On Solutions of Generalized Implicit Equilibrium Problems with Application in Game Theory

Subject Areas : Financial Mathematics

Parastoo Zangenehmehr ¹ ( Department of Mathematics, Kermanshah Branch, Islamic Azad University, Kermanshah, Iran )
Ali Farajzadeh ² ( Department of Mathematics, Razi University, Kermanshah, 67149, Iran. )

Received: 2021-07-11 Accepted : 2021-10-31 Published : 2022-04-01

Keywords: set-valued mapping, Upper semicontinuous mapping, Generalized Nash equilibrium problem, Finite intersection property, KKM mapping,

Abstract :

In this paper, first a brief history of equilibrium problems(EP) and generalized implicit vector equilibrium problems(GIVEP) are given. Then some existence theorems for GIVEP are presented, also some suitable conditions in order the solution set of GIVEP is compact and convex for set-valued mappings whose are a subset of the cartesian product of Hausdorff topological vector space and their range is a subset of a topological space values (not necessarily locally convex or a topological vector space). In almost all of published results for GIVEP the set-valued mappings are considered from a topological vector space(locally convex topological vector space) to a topological vector space while in this paper the range of the set-valued mappings are a subsets of a topological spaces. As applications of our results, we derive some suitable conditions for existing a normalized Nash equilibrium problems when the number of players are finite and the abstract case, that is infinite players. Finally, a numerical result, as an application of the main results, is given. The method used for proving the existence theorems is based on finite intersection theorems and Ky-Fan’s theorem. The results of this paper, can be considered as suitable generalizations of the published paper in this area.

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