Existence Results for Some Equilibrium Problems Involving -Monotone Bifunction

Abstract

This paper deals with some existence results of equilibrium problems on convex and closed sets (either bounded or unbounded) in Banach spaces. Moreover, an application to the existence of solution for a differential inclusion is given.

1. Introduction and Preliminaries

Equilibrium problems provide a unifying framework for many important problems, such as optimization problems, variational inequality problems, complementarity problems, minimax inequality problems, and fixed point problems. They have been widely applied to study real word applications, arising in economics, mechanics, and engineering science. In recent decades, many results concerning the existence of solutions for equilibrium problems and vector equilibrium problems have been established (see Ansari and Yao [1], Bianchi and Schaible [2], and Blum and Oettli [3]). Many problems of practical interest in optimization, economics, physics, mechanics, and engineering sciences involve equilibrium in their description. Because of their wide applicability, equilibrium problems and mixed equilibrium problems have been investigated and generalized by a number of authors (see [4–6]).

The generalized monotonicity plays an important role in the study of equilibrium problems. In recent years, a number of authors have proposed many essential generalizations of monotonicity, such as -monotonicity, relaxed monotonicity, relaxed monotonicity, and quasimonotonicity (see [7–10]).

Let be a nonempty subset of a real reflexive Banach space. Let be three real-valued functions; is equilibrium function; that is, .

We consider the following generalized equilibrium problem (for short, ()) is to find such thatIn particular cases, one can obtain the following:(1)If then problem (1) is reduced to the classical equilibrium problem (for short, EP), which is to find such that ; see [3].(2)If then problem (1) is reduced to the mixed equilibrium problem (for short, MEP); see [11].(3)If then problem (1) is reduced to the generalized equilibrium problem (for short, GEP); see [12].

Here, a generalization of monotone bifunction is introduced.

Definition 1. A bifunction is called -monotone if

The notion of a KKM map and the well-known intersection lemma that is due to Fan [13] will be needed.

Definition 2 (see [14]). Let be a nonempty subset of a Hausdorff topological vector space . A mapping (by we understand the family of all the subsets of ) is said to be a KKM-mapping if, for any finite subset of ; we have , where denotes the convex hull of .

Lemma 3 (see [15]). Let be a nonempty subset of a Hausdorff topological vector space and let be a KKM-mapping. If is closed in for every and compact for some , then .

Definition 4 (see [16]). A real-valued function, defined on a convex subset of , is said to be hemicontinuous, if

Definition 5 (see [17]). A bifunction is called vector 0-diagonally convex if, for any finite subset of and with and , one has

Definition 6. Let be a Banach space. A mapping is said to be
(1) lower semicontinuous (for short, lsc) at , if(2) upper semicontinuous (for short, usc) at , if for any sequence of such that .

For technical reasons, the authors presented the following definition.

Definition 7. Assume that is a Banach space and a proper function. One can say that is -subdifferential of in , if

Throughout this paper, we consider :In the present paper, the first goal is to study the equilibrium problem for -monotone operator in Banach and real reflexive Banach spaces via the KKM technique. In addition, we provide sufficient conditions for existence solution for the case of unbounded sets. The results of this paper can be viewed as generalization of many known results (see [15, 18, 19]). Moreover, we recall an application of existence result for a differential inclusion.

2. Main Results

In this section we establish some existence results for equilibrium problem . Surprisingly, through the results of this section, we prove the existence of a solution of the problem without assumption of boundedness of the set in the last theorem.

Theorem 8. Suppose that is -monotone bifunction, hemicontinuous in the first argument and convex in the second argument. Let be convex in the second argument; then generalized equilibrium problem is equivalent to the following problem.
Find such that

Proof. Suppose that has a solution. There exists such thatSince is -monotone bifunction thenTherefore, is a solution of problem (9). Conversely, assume that is a solution of problem (9) and fix .
Letting , , then , since is convex, soThenSince is convex in the second argumentsoBy the convexity of and in the second argumentThen,From (14), (16), and (18)Since is hemicontinuous in the first argument, so From (8),Therefore, has a solution.

Theorem 9. Let be a nonempty closed bounded convex subset of a real reflexive Banach space . Assume that (1) is monotone bifunction, 0-diagonal convex, hemicontinuous in first argument and lsc, convex in second argument;(2) is convex in second argument, usc in first argument, and 0-diagonal convex;(3) is lsc in first argument and convex in second argument. Then has a solution.

Proof. Define two set valued mappings as follows: Then, the problem has a solution iff . Now, we show that is a KKM-mapping. On the contrary, is not a KKM-mapping. Then, there exists a finite subset of and with such that . Thenand this contradicts the 0-diagonal convex property of and . Then is a KKM-mapping. Next, we will show that . Let ; thenFrom definition of -monotone bifunction, This implies that . Thus . Therefore, is a KKM-mapping. Since , are lsc and is usc, then Therefore, is weakly closed . Since is nonempty, bounded, closed, and convex and is real reflexive, it follows that is weakly compact. Hence, is weakly compact . From Lemma 3 and Theorem 8Therefore, the problem has a solution.

Corollary 10. Assume that is a nonempty convex subset of Banach space and assumptions (1–3) in Theorem 9 hold. Moreover, (1);(2) and are relative compact sets. Then has a solution.

Proof. It suffices to prove is compact for some . Since Since , From conditions (1-2), is a subset of relative compact set or . This implies that is compact for some .

Now, we show that the problem admits one solution without boundedness of and under suitable conditions.

Theorem 11. Assume that the same hypotheses as in Theorem 9 hold without the assumption of boundedness of . Suppose in addition the following conditions are fulfilled: (1)Consider .(2) such that , where and is large enough. Then the problem admits a solution.

Proof. Set . Let us consider the problem. Find such thatFrom the boundedness of and Theorem 8, we obtain that the problem has at least one solution.
Choose . From (31), we getSo because . Let us choose and large enough so that by condition 2 we havewhich contradicts (32). Hence, For each , one can take small enough such that . From (31), for each , one can obtain Dividing by the conclusion follows.

3. Application

In this section, we apply our main results, expressed in the previous section to a partial differential inclusion problem. To do this, we will consider Sobolev space as So, let us consider the following partial differential inclusion problem:where is an open subset of is a continuous and concave function, and is a bounded convex subset of Sobolev space .

Definition 12. One says that has a -weak solution of problem (35) ifBy the integral form of , one can get that Set and . Therefore, we obtainConsidering , then is -monotone bifunction. In Theorem 8 we proved that (9) and (10) are equivalent under some assumptions. Hence, we must prove that and hold all assumptions of Theorem 9. It is clear that the bifunction satisfies all assumptions in Theorem 9.

Claim 1. is convex in second argument. Let , , so

Claim 2. is 0-diagonal convex, so is needed. For this, In a similar way, we prove that is 0-diagonal convex: It is clear that is hemicontinuous in first argument, lsc and convex in second argument, because is concave and continuous function. Moreover, is usc, since is continuous on Sobolev space.

Therefore, all conditions are achieved.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work has been supported by the University of Thi-Qar, Ministry of Higher Education in Iraq.