Solving the Set Equilibrium Problems
© Y.-C. Lin and H.-J. Chen. 2011
Received: 17 September 2010
Accepted: 21 November 2010
Published: 2 December 2010
We study the weak solutions and strong solutions of set equilibrium problems in real Hausdorff topological vector space settings. Several new results of existence for the weak solutions and strong solutions of set equilibrium problems are derived. The new results extend and modify various existence theorems for similar problems.
1. Introduction and Preliminaries
Let , , be arbitrary real Hausdorff topological vector spaces, let be a nonempty closed convex set of , and let be a proper closed convex and pointed cone with apex at the origin and , that is, is proper closed with and satisfies the following conditions:
for all . It has been studied by Chen and Craven .
for all . The vector variational inequalities problem was first introduced by Giannessi  in finite-dimensional Euclidean space.
Summing up the above arguments, they show that for a suitable choice of the mapping and the spaces , , and , we can obtain a number of known classes of vector equilibrium problems, vector variational inequalities, and implicit generalized variational inequalities. It is also well known that variational inequality and its variants enable us to study many important problems arising in mathematical, mechanics, operations research, engineering sciences, and so forth.
In this paper we aim to derive some solvabilities for the set equilibrium problems. We also study some results of existence for the weak solutions and strong solutions of set equilibrium problems. Let be a nonempty subset of a topological vector space . A set-valued function from into the family of subsets of is a KKM mapping if for any nonempty finite set , the convex hull of is contained in . Let us first recall the following results.
Fan's Lemma (see ).
Definition 1.1 (see ).
The following definition can also be found in .
(3)continuous if it is both upper semicontinuous and lower semicontinuous.
We note that is lower semicontinuous at if for any net , , implies that there exists net such that . For other net-terminology properties about these two mappings, one can refer to .
Lemma 1.4 (see ).
is upper semicontinuous with nonempty compact values.
By using similar technique of [11, Proposition 2.1], we can deduce the following lemma that slight-generalized the original one.
2. Existence Theorems for Set Equilibrium Problems
Now, we state and show our main results of solvabilities for set equilibrium problems.
Let , , be real Hausdorff topological vector spaces, let be a nonempty closed convex subset of , and let be a proper closed convex and pointed cone with apex at the origin and . Given mappings , , and , suppose that
Then all conditions of Theorems 2.1 and 2.3 are satisfied. By Theorems 2.1 and 2.3, respectively, the (SEP) I not only has a weak solution, but also has a strong solution. A simple geometric discussion tells us that is a strong solution for (SEP) I .
Let , , , , , , be as in Theorem 2.1. Assume that the mapping is -convex on for each and , such that is continuous with nonempty compact values for each , and is upper semicontinuous with nonempty compact values. Suppose that
Using the technique of the proof in Theorem 2.2 and applying Theorem 2.6, we have the conclusion.
Using the technique of the proof in Theorem 2.3, we have the following result.
Using a similar argument to that of the proof in Theorem 2.10 and combining Theorem 2.6 and Corollary 2.7, respectively, we have the following two results of existence for the strong solution of (SEP) I .
In order to illustrate Theorems 2.10 and 2.12 more precisely, we provide the following concrete example.
We claim that condition () holds. Indeed, We know that the weak solution . For each , if we choose any , then and . Hence condition () and all other conditions of Theorems 2.10 and 2.12 are satisfied. By Theorems 2.10 and 2.12, respectively, the (SEP) I not only has a weak solution, but also has a strong solution. We can see that is a strong solution for (SEP) I .
The authors would like to thank the referees whose remarks helped improving the paper. This work was partially supported by Grant no. 98-Edu-Project7-B-55 of Ministry of Education of Taiwan (Republic of China) and Grant no. NSC98-2115-M-039-001- of the National Science Council of Taiwan (Republic of China) that are gratefully acknowledged.
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