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 [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.

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 , ;

• (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;

• (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;

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

• (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;

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

• (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 [1–6] do not include each other as a special case.