- Research Article
- Open Access
On the Existence Result for System of Generalized Strong Vector Quasiequilibrium Problems
© S. Plubtieng and K. Sitthithakerngkiet 2011
- Received: 3 December 2010
- Accepted: 12 January 2011
- Published: 19 January 2011
We introduce a new type of the system of generalized strong vector quasiequilibrium problems with set-valued mappings in real locally convex Hausdorff topological vector spaces. We establish an existence theorem by using Kakutani-Fan-Glicksberg fixed-point theorem and discuss the closedness of strong solution set for the system of generalized strong vector quasiequilibrium problem. The results presented in the paper improve and extend the main results of Long et al. (2008).
- Equilibrium Problem
- Strong Solution
- Topological Vector Space
- Vector Variational Inequality
- Vector Equilibrium Problem
The equilibrium problem is a generalization of classical variational inequalities. This problem contains many important problems as special cases, for instance, optimization, Nash equilibrium, complementarity, and fixed-point problems (see [1–3] and the references therein). Recently, there has been an increasing interest in the study of vector equilibrium problems. Many results on existence of solutions for vector variational inequalities and vector equilibrium problems have been established (see, e.g., [4–16]).
If is nonempty, and satisfies (1.1), then we call a weak efficient solution for (VEP), and if satisfies (1.2), then we call a strong solution for VEP. Moreover, they also proved an existence theorem for a strong vector equilibrium problem (1.2) (see ).
In 2000, Ansari et al.  introduced the system of vector equilibrium problems (SVEPs), that is, a family of equilibrium problems for vector-valued bifunctions defined on a product set, with applications in vector optimization problems and Nash equilibrium problem  for vector-valued functions. The (SVEP) contains system of equilibrium problems, systems of vector variational inequalities, system of vector variational-like inequalities, system of optimization problems and the Nash equilibrium problem for vector-valued functions as special cases. But, by using (SVEP), we cannot establish the existence of a solution to the Debreu type equilibrium problem  for vector-valued functions which extends the classical concept of Nash equilibrium problem for a noncooperative game. Moreover, Ansari et al.  introduced the following concept of system of vector quasiequilibrium problems.
If for all , then (SVQEP) reduces to (SVEP) (see ) and if the index set is singleton, then (SVQEP) becomes the vector quasiequilibrium problem. Many authors studied the existence of solutions for systems of (vector) quasiequilibrium problems, see, for example, [19–23] and references therein.
On the other hand, it is well known that a strong solution of vector equilibrium problem is an ideal solution, It is better than other solutions such as efficient solution, weak efficient solution, proper efficient solution and supper efficient solution (see ). Thus, it is important to study the existence of strong solution and properties of the strong solution set. In general, the ideal solutions do not exist.
Moreover, they gave an existence theorem for a generalized strong vector quasiequilibrium problem without assuming that the dual of the ordering cone has a weak* compact base.
We call this a strong solution for the (SGSVQEP).
At a quick glance, our required solution seems to be similar to such a thing of Ansari et al. [5, 18], in the case of and . In fact, however, the main different point comes from the independent choice of coordinate. In this paper, we establish an existence theorem of strong solution set for the system of generalized strong vector quasiequilibrium problem by using Kakutani-Fan-Glicksberg fixed-point theorem and discuss the closedness of the solution set. Moreover, we apply our result to obtain the result of Long et al. .
Throughout this paper,we suppose that , , and are real locally convex Hausdorff topological vector spaces, and are nonempty compact convex subsets, and is a nonempty closed convex cone. We also suppose that , , and are set-valued mappings.
Let and be two topological vector spaces and a nonempty subset of and let be a set-valued mapping.
Let and be two topological vector spaces, and be a set-valued mapping.
(i) is said to be upper semicontinuous at if, for any open set containing , there exists an open set containing such that, for all , ; is said to be upper semicontinuous on if it is upper semicontinuous at all .
(ii) is said to be lower semicontinuous at if, for any open set with , there exists an open set containing such that, for all , ; is said to be lower semicontinuous on if it is lower semicontinuous at all .
(iii) is said to be continuous on if, it is at the same time upper semicontinuous and lower semicontinuous on .
(iv) is said to be closed if the graph, , of , that is, , is a closed set in .
Lemma 2.4 (see ).
Let be a nonempty compact subset of locally convex Hausdorff vector topology space . If is upper semicontinuous and for any , is nonempty, convex and closed, then there exists an such that .
Lemma 2.5 (see ).
Let and be two Hausdorff topological vector spaces and be a set-valued mapping. Then, the following properties hold:
(i)if is closed and is compact, then is upper semicontinuous, where and denotes the closure of the set ,
(ii)if is upper semicontinuous and for any , is closed, then is closed,
(iii) is lower semicontinuous at if and only if for any and any net , , there exists a net such that and .
In this section, we apply Kakutani-Fan-Glicksberg fixed-point theorem to prove an existence theorem of strong solutions for the system of generalized strong vector quasiequilibrium problem. Moreover, we also prove the closedness of strong solution set for the system of generalized strong vector quasiequilibrium problem.
For each , let be continuous set-valued mappings such that for any , are nonempty closed convex subsets of . Let be upper semi continuous set-valued mappings such that for any are nonempty closed convex subsets of and be set-valued mappings satisfy the following conditions:
(i)for all , ,
(ii)for all , are properly -quasiconvex,
(iii) are upper -continuous,
(iv)for all , are lower -continuous.
Then, SGSVQEP has a solution. Moreover, the set of all strong solutions is closed.
Show that and are nonempty.
For any , we note that and are nonempty. Thus, for any , we have and are nonempty.
Show that and are convex subsets of .
which contradicts to . Therefore and hence is a convex subset of . Similarly, we have is convex subset of .
Show that and are closed subsets of .
We claim that . Assume that there exists and . Thus, we note that and is closed. Hence is open and . Since is a locally convex space, there exists a neighbourhood of the origin such that is convex and . This implies that , that is, , which is a contradiction. Therefore . This mean that and so is a closed subset of . Similarly, we have is a closed subset of .
Show that and are upper semicontinuous.
which contradicts to . Hence and, therefore, is a closed mapping. Since is a compact set and is a closed subset of , we note that is compact. Then, is also compact. Hence, by Lemma 2.5(i), is an upper semicontinuous mapping. Similarly, we note that is an upper semicontinuous mapping.
Show that SGSVQEP has a solution.
Then, and are upper semicontinuous and, for all , , and are nonempty closed convex subsets of .
Hence SGSVQEP has a solution.
Show that the set of solutions of SGSVQEP is closed.
This means that belongs to the set of solutions of SGSVQEP. Hence the set of solutions of SGSVQEP is closed set. This completes the proof.
If we take , , and . Then, from Theorem 3.1, we derive the following result.
Let be a continuous set-valued mapping such that for any , is nonempty closed convex subset of . Let be an upper semicontinuous set-valued mapping such that for any , is a nonempty closed convex subset of and be set-valued mapping satisfy the following conditions:
(i)for all , ,
(ii)for all , is properly -quasiconvex,
(iii) is an upper -continuous,
(iv)for all , is a lower -continuous,
(v)if and then .
Then, GSVQEP has a solution. Moreover, the set of all solution of GSVQEP is closed.
Now we give an example to explain that Theorem 3.1 is applicable.
The authors would like to thank the referees for the insightful comments and suggestions. S. Plubtieng the Thailand Research Fund for financial support under Grants no. BRG5280016. Moreover, K. Sitthithakerngkiet would like to thanks the Office of the Higher Education Commission, Thailand for supporting by grant fund under Grant no. CHE-Ph.D-SW-RG/41/2550, Thailand.
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