is called upper semicontinuous at if, for any open neighborhood of the zero element in , there exists an open neighborhood of in such that, for all ,

(2.3)

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

(2) is called lower semicontinuous at if, for any open neighborhood V of the zero element in , there exists an open neighborhood of in such that, for all ,

(2.4)

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.

(1) is called convex if, for each ,

(2.5)

and concave if is convex.

(2) is called quasiconvex-like if, for each , ,

(2.6)

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:

(1)for each , ;

1. (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;

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

1. (3)

for each .

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

Proof.

By (3) and compactness of , there exists an open neighborhood of the zero element of such that . By (2), there exist open neighborhood of and open neighborhood of , open neighborhood of such that whenever . Since is compact and , there exist finite such that . Taking

(3.1)

Clearly, and are open neighborhood of and , respectively. Thus for each , we have whenever , By (1), there exist open neighborhood of with and open neighborhood of such that whenever , which implies that

(3.2)

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;

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

1. (4)

for each and each , is convex or quasiconvex-like;

2. (5)

for each and each , if , then , where is the component of .

Then the SGIVQIP has a solution, that is, there exists such that for each and

(3.3)

Proof.

For each , define a set-valued map by

(3.4)

Step 1.

We prove that the set is closed. For any sequence with , we have

(3.5)

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 .

Define a set-valued map by

(3.6)

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 .

For each , we have and . By , there exists such that , where . Since is continuous with convex compact values, then there exists an open neighborhood of such that

(3.7)

whenever , where . Thus whenever , which implies that whenever , that is, whenever . This shows that the set is open. By , we have . By Lemma 3.1, there exists an open neighborhood of such that

(3.8)

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.

For each , define the set-valued map by

(3.9)

Note that is upper semicontinuous when and is upper semicontinuous when , and it is easy to verify that is also upper semicontinuous when , where denotes the boundary of . Thus, is upper semicontinuous with nonempty convex compact values. By [8, Theorem  7.1.15], the set-valued map defined by is closed with nonempty convex values. By Kakutani-Fan-Glicksberg's fixed points theorem (see [9, pages 550]), has a fixed point, that is, there exists . The condition (5) implies that for each , , that is, for each . Thus we have that for each , and

(3.10)

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;

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

1. (4)

for each and each , is convex or quasiconvex-like;

2. (5)

for each and each , if , then .

Then the GIVQIP has a solution, that is, there exists such that ,

(3.11)

Remark 3.4.

Theorem 3.2, Corollary 3.3, and each corresponding result in literatures [16] do not include each other as a special case.

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## Acknowledgments

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

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Correspondence to Zhi Lin.

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Lin, Z. Existence of Solutions to the System of Generalized Implicit Vector Quasivariational Inequality Problems. J Inequal Appl 2009, 654370 (2009). https://doi.org/10.1155/2009/654370

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• DOI: https://doi.org/10.1155/2009/654370

### Keywords

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