Stability Research And Application Of Solution Set For A Class Of Equilibrium Problems Based On Ekeland Variational Principle

Equilibrium problems play a very important role in many fields,and they provide a unified research framework for optimization problems,fixed point problems,variational problems,etc.Based on Ekeland’s variational principle,this paper mainly establishes a class of equilibrium problems and vector equilibrium problems.Using nonlinear analysis methods,we study the general uniqueness of the solution sets of these two equilibrium problems,and study the stability of the solution sets by establishing bounded rational models,and further apply the two equilibrium problems to game theory to discusses the stability of the equilibrium point of two game problems in nonconvex compact spaces.This paper is divided into six chapters.The main contents of each chapter are summarized as follows:In chapter one,we briefly introduce the research background of the equilibrium problems,Ekeland’s variational principle and bounded rationality models,as well as the research status of the existence,general uniqueness and stability of solutions of equilibrium problems and vector equilibrium problems at home and abroad.In addition,the research motivation of a class of equilibrium problems and vector equilibrium problems established in this paper is also briefly described.The second chapter introduces relevant definitions,lemmas and theorems that need to be used in this paper,mainly including several kinds of equilibrium problem models,vector equilibrium problem models,set-valued mapping,bounded rationality models and so on.In chapter 3,the existence and general uniqueness of solutions of a class of equilibrium problems and vector equilibrium problems are studied.First,based on the condition of cyclic antimonotonicity under Ekeland’s variational principle,we establish a more widely applicable equilibrium problem,and extend the function of the equilibrium problem to vector-valued form,thus establishing a class of vector equilibrium problems.Secondly,by using nonlinear analysis method,we study the general uniqueness of the solutions of equilibrium problems and vector equilibrium problems in nonconvex and nonconvex noncompact spaces.In the fourth chapter,we study the stability of solution sets for a class of equilibrium problems and vector equilibrium problems,mainly establishing bounded rationality models to study the stability of solution sets for these two equilibrium problems in nonconvex compact and nonconvex noncompact spaces.The application is given in chapter five.The equilibrium problems and vector equilibrium problems established in chapter 3 are applied to n-person non-cooperative game and multi-objective game respectively,and using bounded rationality method,the stability of Nash equilibrium and weak Pareto-Nash equilibrium are studied.The final chapter summarizes the whole paper and briefly describes the following research direction.