Generalized Equilibrium Problem with Mixed Relaxed Monotonicity

We extend the concept of relaxed α-monotonicity to mixed relaxed α-β-monotonicity. The concept of mixed relaxed α-β-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.


Introduction
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][2][3]. Karamardian and Schaible [4] 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][6][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 [8] and Noor and Oettli [9] 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][11][12][13].

Preliminaries
Let be a nonempty subset of real Banach space . Let : × → be a real-valued function and let : × → be an equilibrium function; that is, ( , ) = 0, 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, [14]. If ≡ 0, then the problem (1) reduces to the classical equilibrium problem, that is, to find ∈ such that ( , ) ≥ 0, with ( , ) = 0, ∀ ∈ . (2) Problem (2) was introduced and studied by Blum and Oettli [8].
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 The Scientific World Journal Definition 2. Let : → 2 be a multivalued mapping. The is said to be a KKM-mapping if, for any finite subset { 1 , 2 , . . . , } of , co{ 1 , 2 , . . . , } ⊂ ⋃ =1 ( ), where co denotes the convex hull.
Lemma 3 (see [15]). Let be a nonempty subset of a topological vector space and let : → 2 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 0 ∈ , if for any sequence { } of such that → 0 .
Definition 5. Let be a Banach space. A mapping : → is said to be weakly upper semicontinuous at 0 ∈ , if for any sequence { } of such that → 0 . Now, we extend the definition of relaxed -monotonicity [11] to mixed relaxed --monotonicity. where and > 1 is a constant. If = 0, then Definition 6 reduces to the definition of generalized relaxed -monotone; that is, where If = 0, then Definition 6 reduces to the definition of generalized relaxed -monotone; that is, where lim → 0 ( , + (1 − ) ) = 0.
If both = 0 = , then Definition 6 coincides with the definition of monotonicity; that is,

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.
Since is mixed relaxed --monotone, we have Hence, ∈ is a solution of problem (14). Conversely, suppose that ∈ is a solution of problem (14); that is, which implies that Also as is convex in the second argument, we have Since is hemicontinuous in the first argument, taking → 0, we have that is, we have Hence ∈ is a solution of generalized equilibrium problem (1). (29) 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 { 1 , 2 , . . . , } of and ≥ 0 ( = 1, 2, . . . ) with ∑ =1 = 1 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 : We will show that ( ) ⊂ ( ), ∀ ∈ . For any given ∈ , let ∈ ( ); then ( , ) + ( , ) − ( , ) ≥ 0.
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