Nonemptiness And Boundedness Of Solution Sets For Vector Equilibrium Problems

Vector equilibrium problem is a kind of mathematical model of universal significance, it contains the vector optimization problems, vector variational inequality problems, fixed point problems and many important mathematical problems, and it has wide applications in the fields of economy, finance, transportation, resource allocation and project management. The existencess and boundedness of solutions for vector equilibrium problem is the basic problem of vector equilibrium problem.This paper mainly studies the nonemptiness and boundedness of solution sets for vector equilibrium problems and vector optimization problems, vector variational inequality. We transform vector equilibrium problem into its corresponding scalar problem, then using the connectedness of weakly* compact base of its dual cone, we prove that the boundedness and nonemptiness of solution sets for each scalar problem is equivalent to the boundedness and nonemptiness of solution sets for vector optimization problems, vector variational inequalities and vector equilibrium problems respectively.It is organized as follows:In Chapter 1, we introduce the background of vector equilibrium, vector optimization, and vector variational inequalities, and list some basic conceptions and lemmas which are used is this dissertation.In Chapter 2, vector equilibrium problems and its scalar problems of equivalence re-lation are given. We get the necessary and sufficient conditions of the boundedness and nonemptiness of solution sets for vector equilibrium problems by using the connectedness of weakly* compact base of its dual cone, and the Mosco convergence convex function sequence in the infinite dimensional and finite dimensional space, respectively. The result is applied to the vector optimization problems, we prove the boundedness and nonemptiness of solution sets for vector optimization problems is equivalent to the boundedness and nonemptiness of solution sets for its corresponding scalar optimization problem.In Chapter 3, some characterizations of nonemptiness and boundedness of solution sets for vector variational inequalities are studied in finite and infinite dimensional spaces, respectively. Some sufficient and necessary conditions for the nonemptiness and boundedness of solution sets is established.The main feature of this paper is to give a new method to prove the nonemptiness and boundedness of solution sets for vector equilibrium problems. We first use the connectedness of weakly* compact base of its dual cone for vector equilibrium problems and obtain some new results of solution set for vector equilibrium problems under weaker conditions.