Solving the Set Equilibrium Problems

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.

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 .

The trimapping and mapping are given. The set equilibrium problem (SEP) _I is to find an such that

(1.1)

for all and for some . Such solution is called a weak solution for (SEP) _I . We note that (1.1) is equivalent to the following one:

(1.2)

for all and for some .

For the case when does not depend on , that is, to find an with some such that

(1.3)

for all , we will call this solution a strong solution of (SEP) _I . We also note that (1.3) is equivalent to the following one:

(1.2x02032)

for all .

We note that if is a vector-valued function and the mapping is constant for each , then (SEP) _I reduces to the vector equilibrium problem (VEP), which is to find such that

(1.4)

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

If is a vector-valued function and which is denoted the space of all continuous linear mappings from to and , where denotes the evaluation of the linear mapping at , then (SEP) _I reduces to (GVVIP): to find and such that

(1.5)

for all . It has been studied by Chen and Craven [3].

If we consider , , , and , where denotes the evaluation of the linear mapping at , then (SEP) _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

(1.6)

If , is a single-valued mapping, , then (SEP) _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

(1.7)

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]).

Let , , and be real topological vector spaces, and let and be nonempty subsets of and , respectively. Let , be set-valued mappings. If both and are upper semicontinuous with nonempty compact values, then the set-valued mapping defined by

(1.8)

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.

Let , be two Hausdorff topological vector spaces, and let , be nonempty compact convex subsets of and , respectively. Let be continuous mapping with nonempty compact valued on ; the mapping is naturally quasi -convex on for each , and the mapping is -convex on for each . Assume that for each , there exists such that

(1.9)

Then, one has

(1.10)

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 ;

(2)for each , there is an such that for all ,

(2.1)

(3)for each , the set is convex;

(4)there is a nonempty compact convex subset of , such that for every , there is a such that for all ,

(2.2)

• (5) for each , the set is open in .

Then there exists an which is a weak solution of (SEP)_I. That is, there is an such that

(2.3)

for all and for some .

Proof.

Define by

(2.4)

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 .

Next, we claim that the family has the finite intersection property, and then the whole intersection is nonempty and any element in the intersection is a solution of (SEP) _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

(2.5)

for each . From conditions (1) and (2), we have

(2.6)

and for each , there is an such that

(2.7)

Hence , and then for all .

We can easily see that has closed values in . Since, for each , , if we prove that the whole intersection of the family is nonempty, we can deduce that the family has finite intersection property because and due to condition (4). In order to deduce the conclusion of our theorem, we can apply Fan's lemma if we claim that is a KKM mapping. Indeed, if is not a KKM mapping, neither is since for each . Then there is a nonempty finite subset of such that

(2.8)

Thus there is an element such that for all , that is, for all . By (3), we have

(2.9)

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.

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 . Let the mapping be such that for each , the mappings and are upper semicontinuous with nonempty compact values and . Suppose that conditions (1)–(4) of Theorem 2.1 hold. Then there exists an which is a solution of (SEP)_I. That is, there is an such that

(2.10)

for all and for some .

Proof.

For any fixed , we define the mapping by

(2.11)

for all and . Since the mappings and are upper semicontinuous with nonempty compact values, by Lemma 1.4, we know that is upper semicontinuous on with nonempty compact values. Hence, for each , the set

(2.12)

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.

Under the framework of Theorem 2.2, one has a weak solution of (SEP)_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

(Y)

then is a strong solution of (SEP)_I; that is, there exists such that

(2.13)

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 .

From condition () and the convexity of , Lemma 1.5 tells us that . Then there is an such that . Thus for all , we have . Hence there exists such that

(2.14)

for all . Such an is a strong solution of (SEP) _I .

Finally, to see that the solution set of (SEP) _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

(2.15)

Since is upper semicontinuous with compact values and the set is compact, it follows that is compact. Therefore without loss of generality, we may assume that the sequence converges to some . Then and . Let . Since the mapping is upper semicontinuous with nonempty compact values, the set is open in . Hence is closed in . By the facts and , we have . This implies that . We then obtain

(2.16)

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.

Let , , , , and . Choose to be defined by for every and is defined by , where , with , for some , , and is defined by

(2.17)

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.

Under the framework of Theorem 2.1, one has a weak solution of (SEP)_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

(2.18)

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 ;

(2)there is a nonempty compact convex subset of , such that for every , there is a such that for all ,

(2.19)

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

Then there is an which is a weak solution of (SEP)_I.

Proof.

For any given nonempty finite subset of . Letting , then is a nonempty compact convex subset of . Define as in the proof of Theorem 2.1, and for each , let

(2.20)

We note that for each , is nonempty and closed since by conditions (1) and (3). For each , is compact in . Next, we claim that the mapping is a KKM mapping. Indeed, if not, there is a nonempty finite subset of , such that . Then there is an such that

(2.21)

for all and . Since the mapping

(2.22)

is -convex on , we can deduce that

(2.23)

for all . This contradicts condition (1). Therefore, is a KKM mapping, and by Fan's lemma, we have . Note that for any , we have by condition (2). Hence, we have

(2.24)

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 ;

(2)there is a nonempty compact convex subset of , such that for every , there is a such that for all ,

(2.25)

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.

Under the framework of Theorem 2.6, on has a weak solution of (SEP)_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

(2.26)

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.

Under the framework of Corollary 2.7, one has a weak solution of (SEP)_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

(2.27)

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.

Letting be a finite-dimensional real Banach space, under the framework of Theorem 2.1, one has a weak solution of (SEP)_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

(Yx02032)

is satisfied, where . Then is a strong solution of (SEP)_I; that is, there exists such that

(2.28)

for all . Furthermore, the set of all strong solutions of (SEP)_I is compact.

Proof.

Let us choose such that condition () holds. Letting , then the set is nonempty and compact in . We replace by in Theorem 2.3; all conditions of Theorem 2.3 hold. Hence by Theorem 2.3, we have such that

(2.29)

for all . For any , choose small enough such that . Putting in (2.29), we have

(2.30)

We note that

(2.31)

which implies that

(2.32)

for all . This completely proves the theorem.

Corollary 2.11.

Letting be a finite-dimensional real Banach space, under the framework of Theorem 2.2, one has a weak solution of (SEP)_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

(2.33)

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.

Let be a finite-dimensional real Banach space, under the framework of Theorem 2.6, one has a weak solution of (SEP)_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

(2.34)

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.

Let , , , , and . Choose to be defined by for every and is defined by , where , , and is defined by

(2.35)

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.

Letting be a finite-dimensional real Banach space, under the framework of Corollary 2.7, one has a weak solution of (SEP)_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

(2.36)

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?