On Generalized Implicit Vector Equilibrium Problems in Topological Ordered Spaces

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.

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

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

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

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

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

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

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

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

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

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.

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

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

Proof.

Define by

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

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

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

Since , we know that

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

By and Lemma 1.6, . Define by

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

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

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

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

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

Proof.

Define by

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

Proof.

Define by

where

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

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

Since , . Hence,

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

For any ,

It follows from that

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

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

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 ,

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

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

If and there exists such that

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

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

This completes the proof.

This research was supported by the Natural Science Foundations of Guangdong Province (9251064101000015). The author is grateful to the referees for the valuable comments and suggestions.

1. Li, J, Huang, N-J, Kim, JK: On implicit vector equilibrium problems. Journal of Mathematical Analysis and Applications. 283(2), 501–512 (2003). Publisher Full Text

2. Konnov, IV, Yao, JC: Existence of solutions for generalized vector equilibrium problems. Journal of Mathematical Analysis and Applications. 233(1), 328–335 (1999). Publisher Full Text

3. Ceng, L-C, Guu, S-M, Yao, J-C: On generalized implicit vector equilibrium problems in Banach spaces. Computers & Mathematics with Applications. 57(10), 1682–1691 (2009). PubMed Abstract | Publisher Full Text

4. Chen, XH: Existence of solution for the implicit multi-valued vector equilibrium problem. Journal of Applied Mathematics and Computing. 30(1-2), 469–478 (2009). Publisher Full Text

5. Huang, NJ, Li, J, Thompson, HB: Implicit vector equilibrium problems with applications. Mathematical and Computer Modelling. 37(12-13), 1343–1356 (2003). Publisher Full Text

6. Li, J, Huang, N-J: Implicit vector equilibrium problems via nonlinear scalarisation. Bulletin of the Australian Mathematical Society. 72(1), 161–172 (2005). Publisher Full Text

7. Chen, G-Y, Huang, XX, Yang, XQ: Vector Optimization: Set-Valued and Variational Analysis, Lecture Notes in Economics and Mathematical Systems, Springer, Berlin, Germany (2005)

8. Horvath, CD, Llinares Ciscar, JV: Maximal elements and fixed points for binary relations on topological ordered spaces. Journal of Mathematical Economics. 25(3), 291–306 (1996). Publisher Full Text

9. Aubin, J-P, Ekeland, I: Applied Nonlinear Analysis, Pure and Applied Mathematics, John Wiley & Sons, New York, NY, USA (1984)

10. Luo, Q: Ky Fan's section theorem and its applications in topological ordered spaces. Applied Mathematics Letters. 17(10), 1113–1119 (2004). Publisher Full Text