Mixed Equilibrium Problems with Weakly Relaxedα-Monotone Bifunction in Banach Spaces

Abstract

We introduce the class of mixed equilibrium problems with the weakly relaxedα-monotone bi-function in Banach spaces. Using the KKM technique, we obtain the existence of solutions for mixed equilibrium problem with weakly relaxedα-monotone bi-function in Banach spaces. The results presented in this paper extend and improve the corresponding results in the existing literature.

Generalized mixed equilibrium problems with generalized α -η -monotone bifunction in topological vector spaces

“…Moreover, the compactness and convexness of the solution set are investigated. The results in the paper extend and generalize the corresponding results, especially Sintunavarat (2013) [20] in this area by providing mild assumptions in order to guarantee the existence of a solution for the generalized mixed equilibrium problem. …”

“…Definition 2.7 (see [20]). A bi-function f : K × K → R is said to be a weakly relaxed α-monotone if there exists a function α :…”

“…If we define τ (x, y) ≡ ϕ(y) − ϕ(x) for all x, y ∈ K and A ≡ 0 (The zero function in E * ), then the GMEP reduces to the mixed equilibrium problem (for short, MEP) given in [20], which consists of findingx ∈ K such that f (x, y) + ϕ(y) − ϕ(x) ≥ 0, ∀y ∈ K. (1.2) In particular, if τ ≡ 0, then MEP reduces to the classical equilibrium problem (for short, EP), which is to findx ∈ K such that f (x, y) ≥ 0, ∀y ∈ K.…”

“…Very recently, Sintunavarat [20] and Kutbi and Sintunavarat [16] introduced some generalizations of relaxed α-monotonicity for bi-functions. They also obtained the existence of solutions in reflexive Banach spaces by using the KKM technique for mixed equilibrium problems and variational-like inequalities problems, respectively.…”

“…The results in the paper can be viewed as some improvements and generalizations of Sintunavarat [20] in this area by providing mild assumptions in order to guarantee the existence of a solution for the generalized mixed equilibrium problem.…”