Abstract

In a topological sup-semilattice, we established a new existence result for vector quasiequilibrium problems. By the analysis of essential stabilities of maximal elements in a topological sup-semilattice, we prove that for solutions of each vector quasi-equilibrium problem, there exists a connected minimal essential set which can resist the perturbation of the vector quasi-equilibrium problem.

1. Introduction

Vector equilibrium problems can unify many nonlinear problems such as vector optimization, vector variational inequality [1], and vector complementarity problems [2]. Recently, not only vector equilibrium problems [37] but also vector quasiequilibrium problems [814] and the system of vector quasi-equilibrium problems have attracted much attention [1519].

Topological vector spaces provide the usual mathematical framework in the study of many problems. To avoid the linear feature, sup-semilattices may be good choices. In fact, some results like the existence of KKM points in topological spaces were established in topological sup-semilattices [20], where a two-tuple is said to be a sup-semilattice, if is a partially ordered set with the partial ordering , in which every pair has a least upper bound .

The aim of this paper is to study the existence and essential stability of vector quasi-equilibrium problems in topological sup-semilattices. In order to achieve this, firstly, we give a stability result in relation to maximal elements in a topological sup-semilattice. Secondly, a new existence result for vector quasi-equilibrium problems is established, and we show that each vector quasi-equilibrium problem has a connected minimal essential set in its solution set.

2. Preliminaries

Let be a sup-semilattice. If and are two elements in and , the set is called an order interval. Let , be two nonempty finite subsets of . Then the set is well defined and has the properties: and if .

Definition 1 (see [20]). A subset is -convex, if for any nonempty finite subset , we have . being a -convex set is equivalent to the following conditions:(a)if , , then its least upper bound .(b)if , and , then the order interval .

It is easy to check that the intersection of two -convex sets is -convex as well.

A topological space is said to be a topological sup-semilattice if is equipped with a sup-semilattice as its partial ordering denoted by , for which with is a continuous function.

Let be a topological vector space and the zero element in . A subset is called a cone if, for any and real number . A cone is convex if is a convex set. If , it is called a pointed cone.

Definition 2 (see [18]). Let be a topological sup-semilattice, a topological vector space with a cone , a vector-valued function.(a) is -quasiconcave if, for any nonempty two points subset and , .(b) is said to be -quasiconcave-like if, for any , or .

Remark 3. In general cases, -quasiconcave, -quasiconcave-like, and usual quasiconcave functions are independent of each other. See examples in [18]. Let , . Then the partial order on is “” (less than or equal to);, hence, the -quasiconcave, -quasiconcave-like, and usual quasiconcave property of coincide (the usual quasiconcave function means that for any , , , and , ).

Lemma 4 (see [18]). Let be a topological sup-semilattice, a Hausdorff locally convex topological vector space with a closed, convex, and pointed cone . If the vector-valued function is -quasiconcave or -quasiconcave-like, then the set is -convex.

Now we introduce the vector quasiequilibrium problem (VQEP) that we will consider in this paper.

Let be a topological sup-semilattice and a topological vector space. is a closed, convex, and pointed cone with . is a vector-valued function, and is a multivalued mapping on . The vector quasi-equilibrium problem with is to find , such that

Let , ; then the VQEP is just a vector equilibrium problem (VEP). That is to find , such that

Definition 5 (see [21, 22]). A vector-valued function is said to be C-continuous on if, for each and any open neighborhood of in , there exists an open neighborhood of in such that

Remark 6. For a function , is - on if and only if is lower semicontinuous on .

A maximal element version of the Browder fixed point theorem in a topological sup-semilattice can be found in [18]. We limit it in a metric space as the following lemma.

Lemma 7 (see [18]). Let be a compact sup-semilattice with path connected interval, where is the metric on , a multivalued map on with the conditions: (i), is -convex; (ii) is open in ; (iii), . Then there exists an , such that .

Remark 8. The existence of a metric space with a sup-semilattice can be guaranteed. For instance, let , , if means that , then , where . Clearly, with the usual Euclidean metric is a topological sup-semilattice.

Let denote the collection of satisfying all the conditions of Lemma 7. For any , define the metric between and as where is the Hausdorff metric induced by . For each and each , since , we have , that is, . Noting that is closed, the metric on is well defined. Then is a metric space.

For each , denote by the set of all maximal elements of . Then defines a multivalued mapping from to and .

Definition 9. For each , a set is called an essential set of if it satisfies the following conditions:(1) is closed subset of .(2)For any open set , there exists an open neighborhood of such that , for any .
A set is called a minimal essential set of if it is a minimal element of all essential sets ordered by set inclusion in . A connected component in is called an essential component, if it includes at least one minimal essential set of .

We recall some notions about multi-valued mappings. Let be a multi-valued mapping, where , are two topological vector spaces. Then (i) is said to be upper semicontinuous at , if for each open set , there exists an open neighborhood of such that for any . (ii) is lower semi-continuous at , if for each open set , there exists an open neighborhood of such that for any .

Remark 10. For each , a set is essential if is lower semi-continuous at . If is upper semi-continuous at , then itself is an essential set. For any two closed sets , with , if is essential, then is also essential. For each and each , if is open, then is closed; hence, is closed because we have ; consequently, is compact.

Lemma 11 (see [23]). Let be a metric space, and two nonempty compact subsets of , , and two nonempty disjoint open subsets of . If , then , where is the Hausdorff metric defined on .

3. The Stability of Maximal Elements on Topological Semilattices

Theorem 12. is an upper semi-continuous mapping with compact values.

Proof. For each , by Remark 10, is compact. Suppose that is not upper semi-continuous. Then there is a , an open set with and , such that and , . That is, there exists a point such that . Without loss of generality, we may assume that . Since , it holds that , . Since , we have , . As gets close to infinity, we can obtain that , , that is, . This results in the fact that while is large enough, a contradiction with . Therefore, is definitely upper semi-continuous.

Theorem 13. For each , there exists at least a minimal essential set of . If is a minimal essential set of , then is connected.

Proof. For the existence, by Remark 10, each decreasing chain, consisting of essential subsets of , has a minimal element, which is the intersection of the chain. By the Zorn’s lemma, the minimal element is just a minimal essential set. For the connectedness, by way of contradiction, suppose that is not connected. There exist two disjoint closed sets , such that .
Since is not essential, there is an open set with such that for any , there exists a with and , . Clearly, is compact, then there is an open set with , , such that . For , because is essential, there is a number , such that for each satisfying . Therefore, we can select a such that and , . Then .
Define a multi-valued mapping as
We show that .(a)For each , since and , we have ;(b)for each , because and are -convex sets, it follows that is -convex;(c)for each , we have Noting that , , and are open sets, it follows that is open.
Through a direct calculation, can be written as Take any . Note that if , then ; if , then ; if , then . Consequently, we can obtain that if , then Therefore, we have Since , by Lemma 11, we have That is, . This results in the fact that Consequently, we have . If there is a point , then and , , that is, and which contradicts with , . Therefore, is connected.

4. The Existence and Stability of Solutions for VQEP

This section gives an existence result in relation to VQEP in topological sup-semilattices and induces the existence of minimal essentially stable sets for each VQEP in the set of its solutions.

Theorem 14. Let be a VQEP, where is a compact topological sup-semilattice with path connected intervals, is a Hausdorff locally convex topological vector space, and is a multi-valued mapping with nonempty and -convex values. If the VQEP satisfies that(i), ;(ii), is -continuous;(iii), is -quasiconcave-like or -quasiconcave;(iv), is open in ,(v) is closed in ,
then the VQEP has a solution.

Proof. Denote by . Let such that , . Define Then for each , if , we have , by the condition (i), , hence, ; if , from the definition of , we have .
Since is -quasiconcave-like or -quasiconcave, by Lemma 4, we have that is -convex. Then is a -convex set, noting that has -convex values, we have that is also -convex.
For each , we can check that Take a point , since is open, there is an open set such that , then, by the condition (ii), there exists an open neighborhood in such that for all , That is, ; hence, is open. Noting that and are open sets in . We can obtain that is also open in .
Thus, there is an such that by Lemma 7. If , then , a contradiction to the fact that has nonempty values. Therefore, and , that is, , , .

By Theorem 14 and its proof, we can also obtain the existence result for VQEP as the following.

Corollary 15. Let be a VQEP, where is a compact topological sup-semilattice with path connected intervals, is a Hausdorff topological vector space, and is a multi-valued mapping with nonempty and -convex values. If the VQEP satisfies that(i), ;(ii), is open in ;(iii), is -convex;(iv), is open in ,(v) is closed in ,
then the VQEP has a solution.

By Theorem 14, for the special case of VQEP without the feasible mapping , we can obtain the existence result concerning VEP as the following.

Corollary 16. Let be a vector equilibrium problem, where is a compact topological sup-semilattice with path connected intervals, is a Hausdorff locally convex topological vector space. If the VEP satisfies the following conditions:(i), is -quasiconcave or -quasiconcave-like;(ii), is -continuous;(iii), ,
then this VEP has a solution.

Example 17. Let , . The is a sup-semilattice, in which means that , .
  (a) For any , , the function is defined as
It can be easily checked that for each , is quasiconcave and quasiconcave-like but not a usual quasiconcave function.
Denote by the set . For each , the multi-valued mapping satisfies that Note that is not a usual convex but a -convex multi-valued mapping. For each , if , then ; if , then . Thus, is open in for each . Then and satisfy all the conditions in Theorem 14. We can find that is the unique solution for the VQEP, .
   (b) For any , , let , where is the -neighborhood of in , and is the same as the setting in (a). Then the function and the mappings and meet all the conditions in Theorem 14. The set of solutions for the VEP is the overall which is also the set of solutions for . The solution of is just one point .
To study the stability of vector quasi-equilibrium problems, let be a metric space and define the set as
For each , by the proof of Theorem 14, we can find that a point is a solution of if and only if is a maximal element of defined in the proof. Let denote all the solutions of . Then is a multi-valued mapping from to . For any two , , define the metric between and as where and are multi-valued mappings corresponding to and in the proof of Theorem 14. Then is a metric space. Instead of , and by , and in Definition 9, we can also define essential sets , minimal essential sets of , and essential component in . If an essential set is singleton set , is called an essential solution of .

From Theorems 12 and 13, we have the following results.

Theorem 18. is an upper semi-continuous mapping with compact values. For each VQEP , there exists at least a connected minimal essential set of .

Remark 19. For each , , let
For any , from the definition of the metric between and , then which gives an overall consideration of and . If and are two VEP, then
For the essential stability of solutions for VQEP, clearly, the class of perturbations induced by the metric is different from the perturbation of uniform topology in [3, 14] and also different from the perturbation of best response defined in [16]. For example, the existence of essential sets of solutions for VQEP in topological vector spaces is proved in [3], and the uniform metric for two VQEP and is defined as where is a compact convex subset of a Banach space. Naturally, the feasible mapping requires closed values, which is not a requirement in Theorem 14, however, where each inverse image being open is necessary.

By Theorem 18, each connected component including a connected minimal essential set of solutions is essential; that is, the existence of essential components can be induced.

Corollary 20. Let . There is an essential component in . If is a singleton, then is an essential solution of .

Acknowledgments

This project is supported by Guangxi Natural Science Foundation (2012GXNSFBA053013, 2013GXNSFBA19004), NNSF (61164020), and Doctoral Research Fund of GUT.