On Study Of Existence And Applications For Vector Equilibrium Problems

In this paper,we focus on the existence of solution and compactness of solution set for set-valued symmetric and bilevel vector quasi-equilibrium problems with variable control structure,and apply them to saddle point and traffic network problems.The main results improve the related research results in the existing literature.This paper is divided into four chapters:Chapter 1,this chapter gives an overview of the historical background of the vector equilibrium problem,and then analyzes the research status of the vector equilibrium problem,and puts forward the main research content and significance of this paper.In the second half of this chapter,it also gives some concepts and known conclusions to be used in this paper.Chapter 2,we mainly discuss two classes of set-valued symmetric generalized strong vector quasi-equilibrium problems with variable control structures.Under suitable assumptions of cone continuity and cone convexity,we use the concept of cosmically upper continuity rather than upper semicontinuity of cone valued mappings,and make full use of the famous Fan-KKM theorem and Kakutani-Fan-Glicksberg fixed point theorem,the existence theorem of the solution is proved.Furthermore,the compactness of the solution set is obtained.As an application,the existence of solution and compactness of solution set for strong saddle point problem are also obtained.Chapter 3,we study the bilevel strong(weak)vector quasi-equilibrium problem with variable control structure in the framework of locally convex Hausdorff topological vector space.By using Kakutani-Fan-Glicksberg fixed point theorem and under the suitable assumption of cone convexity of equilibrium mapping,we obtain the existence of its solution.Furthermore,we obtain the compactness of its solution set.In this chapter,we also establish the existence results of solutions for traffic network problems with equilibrium constraints.Chapter 4,we summarizes the research work of this paper and puts forward some problems that can be further studied in the future.