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Fixed Point Theory and Applications

, 2011:945413 | Cite as

Solving the Set Equilibrium Problems

Open Access
Research Article
Part of the following topical collections:
  1. Equilibrium Problems and Fixed Point Theory

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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 .

The trimapping Open image in new window and mapping Open image in new window are given. The set equilibrium problem (SEP) I is to find an Open image in new window such that
for all Open image in new window and for some Open image in new window . Such solution is called a weak solution for (SEP) 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 .

For the case when Open image in new window does not depend on Open image in new window , that is, to find an Open image in new window with some Open image in new window such that
for all Open image in new window , we will call this solution a strong solution of (SEP) I . We also note that (1.3) is equivalent to the following one:

for all Open image in new window .

We note that if Open image in new window is a vector-valued function and the mapping Open image in new window is constant for each Open image in new window , then (SEP) 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].

If Open image in new window is a vector-valued function and Open image in new window which is denoted the space of all continuous linear mappings from Open image in new window to Open image in new window and Open image in new window , where Open image in new window denotes the evaluation of the linear mapping Open image in new window at Open image in new window , then (SEP) 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].

If we consider Open image in new window , Open image in new window , Open image in new window , and Open image in new window Open image in new window , where Open image in new window denotes the evaluation of the linear mapping Open image in new window at Open image in new window , then (SEP) 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
If Open image in new window , Open image in new window is a single-valued mapping, Open image in new window , 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 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]).

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

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 Open image in new window , Open image in new window be two Hausdorff topological vector spaces, and let Open image in new window , Open image in new window be nonempty compact convex subsets of Open image in new window and Open image in new window , respectively. Let Open image in new window be continuous mapping 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 . Assume that for each Open image in new window , there exists Open image in new window such that
Then, one has

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;

(4)there is a nonempty compact convex subset Open image in new window of Open image in new window , such that for every Open image in new window , there is a Open image in new window such that for all Open image in new window ,
Then there exists an Open image in new window which is a weak solution of (SEP)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 .

Next, we claim that the family Open image in new window has the finite intersection property, and then the whole intersection Open image in new window is nonempty and any element in the intersection Open image in new window is a solution of (SEP) 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
for each Open image in new window . From conditions (1) and (2), we have

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

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

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.

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 . Let the mapping Open image in new window be such that for each Open image in new window , the mappings Open image in new window and Open image in new window are upper semicontinuous with nonempty compact values and Open image in new window . Suppose that conditions (1)–(4) of Theorem 2.1 hold. Then there exists an Open image in new window which is a solution of (SEP)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 any fixed Open image in new window , we define the mapping Open image in new window by
for all Open image in new window and Open image in new window . Since the mappings Open image in new window and Open image in new window are upper semicontinuous with nonempty compact values, by Lemma 1.4, we know that Open image in new window is upper semicontinuous on Open image in new window with nonempty compact values. Hence, for each Open image in new window , the set

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.

Under the framework of Theorem 2.2, one has a weak solution Open image in new window of (SEP)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
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.

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 .

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

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

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.

Under the framework of Theorem 2.1, one has a weak solution Open image in new window of (SEP)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 ;

(2)there is a nonempty compact convex subset Open image in new window of Open image in new window , such that for every Open image in new window , there is a Open image in new window such that for all 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 any given nonempty finite subset Open image in new window of Open image in new window . Letting Open image in new window , then Open image in new window is a nonempty compact convex subset of Open image in new window . Define Open image in new window as in the proof of Theorem 2.1, and for each Open image in new window , let
We note that for each Open image in new window , Open image in new window is nonempty and closed since Open image in new window by conditions (1) and (3). For each Open image in new window , Open image in new window is compact in Open image in new window . Next, we claim that the mapping Open image in new window is a KKM mapping. Indeed, if not, there is a nonempty finite subset Open image in new window of Open image in new window , such that Open image in new window . Then there is an Open image in new window such that
for all Open image in new window . This contradicts condition (1). Therefore, Open image in new window is a KKM mapping, and by Fan's lemma, we have Open image in new window . Note that for any Open image in new window , we have Open image in new window by condition (2). Hence, we have

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 ;

(2)there is a nonempty compact convex subset Open image in new window of Open image in new window , such that for every Open image in new window , there is a Open image in new window such that for all 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.

Under the framework of Theorem 2.6, on has a weak solution Open image in new window of (SEP)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.

Under the framework of Corollary 2.7, one has a weak solution Open image in new window of (SEP)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

for all Open image in new window . 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 Open image in new window without compactness setting from Theorems 2.10 to 2.14 below.

Theorem 2.10.

Letting Open image in new window be a finite-dimensional real Banach space, under the framework of Theorem 2.1, one has a weak solution Open image in new window of (SEP)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
is satisfied, where Open image in new window . 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.

Proof.

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

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

Corollary 2.11.

Letting Open image in new window be a finite-dimensional real Banach space, under the framework of Theorem 2.2, one has a weak solution Open image in new window of (SEP)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.

Let Open image in new window be a finite-dimensional real Banach space, under the framework of Theorem 2.6, one has a weak solution Open image in new window of (SEP)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

for all Open image in new window . 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 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.

Letting Open image in new window be a finite-dimensional real Banach space, under the framework of Corollary 2.7, one has a weak solution Open image in new window of (SEP)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

for all Open image in new window . 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 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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Copyright information

© Y.-C. Lin and H.-J. Chen. 2011

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Department of Occupational Safety and HealthChina Medical UniversityTaichungTaiwan

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