Gap Functions And Existence Of Solutions To Some Vector Equilibrium Problems Via Nonlinear Scalarization Function

Scalaxization method and gap function are important tools in the study of some vector equilibrium problems, the solutions of vector equilibrium problems can be found by solving scalar optimization problems. In this paper we introduce a nonlinear scalarization function. Several important properties, such as subadditiveness, strict monotone and continuity, of this nonlinear scalarization function are established. Using the so-called nonlinear scalarization function, existence theorems and gap functions for some equilibriums problems in the cases of vector-valued and set-valued are established. This paper includes four chapters. Now we will describe them briefly one by one.In Chapter 1, we introduce many other authors' work on equilibrium problem (EP), and show that equilibrium theory has been applied widely. In this chapter, we also recall some definitions and lemmas which needed in the main results of this paper.In Chapter 2, we introduce a nonlinear scalarization function and a gap function of vector equilibrium problems. Several important properties of this nonlinear scalarization function are established.In Chapter 3, this nonlinear scalarization function is applied to study the existence of solutions and gap functions for vector equilibrium problems, quasi-vector equilibrium problems and implicit vector equilibrium problems.In Chapter 4, this nonlinear scalarization function is applied to study the existence of solutions and gap functions for generalized vector equilibrium problems and generalized quasi-vector equilibrium problems.