Existence of Nash equilibrium point in generalized game within an abstract convex space

In order to solve the existence problem of Nash equilibrium point for the generalized maximal element game and the restricted generalized maximal element game in an abstract convex game space which satisfies H0-condition,this paper defines the optimal reflection set-valued mappings of two strategies respectively based on two relevant strategy players.In addition,the paper proves the existence of the fixed point in optimal reflection set-valued mappings of two strategies and,therefore,obtains the existence conclusion of Nash equilibrium point for the generalized maximal element game and the restricted generalized maximal element game by applying the Fan-Glicksberg-kakutani fixed point theorem in an abstract convex space.The study results are valuable for future study in the area.