Existence Results for New Weak and Strong Mixed Vector Equilibrium Problems on Noncompact Domain

Abstract

We introduce and consider two new mixed vector equilibrium problems, that is, a new weak mixed vector equilibrium problem and a new strong mixed vector equilibrium problem which are combinations of certain vector equilibrium problems, and vector variational inequality problems. We prove existence results for the problems in noncompact setting.

1. Introduction

There are several problems of applied and substantial interest in optimization, economics, and engineering that are related to equilibrium in their nature. The equilibrium problem was introduced and studied by [1] as a generalization of variational inequality problem. It has been shown that the equilibrium problem provides a natural, novel, and unified framework to study a wide class of problems arising in nonlinear analysis, optimization, economics, finance, and game theory. The equilibrium problem includes many mathematical problems as particular cases such as mathematical programming problems, complementarity problems, variational inequality problems, fixed point problems, minimax inequality problems, and Nash equilibrium problems in noncooperative games; see [1–4].

Let be a Hausdorff topological vector space, let be a subset of , and let be a mapping with . The classical, scalar-valued equilibrium problem deals with the existence of such that Moreover, in the case of vector valued mappings, let be another Hausdorff topological vector space, , a convex cone with nonempty interior. Given a vector mapping , then the problem of finding such that is called weak equilibrium problem and the point is called weak equilibrium point, where denotes the interior of the cone in . In 2014, Rahaman and Ahmad [5] considered two types of mixed vector equilibrium problems which were combinations of a vector equilibrium problem and a vector variational inequality problem. Remark that is a pointed closed convex cone with nonempty interior; that is, . The partial ordering induced by on is denoted by and is defined by if and only if . Let and be two mappings, where is the space of all linear continuous mappings from to . Here denotes the evaluation of the linear mapping at . They considered the following two problems.

Find such that

It is clear that the solution set of (4) is a subset of the solution set of (3). Also if we consider and , then and the solution set of (3) is always the whole set . They called problem (3) weak mixed vector equilibrium problem and problem (4) strong mixed vector equilibrium problem. Problems (3) and (4) are unified models of several known problems used in applied sciences, for instance, vector variational inequality problem, vector complementarity problem, vector optimization problem, and vector saddle point problem; see, for example, [3, 6–10] and references therein.

With the inspiration from the notice of some characteristics of the mappings of the original problem, we are interested and motivated in the development of the existing problems to the new weak mixed vector equilibrium problem and the new strong mixed vector equilibrium as follows.

Find such that where is a bifunction, , and . For a more comprehensive bibliography on vector equilibrium problems, vector variational inequality problems, and their generalizations, we refer to volume edited by [3]. Our results generalize the results obtained by [1] and therefore the results of Fan [11] for vector valued mappings. For more details, we refer to [6, 12, 13]. As the underlying set is noncompact, therefore we use only a very weak coercivity condition, that is, coercing family.

2. Preliminaries

The following definitions and results are needed in the sequel. Let and be two Hausdorff topological vector spaces, a subset of , and a pointed convex cone of .

Definition 1. Let be a mapping. Then is said to be -convex, if for all and which implies that

Definition 2. Let be a mapping.(i)is said to be lower semicontinuous with respect to at a point , if for any neighborhood of in there exists a neighborhood of such that ;(ii) is said to be upper semicontinuous with respect to at a point , if ;(iii) is said to be continuous with respect to at a point , if it is lower semicontinuous and upper semicontinuous with respect to at that point.

Remark 3. If is lower semicontinuous (upper semicontinuous or continuous, resp.) with respect to at any point of , then is lower semicontinuous (upper semicontinuous or continuous, resp.) with respect to on .

Definition 4. A mapping is said to be -monotone, if for all

Lemma 5 (see [10]). If is a lower semicontinuous mapping with respect to , then the set is closed in .

Lemma 6 (see [14]). Let be an ordered topological vector space with a pointed closed convex cone . Then for all we have that(i) and imply ;(ii) and imply ;(iii) and imply ;(iv) and imply .

Definition 7 (see [15]). Consider a subset of a topological vector space and a topological space A family of pair of sets is said to be coercing for a multivalued mapping if and only if(i)for each , is contained in a compact convex subset of and is a compact subset of ;(ii)for each there exists such that ;(iii)for each there exists with .

Definition 8. Let be a nonempty convex subset of a topological vector space A multivalued mapping is said to be KKM mapping, if, for every finite subset of , where denotes the convex hull of .

Theorem 9 (see [15]). Let be a Hausdorff topological vector space, a convex subset of , a nonempty subset of , and a KKM mapping with compactly closed values in (i.e., for all , is closed for every compact set of ). If admits a coercing family, then

: we say that the cone satisfies , if there is a pointed convex closed cone such that .

3. Main Results

In this section, we prove the following existence results for new weak and strong mixed vector equilibrium problems (5) and (6) for noncompact domain.

Theorem 10. Let be a nonempty closed convex subset of a Hausdorff topological vector space , a Hausdorff topological vector space, and a closed convex pointed cone in with . Let , , , and be four mappings satisfying the following conditions:(i) and are -monotone.(ii), and for all .(iii)For any fixed , and are upper semicontinuous with respect to at .(iv)For any fixed , are -convex, lower semicontinuous with respect to on .(v) and are upper semicontinuous with respect to with nonempty closed values.(vi)There exists a family satisfying conditions and of Definition 7 and the following condition: For each , there exists such that Then, there exists a point such that

For the proof of Theorem 10, we need the following proposition, for which the assumptions remain the same as in Theorem 10.

Proposition 11. The following two problems are equivalent:(i)Find such that ; .(ii)Find such that ; .

Proof. Suppose (i) holds. Then, for fixed , set , for . It is clear that for all and hence Since and is -convex, we have On the other hand, the convexity of in the second variable implies that Also, Combining (16), (17), and (18) we obtain for all . It is not hard to see that (19) is equivalent to By using (15) and (20) and (ii) of Lemma 6, we have By condition (iii) of Theorem 10 as and are upper semicontinuous with respect to at , therefore from (21) we have and hence (ii) holds.
Conversely, we assume that (ii) holds. In order to prove (i), on the contrary suppose that there exists a point such that for some .
On the other hand, since and are -monotone, we have for some and for some . Combining (23), (24), and (25), we have which contradicts assumption (ii). Therefore (i) holds.

Now, we are able to prove Theorem 10 which has the following details.

Proof. For each , consider the set By Lemma 5, is closed in and hence has compactly closed values in .
Now, we show that is a KKM map. For this, let be a finite subset of and .
We claim that On the contrary, suppose that . As , we have with and . This follows that Since is convex, therefore Since is -convex and -monotone, we have On the other hand, the convexity of in the second variable and -monotone imply that Furthermore, Combining (31), (32), and (33), we have From (31) and (34), we conclude that which is a contradiction. This follows that and hence . Thus, is a KKM mapping. From assumption , we can see that the family satisfies the condition which is, for all , there exists such that and therefore it is a coercing family for . We deduce that satisfies all the hypothesis of Theorem 9. Therefore, we have Hence, there exists such that for any Now applying Proposition 11, we obtain that there exists such that for all Hence problem (5) admits a solution. This completes the proof.

Corollary 12. Let , , , , and satisfy all the assumptions of Theorem 10. In addition, if satisfies Condition, then problem (6) is solvable; that is, there exists such that for any

Proof. Suppose that satisfies . Then there is a pointed convex and closed cone in such that . Therefore, it is not hard to see that , , , , and satisfy all the assumptions of Theorem 10. It follows from Theorem 10 that Since , (41) yields the fact that there exists such that Therefore, problem (6) admits a solution. This completes the proof.

In the case of and , we obtain the following corollaries.

Corollary 13 (see [5]). Let be a nonempty closed convex subset of a Hausdorff topological vector space , a Hausdorff topological vector space, and a closed convex pointed cone in with . Let and be two mappings satisfying the following conditions:(i) is -monotone.(ii), .(iii)For any fixed , is upper semicontinuous with respect to at .(iv)For any fixed , are -convex, lower semicontinuous with respect to on .(v) is upper semicontinuous with respect to with nonempty closed values.(vi)There exists a family satisfying conditions and of Definition 7 and the following condition: For each , there exists such that Then, there exists a point such that

Corollary 14 (see [5]). Let , and satisfy all the assumptions of Corollary 13. In addition, if satisfies Condition, then problem (4) is solvable; that is, there exists such that for any

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank Professor Dr. Tomonari Suzuki and two anonymous reviewers for their valuable comments and suggestions, which were helpful in improving the paper. The second author would like to thank Naresuan University. This research is supported by Naresuan University and the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.