- Research Article
- Open Access

# Solving the Set Equilibrium Problems

- Yen-Cherng Lin
^{1}Email author and - Hsin-Jung Chen
^{1}

**2011**:945413

https://doi.org/10.1155/2011/945413

© Y.-C. Lin and H.-J. Chen. 2011

**Received:**17 September 2010**Accepted:**21 November 2010**Published:**2 December 2010

## 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 , , 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:

(1) , for all ;

(2) ;

(3) .

Letting , be two sets of , we can define relations " " and " " as follows:

(1) ;

(2) .

Similarly, we can define the relations " " and " " if we replace the set by .

_{ I }is to find an such that

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

for all and for some .

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

for all .

_{ I }reduces to the vector equilibrium problem (VEP), which is to find such that

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

_{ I }reduces to (GVVIP): to find and such that

for all . 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 and 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 such that

for all . 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 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 [10]).

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

Definition 1.1 (see [11]).

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

(1)
*-convex* if
for every
and
;

(2)*naturally quasi*
*-convex* if
for every
and
.

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

Definition 1.2.

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

(1)a point
is called a *minimal point* of
if
;
is the set of all minimal points of
;

(2)a point
is called a *maximal point* of
if
;
is the set of all maximal points of
;

(3)a point
is called a *weakly minimal point* of
if
;
is the set of all weakly minimal points of
;

(4)a point
is called a *weakly maximal point* of
if
;
is the set of all weakly maximal points of
.

Definition 1.3.

Let , be two topological spaces. A mapping is said to be

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

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

(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 [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 , , 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

(1) for all ;

(3)for each , the set is convex;

- (5)
for each , the set is open in .

for all and for some .

Proof.

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

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

Hence , and then for all .

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

Theorem 2.2.

_{I}. That is, there is an such that

for all and for some .

Proof.

is open in
. 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
,
is single-valued function.

Theorem 2.3.

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

for all
. Furthermore, the set of all strong solutions of (SEP)_{I} is compact.

Proof.

From Theorem 2.2, we know that such that (1.1) holds for all and for some . Then we have .

for all
. Such an
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 denote the solution set of (SEP)

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

Hence and is closed.

We would like to point out that condition () is fulfilled if we take and 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
is a strong solution for (SEP)_{
I
}.

Corollary 2.5.

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

_{I}; that is, there exists such that

for all
. Furthermore, the set of all strong solutions of (SEP)_{I} is compact.

Theorem 2.6.

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

(1)for each , there is an such that ;

(3)for each , the set is open in .

Then there is an
which is a weak solution of (SEP)_{I}.

Proof.

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

Corollary 2.7.

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

(1)for each , there is an such that ;

Then there is an
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) . We need to combine Theorem 2.6 and use the technique of the proof in Theorem 2.3.

Theorem 2.8.

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

_{I}; that is, there exists such that

for all
. 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 . In addition, if and , is compact, is convex, and the mapping is naturally quasi -convex on for each . Assuming that condition () holds, then is a strong solution of (SEP)

_{I}; that is, there exists such that

for all
. Furthermore, the set of all strong solutions of (SEP)_{I} is compact.

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

Theorem 2.10.

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

_{I}; that is, there exists such that

for all
. Furthermore, the set of all strong solutions of (SEP)_{I} is compact.

Proof.

for all . This completely proves the theorem.

Corollary 2.11.

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

_{I}; that is, there exists such that

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

_{I}; that is, there exists such that

for all
. Furthermore, the set of all strong solutions of (SEP)_{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
. 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
}.

Theorem 2.14.

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

_{I}; that is, there exists such that

for all
. Furthermore, the set of all strong solutions of (SEP)_{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 or more general spaces, such as Hausdorff topological vector spaces?

## Declarations

### 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.

## Authors’ Affiliations

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