On Generalized Implicit Vector Equilibrium Problems in Topological Ordered Spaces

Abstract

We discuss three classes of generalized implicit vector equilibrium problems in topological ordered spaces. Under some conditions, we prove three new existence theorems of solutions for the generalized implicit vector equilibrium problems in topological ordered spaces by using the Fan-Browder fixed point theorem.

1. Introduction and Preliminaries

It is well known that the vector equilibrium problem is closely related to vector variational inequality, vector optimization problem, and many others (see, e.g., [1–6] and the references therein).

Recently, a large of generalized vector equilibrium problems have been studied in different conditions by many authors and a lot of results concerned with the existence of solutions and properties of solutions have been given in finite and infinite dimensional spaces (see [7] and the references therein).

The main purpose of this paper is to extend some known results for vector equilibrium problems to topological ordered spaces (see [8]). We discuss three classes of generalized implicit vector equilibrium problems in topological ordered spaces. Under some conditions, we prove three new existence theorems of solutions for the generalized implicit vector equilibrium problems in topological ordered spaces by using the Fan-Browder fixed point theorem.

A semilattice is a partially ordered set , with the partial ordering denoted by , for which any pair of elements has a least upper bound, denoted by . It is easy to see that any nonempty finite subset of has a least upper bound, denoted by sup . In the case , the set is called an order interval. Now assume that is a semilattice and is a nonempty finite subset. Thus, the set

(1.1)

is well defined and it has the following properties:

(a),

(b)if , then .

A subset is said to be -convex if, for any nonempty finite subset , we have .

For any denotes the family of all finite subsets of and

(1.2)

Let be a topological semilattice, a nonempty -convex subset, a Hausdorff topological vector space. Assume that , , , and such that, for any , is a closed, pointed, and convex cone in and .

In this paper, we consider the following three classes generalized implicit vector equilibrium problems:

weak generalized implicit vector equilibrium problem (WGIVEP): find , such that and for any , there exists an such that

(1.3)

strong generalized implicit vector equilibrium problem (SGIVEP): find such that and

(1.4)

uniform generalized implicit vector equilibrium problem (UGIVEP): find such that and there exists such that

(1.5)

Definition 1.1.

Let and be two topological spaces.

A mapping is called upper semicontinuous (usc) at if, for any neighborhood of , there exists a neighborhood of such that

(1.6)

F is called usc on if it is usc at each point of .

A mapping is called lower semicontinuous (lsc) at if, for any net in such that and for any , there exists such that . is called lsc on if it is lsc at each point of .

A mapping is called complement pseudo-upper semicontinuous (c p-usc) at if, for with , we have . is called c p-usc on if is c p-usc at each point of .

Remark 1.2.

By [9], if is usc with closed values, then for any net in such that and for any net in with such that in , we have .

If is usc with closed values, then is c p-usc, where

(1.7)

Lemma 1.3 (see [9]).

Let and be two topological spaces. Let be compact and be usc such that is compact for each . Then is compact.

Definition 1.4.

Let be a topological semilattice or a -convex subset of a topological semilattice, let be a Hausdorff topological vector space, and let be a closed, pointed, and convex cone with .

A mapping is called a -convex mapping (or a -concave mapping) with respect to if, for any nonempty finite subset , , , with and , there exists such that

(1.8)

A mapping is called to have -inheritance if, for any nonempty finite subset of , , with and , we have .

Let . A mapping is called --convex (or --concave) with respect to in second argument if, for any nonempty finite subset , , , , with and , there exists such that

(1.9)

Remark 1.5.

If is a --convex mapping (or a --concave mapping) with respect to in second argument, then for any , is a -convex mapping (or a -concave mapping) with respect to .

Lemma 1.6 (see [10]).

Let be a nonempty compact -convex subset of a topological semilattice with path-connected intervals , let be a mapping with nonempty -convex values such that, for each , is an open set in . Then has a fixed point.

2. Existence Theorems

Theorem 2.1.

Let be a nonempty compact -convex subset of a topological semilattice with path-connected intervals , let be a Hausdorff topological vector space. Let be a mapping with nonempty -convex values, and let and be mappings and be a mapping such that, for each , is a closed, pointed, and convex cone in with . Assume that

(1)For any , is open;

(2) is closed;

(3) is usc with compact values;

(4) for any and ;

(5)for any and , is -concave with respect to ;

(6)for any , is lsc;

(7) is usc, where for each .

Then there exists an such that and for any , there exists an such that

(2.1)

Furthermore, the solution set of (WGIVEP) is closed, and hence is compact.

Proof.

Define by

(2.2)

We first prove that for any , is open, that is,

(2.3)

is closed. Let a net and . Then there exists such that , , for any . By and Lemma 1.3, we know that

(2.4)

is compact and so has a cluster point . We may assume that and thus, . For any , by , there exists such that and so . It follows from that and hence . Thus, is closed and so is open.

Suppose that there exists an such that is not -convex, that is, there exist such that . Hence, there exists , , that is, there exists such that . For each , , take , . Let , . For any , and , we have . By , there exists such that

(2.5)

Since , we know that

(2.6)

which is a contradiction. Therefore, for any , is -convex.

By and Lemma 1.6, . Define by

(2.7)

Then is -convex for each . It follows from and that

(2.8)

is open.

Suppose that for all , is nonempty. Then, by Lemma 1.6   has a fixed point, that is, there exists , such that . If , then , hence , for all , which contradicts to assumption ; If , then , hence which contradicts with . Therefore, there exists , such that . Since is nonempty for any , then , , that is, and for any , . Therefore, and for any , there exists an such that

(2.9)

Let denote the solution set of (WGIVEP) and with . We show that , that is, , and for all , there exists such that

(2.10)

In fact, it follows from that . For any , . By , there exists an open neighborhood of such that . Since , there exists such that for any , . Thus, and so there exists such that , that is, . Since is compact, has a cluster point . We may assume that . From , we have . By , for any , there exists such that . It follows from that , that is, . Thus, is closed, and hence is compact. This completes the proof.

Theorem 2.2.

Let be a nonempty compact -convex subset of a topological semilattice with path-connected intervals , let be a Hausdorff topological vector space. Let be with nonempty -convex values, and let and be mappings and be a mapping such that, for each , is a closed, pointed, and convex cone in with . Assume that

(1)For any , is open;

(2) is closed;

(3) is lsc;

(4)for all and , ;

(5)for all , is -concave with respect to in second argument;

(6)for all , is lsc;

(7) is usc, where for all .

Then there exists an such that and

(2.11)

Furthermore, the solution set of (SGIVEP) is closed, and hence is compact.

Proof.

Define by

(2.12)

Then the proof is similar to that of Theorem 2.1 and so we omit it.

Theorem 2.3.

Let be a nonempty compact -convex subset of a topological semilattice with path-connected intervals and let be a Hausdorff topological vector space. Let be with nonempty -convex values, let and be mappings, and let be a mapping such that, for each , is a closed, pointed, and convex cone in with . Assume that

(1) is usc with compact values;

(2)For any , is nonempty -convex;

(3) is c p-usc on ;

(4) is usc with nonempty compact values;

(5)for all , ;

• (6) for all , is -concave with respect to ;

(7) has -inheritance;

(8) is c p-usc on .

Then there exists an such that and there exists such that

(2.13)

Proof.

Define by

(2.14)

where

(2.15)

The proof is divided into the following five steps.

• (I) For any , is nonempty.

If it is false, then there exists such that , that is, for any , there exists such that . Let

(2.16)

Then is nonempty values. If there exists such that is not -convex, then there exist such that , that is, there exists with . Thus, . For each , , take . For any , and , we have . By , there exists such that

(2.17)

Since , . Hence,

(2.18)

which is a contradiction. Thus, for any , is nonempty -convex.

For any ,

(2.19)

It follows from that

(2.20)

is closed and so is open. Since is nonempty compact and -convex, by Lemma 1.6   has a fixed point. Thus, there exists such that , that is, which contradicts with Assumption . Hence for any .

• (II) For any , is -convex. If it is false, then there exists such that is not -convex, that is, there exist such that

(2.21)

Thus, there exists such that . Then and for all , , . Since is -convex, . By , . Since , there exists such that

(2.22)

which is a contradiction. Therefore, for any , is -convex.

• (III) is nonempty -convex for any . By steps (I) and (II), the conclusion follows directly from .

• (IV) For any ,

(2.23)

is open. In fact, we only need to show that

(2.24)

is closed. Let a net and . If , then and hence

(2.25)

If and there exists such that

(2.26)

Take . By and Lemma 1.3, is compact and hence has a cluster point . We may assume that and so . Similarly, by , has a cluster point . We assume that and hence . Since closed, . Thus, , and . Hence, is closed. Let a net

(2.27)

and , then . By , we have , and hence is closed. Thus, is closed and so is open.

• (V) The UGIVEP has a solution. By Lemma 1.6, has a fixed point. Thus, there exists such that , that is, and such that and

(2.28)

This completes the proof.