Recent Developments on Sequence Spaces and Compact Operators with ApplicationsView this Special Issue
Generalized Equilibrium Problem with Mixed Relaxed Monotonicity
We extend the concept of relaxed -monotonicity to mixed relaxed --monotonicity. The concept of mixed relaxed --monotonicity is more general than many existing concepts of monotonicities. Finally, we apply this concept and well known KKM-theory to obtain the solution of generalized equilibrium problem.
Generalized monotonicities provide a way of finding parameter moves that yield monotonicity of model solutions and allow studying the monotonicity of functions or subset of variables. In recent past, many researchers have proposed many important generalizations of monotonicity such as pseudomonotonicity, relaxed monotonicity, relaxed --monotonicity, quasimonotonicity, and semimonotonicity; see [1–3]. Karamardian and Schaible  introduced various kinds of monotone mappings which in the case of gradient mappings are related to generalized convex functions. For more details, we refer to [5–7].
Many problems of practical interest in optimization, economics, and engineering involve equilibrium in their description. The techniques involved in the study of equilibrium problems are applicable to a variety of diverse areas and proved to be productive and innovative. Blum and Oettli  and Noor and Oettli  have shown that the mathematical programming problem can be viewed as special realization of abstract equilibrium problems.
Inspired and motivated by the recent development of equilibrium problems and their solutions methods, in this paper, we extend the concept of relaxed -monotonicity to mixed relaxed --monotonicity. Finally, this concept is applied with KKM-theory to solve a generalized equilibrium problem. The results of this paper can be viewed as generalization of many known results; see [10–13].
Let be a nonempty subset of real Banach space . Let be a real-valued function and let be an equilibrium function; that is, , for all . We consider the following generalized equilibrium problem: find such that Problem (1) has been studied by many authors in different settings; see, for instance, .
We need the following definition and results in the sequel.
Definition 1. A real-valued function defined on a convex subset of is said to be hemicontinuous if
Definition 2. Let be a multivalued mapping. The is said to be a KKM-mapping if, for any finite subset of , , where denotes the convex hull.
Lemma 3 (see ). Let be a nonempty subset of a topological vector space and let be a KKM-mapping. If is closed in for all and compact for at least one , then .
Definition 4. Let be a Banach space. A mapping is said to be lower semicontinuous at , if for any sequence of such that .
Definition 5. Let be a Banach space. A mapping is said to be weakly upper semicontinuous at , if for any sequence of such that .
Now, we extend the definition of relaxed -monotonicity  to mixed relaxed --monotonicity.
Definition 6. A mapping is said to be mixed relaxed --monotone, if there exist mappings with , for all and , such that
and is a constant.
If , then Definition 6 reduces to the definition of generalized relaxed -monotone; that is, where If , then Definition 6 reduces to the definition of generalized relaxed -monotone; that is, where
If both , then Definition 6 coincides with the definition of monotonicity; that is,
Definition 7. A mapping is said to be -diagonally convex if, for any finite subset of and with and , one has
3. Existence of Solution for Generalized Equilibrium Problem
We establish this section with the discussion of existence of solution for generalized equilibrium problem by using mixed relaxed --monotonicity.
Theorem 8. Suppose is mixed relaxed --monotone, hemicontinuous in the first argument and convex in the second argument with , for all . Let be convex in the second argument. Then, generalized equilibrium problem (1) is equivalent to the following problem.
Find such that where and is a constant.
Proof. Suppose that the generalized equilibrium problem (1) admits a solution; that is, there exists such that
Since is mixed relaxed --monotone, we have
Adding on both sides of (17), we have
Hence, is a solution of problem (14).
Conversely, suppose that is a solution of problem (14); that is, Let , , and ; then clearly as is convex. Thus from (17), we have Since is convex in the second argument, we have which implies that
Also as is convex in the second argument, we have
Adding (22) and (24), we have It follows that Since is hemicontinuous in the first argument, taking , we have that is, we have Hence is a solution of generalized equilibrium problem (1).
Theorem 9. Let be a nonempty bounded closed convex subset of a real Banach space . Let be a mixed relaxed --monotone, hemicontinuous in the first argument, convex in the second argument with , -diagonally convex, and lower semicontinuous. Let be convex in the second argument, -diagonally convex, and lower semicontinuous; is weakly upper semicontinuous and is weakly upper semicontinuous in the second argument. Then the mixed equilibrium problem (1) admits a solution.
Proof. Consider a multivalued mapping such that
We show that ; that is, is a solution of generalized equilibrium problem (1).
Our claim is that is a KKM-mapping. Suppose to contrary that is is not a KKM-mapping; then there exists a finite subset of and with such that It follows that Also we have which contradicts the -diagonal convexity of and . Hence is a KKM-mapping.
Now consider another multivalued mapping such that We will show that , . For any given , let ; then It follows from the mixed relaxed --monotonicity of that that is, . Thus and consequently is also KKM-mapping.
Since and both are convex in the second argument and lower semicontinuous, thus they both are weakly lower semicontinuous. From weakly upper semicontinuity of , weakly upper semicontinuity of in the second argument, and the construction of , it is accessible to see that is weakly closed for all . Since is closed, bounded, and convex, it is weakly compact and consequently is weakly compact in for all . Therefore, from Lemma 3 and Theorem 8, we have that is, there exists such that Thus, the generalized equilibrium problem (1) admits a solution.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors express their sincere thanks to the referees for the careful and detailed reading of the paper and the very helpful suggestions that improved the paper substantially. The authors also acknowledge that this research was part of the research project and was partially supported by Universiti Putra Malaysia under ERGS 1-2013/5527179.