# Solving the Set Equilibrium Problems

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## Abstract

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.

### Keywords

Weak Solution Strong Solution Lower Semicontinuous Nonempty Closed Convex Subset Vector Equilibrium Problem## 1. Introduction and Preliminaries

Let Open image in new window , Open image in new window , Open image in new window be arbitrary real Hausdorff topological vector spaces, let Open image in new window be a nonempty closed convex set of Open image in new window , and let Open image in new window be a proper closed convex and pointed cone with apex at the origin and Open image in new window , that is, Open image in new window is proper closed with Open image in new window and satisfies the following conditions:

(1) Open image in new window , for all Open image in new window ;

(2) Open image in new window ;

(3) Open image in new window .

Letting Open image in new window , Open image in new window be two sets of Open image in new window , we can define relations " Open image in new window " and " Open image in new window " as follows:

(1) Open image in new window Open image in new window Open image in new window ;

(2) Open image in new window Open image in new window Open image in new window .

Similarly, we can define the relations " Open image in new window " and " Open image in new window " if we replace the set Open image in new window by Open image in new window .

_{ I }is to find an Open image in new window such that

_{ I }. We note that (1.1) is equivalent to the following one:

for all Open image in new window and for some Open image in new window .

_{ I }. We also note that (1.3) is equivalent to the following one:

for all Open image in new window .

_{ I }reduces to the vector equilibrium problem (VEP), which is to find Open image in new window such that

for all Open image in new window . Existence of a solution of this problem is investigated by Ansari et al. [1, 2].

_{ I }reduces to (GVVIP): to find Open image in new window and Open image in new window such that

for all Open image in new window . It has been studied by Chen and Craven [3].

_{ I }reduces to the (GVVIP) which is discussed by Huang and Fang [4] and Zeng and Yao [5]: to find a vector Open image in new window and Open image in new window such that

_{ I }reduces to the (weak) vector variational inequalities problem which is considered by Fang and Huang [6], Chiang and Yao [7], and Chiang [8] as follows: to find a vector Open image in new window such that

for all Open image in new window . The vector variational inequalities problem was first introduced by Giannessi [9] in finite-dimensional Euclidean space.

Summing up the above arguments, they show that for a suitable choice of the mapping Open image in new window and the spaces Open image in new window , Open image in new window , and Open image in new window , 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 Open image in new window be a nonempty subset of a topological vector space Open image in new window . A set-valued function Open image in new window from Open image in new window into the family of subsets of Open image in new window is a KKM mapping if for any nonempty finite set Open image in new window , the convex hull of Open image in new window is contained in Open image in new window . Let us first recall the following results.

Fan's Lemma (see [10]).

Let Open image in new window be a nonempty subset of Hausdorff topological vector space Open image in new window . Let Open image in new window be a KKM mapping such that for any Open image in new window , Open image in new window is closed and Open image in new window is compact for some Open image in new window . Then there exists Open image in new window such that Open image in new window for all Open image in new window .

Definition 1.1 (see [11]).

Let Open image in new window be a vector space, let Open image in new window be a topological vector space, let Open image in new window be a nonempty convex subset of Open image in new window , and let Open image in new window be a proper closed convex and pointed cone with apex at the origin and Open image in new window , and Open image in new window is said to be

(1) Open image in new window *-convex* if Open image in new window for every Open image in new window and Open image in new window ;

(2)*naturally quasi* Open image in new window *-convex* if Open image in new window for every Open image in new window and Open image in new window .

The following definition can also be found in [11].

Definition 1.2.

Let Open image in new window be a Hausdorff topological vector space, let Open image in new window be a proper closed convex and pointed cone with apex at the origin and Open image in new window , and let Open image in new window be a nonempty subset of Open image in new window . Then

(1)a point Open image in new window is called a *minimal point* of Open image in new window if Open image in new window ; Open image in new window is the set of all minimal points of Open image in new window ;

(2)a point Open image in new window is called a *maximal point* of Open image in new window if Open image in new window ; Open image in new window is the set of all maximal points of Open image in new window ;

(3)a point Open image in new window is called a *weakly minimal point* of Open image in new window if Open image in new window ; Open image in new window is the set of all weakly minimal points of Open image in new window ;

(4)a point Open image in new window is called a *weakly maximal point* of Open image in new window if Open image in new window ; Open image in new window is the set of all weakly maximal points of Open image in new window .

Definition 1.3.

Let Open image in new window , Open image in new window be two topological spaces. A mapping Open image in new window is said to be

(1)upper semicontinuous if for every Open image in new window and every open set Open image in new window in Open image in new window with Open image in new window , there exists a neighborhood Open image in new window of Open image in new window such that Open image in new window ;

(2)lower semicontinuous if for every Open image in new window and every open neighborhood Open image in new window of every Open image in new window , there exists a neighborhood Open image in new window of Open image in new window such that Open image in new window for all Open image in new window ;

(3)continuous if it is both upper semicontinuous and lower semicontinuous.

We note that Open image in new window is lower semicontinuous at Open image in new window if for any net Open image in new window , Open image in new window , Open image in new window implies that there exists net Open image in new window such that Open image in new window . For other net-terminology properties about these two mappings, one can refer to [12].

Lemma 1.4 (see [13]).

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.

Lemma 1.5.

## 2. Existence Theorems for Set Equilibrium Problems

Now, we state and show our main results of solvabilities for set equilibrium problems.

Theorem 2.1.

Let Open image in new window , Open image in new window , Open image in new window be real Hausdorff topological vector spaces, let Open image in new window be a nonempty closed convex subset of Open image in new window , and let Open image in new window be a proper closed convex and pointed cone with apex at the origin and Open image in new window . Given mappings Open image in new window , Open image in new window , and Open image in new window , suppose that

(1) Open image in new window for all Open image in new window ;

(3)for each Open image in new window , the set Open image in new window is convex;

- (5)
for each Open image in new window , the set Open image in new window is open in Open image in new window .

_{I}. That is, there is an Open image in new window such that

for all Open image in new window and for some Open image in new window .

Proof.

for all Open image in new window . From condition (5) we know that for each Open image in new window , the set Open image in new window is closed in Open image in new window , and hence it is compact in Open image in new window because of the compactness of Open image in new window .

_{ I }, for any given nonempty finite subset Open image in new window of Open image in new window . Let Open image in new window , the convex hull of Open image in new window . Then Open image in new window is a compact convex subset of Open image in new window . Define the mappings Open image in new window , respectively, by

Hence Open image in new window , and then Open image in new window for all Open image in new window .

and hence Open image in new window which contradicts (2.6). Hence Open image in new window is a KKM mapping, and so is Open image in new window . Therefore, there exists an Open image in new window which is a solution of (SEP)_{ I }. This completes the proof.

Theorem 2.2.

_{I}. That is, there is an Open image in new window such that

for all Open image in new window and for some Open image in new window .

Proof.

is open in Open image in new window . Then all conditions of Theorem 2.1 hold. From Theorem 2.1, (SEP)_{ I } has a solution.

In order to discuss the results of existence for the strong solution of (SEP)_{ I }, we introduce the condition (). It is obviously fulfilled that if Open image in new window , Open image in new window is single-valued function.

Theorem 2.3.

_{I}with Open image in new window . In addition, if Open image in new window , Open image in new window , and Open image in new window is compact, Open image in new window is convex, the mapping Open image in new window is continuous with nonempty compact valued on Open image in new window , the mapping Open image in new window is naturally quasi Open image in new window -convex on Open image in new window for each Open image in new window , and the mapping Open image in new window is Open image in new window -convex on Open image in new window for each Open image in new window . Assuming that for each Open image in new window , there exists Open image in new window such that

_{I}; that is, there exists Open image in new window such that

for all Open image in new window . Furthermore, the set of all strong solutions of (SEP)_{I} is compact.

Proof.

From Theorem 2.2, we know that Open image in new window such that (1.1) holds for all Open image in new window and for some Open image in new window . Then we have Open image in new window .

for all Open image in new window . Such an Open image in new window is a strong solution of (SEP)_{ I }.

_{ I }is compact, it is sufficient to show that the solution set is closed due to the coercivity condition (4) of Theorem 2.2. To this end, let Open image in new window denote the solution set of (SEP)

_{ I }. Suppose that net Open image in new window which converges to some Open image in new window . Fix any Open image in new window . For each Open image in new window , there is an Open image in new window such that

Hence Open image in new window and Open image in new window is closed.

We would like to point out that condition () is fulfilled if we take Open image in new window and Open image in new window is a single-valued function. The following is a concrete example for both Theorems 2.1 and 2.3.

Example 2.4.

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 Open image in new window is a strong solution for (SEP)_{ I }.

Corollary 2.5.

_{I}with Open image in new window . In addition, if Open image in new window and Open image in new window , Open image in new window is compact, Open image in new window is convex, Open image in new window -convex on Open image in new window for each Open image in new window and the mapping Open image in new window is Open image in new window -convex on Open image in new window for each Open image in new window , Open image in new window such that Open image in new window is continuous with nonempty compact values for each Open image in new window , and Open image in new window is upper semicontinuous with nonempty compact values. Assume that condition () holds, then Open image in new window is a strong solution of (SEP)

_{I}; that is, there exists Open image in new window such that

for all Open image in new window . Furthermore, the set of all strong solutions of (SEP)_{I} is compact.

Theorem 2.6.

Let Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window be as in Theorem 2.1. Assume that the mapping Open image in new window is Open image in new window -convex on Open image in new window for each Open image in new window and Open image in new window such that

(1)for each Open image in new window , there is an Open image in new window such that Open image in new window ;

(3)for each Open image in new window , the set Open image in new window is open in Open image in new window .

Then there is an Open image in new window which is a weak solution of (SEP)_{I}.

Proof.

for each nonempty finite subset Open image in new window of Open image in new window . Therefore, the whole intersection Open image in new window is nonempty. Let Open image in new window . Then Open image in new window is a solution of (SEP)_{ I }.

Corollary 2.7.

Let Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window be as in Theorem 2.1. Assume that the mapping Open image in new window is Open image in new window -convex on Open image in new window for each Open image in new window and Open image in new window , Open image in new window such that Open image in new window is continuous with nonempty compact values for each Open image in new window , and Open image in new window is upper semicontinuous with nonempty compact values. Suppose that

(1)for each Open image in new window , there is an Open image in new window such that Open image in new window ;

Then there is an Open image in new window which is a weak solution of (SEP)_{I}.

Proof.

Using the technique of the proof in Theorem 2.2 and applying Theorem 2.6, we have the conclusion.

The following result is another existence theorem for the strong solutions of (SEP) Open image in new window . We need to combine Theorem 2.6 and use the technique of the proof in Theorem 2.3.

Theorem 2.8.

_{I}with Open image in new window . In addition, if Open image in new window and Open image in new window , Open image in new window is compact, Open image in new window is convex and the mapping Open image in new window is naturally quasi Open image in new window -convex on Open image in new window for each Open image in new window , Open image in new window such that Open image in new window is continuous with nonempty compact values for each Open image in new window , and Open image in new window is upper semicontinuous with nonempty compact values. Assuming that condition () holds, then Open image in new window is a strong solution of (SEP)

_{I}; that is, there exists Open image in new window such that

for all Open image in new window . Furthermore, the set of all strong solutions of (SEP)_{I} is compact.

Using the technique of the proof in Theorem 2.3, we have the following result.

Corollary 2.9.

_{I}with Open image in new window . In addition, if Open image in new window and Open image in new window , Open image in new window is compact, Open image in new window is convex, and the mapping Open image in new window is naturally quasi Open image in new window -convex on Open image in new window for each Open image in new window . Assuming that condition () holds, then Open image in new window is a strong solution of (SEP)

_{I}; that is, there exists Open image in new window such that

_{I} is compact.

Next, we discuss the existence results of the strong solutions for (SEP)_{ I } with the set Open image in new window without compactness setting from Theorems 2.10 to 2.14 below.

Theorem 2.10.

_{I}with Open image in new window . In addition, if Open image in new window and Open image in new window , Open image in new window is convex, Open image in new window for all Open image in new window and for all Open image in new window , the mapping Open image in new window is Open image in new window -convex on Open image in new window for each Open image in new window and Open image in new window and the mapping Open image in new window is naturally quasi Open image in new window -convex on Open image in new window for each Open image in new window , Open image in new window such that Open image in new window is continuous for each Open image in new window , and Open image in new window is upper semicontinuous with nonempty compact values. Assume that for some Open image in new window , such that for each Open image in new window , there is a Open image in new window such that the condition

_{I}; that is, there exists Open image in new window such that

_{I} is compact.

Proof.

for all Open image in new window . This completely proves the theorem.

Corollary 2.11.

_{I}with Open image in new window . In addition, if Open image in new window and Open image in new window , Open image in new window is convex, Open image in new window for all Open image in new window and for all Open image in new window , the mapping Open image in new window is Open image in new window -convex on Open image in new window for each Open image in new window and Open image in new window , and the mapping Open image in new window is naturally quasi Open image in new window -convex on Open image in new window for each Open image in new window . Assume that for some Open image in new window , condition () holds. Then Open image in new window is a strong solution of (SEP)

_{I}; that is, there exists Open image in new window such that

for all Open image in new window Furthermore, the set of all strong solutions of (SEP)_{I} is compact.

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 }.

Theorem 2.12.

_{I}with Open image in new window . In addition, if Open image in new window and Open image in new window , Open image in new window is convex, Open image in new window for all Open image in new window and for all Open image in new window , the mapping Open image in new window is naturally quasi Open image in new window -convex on Open image in new window for each Open image in new window , Open image in new window such that Open image in new window is continuous for each Open image in new window , and Open image in new window is upper semicontinuous with nonempty compact values. Assume that for some Open image in new window , condition () holds. Then Open image in new window is a strong solution of (SEP)

_{I}; that is, there exists Open image in new window such that

_{I} is compact.

In order to illustrate Theorems 2.10 and 2.12 more precisely, we provide the following concrete example.

Example 2.13.

We claim that condition () holds. Indeed, We know that the weak solution Open image in new window . For each Open image in new window , if we choose any Open image in new window , then Open image in new window and Open image in new window Open image in new window . 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 Open image in new window is a strong solution for (SEP)_{ I }.

Theorem 2.14.

_{I}with Open image in new window . In addition, if Open image in new window and Open image in new window , Open image in new window is convex, Open image in new window for all Open image in new window and for all Open image in new window , and the mapping Open image in new window is naturally quasi Open image in new window -convex on Open image in new window for each Open image in new window . Assume that for some Open image in new window , condition () holds. Then Open image in new window is a strong solution of (SEP)

_{I}; that is, there exists Open image in new window such that

_{I} is compact.

We would like to point out an open question naturally arising from Theorem 2.3: is Theorem 2.3 extendable to the case of Open image in new window or more general spaces, such as Hausdorff topological vector spaces?

## Notes

### Acknowledgments

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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