Study On Vector Equilibrium Problems

Equilibrium theory is important in mathematical economics. Since the existence of equilibrium in an abstract economy which compact strategy set in Rn was proved in a seminal paper of Debreu, many famous scholars have studied it, and there have been many generalization of Debreu's theorem. At the same time, the development of equilibrium theory stimulates the study of many fundamental mathematical fields such as variational inequality theory, optimization theory and the fixed point theory.The solution existence of equilibrium problems is the kernel of the study of the theory of equilibrium. According to the existence results of general equilibrium problems and vector equilibrium problems have been studied more and more. Inspired and motivated by these research results, this paper is devoted to study systematically a class of equilibrium problems.1. Many scholars usually utilize the maximal element theorems and the fixed point theorems to study existence of solutions for vector equilibrium problems. In this paper, topological pseudomonotone mapping is introduced and vector equilibrium problems of pseudomonotonicity and nonmontonicity are furtherly discussed on the basis of the fixed theorems. By using(*)-quasiconcavity of vector mappings, connectedness of solutions is presented. At last, we discuss the existence of solutions for vector variational inequalities and vector optimization problems to serve for application.2. According to the existence results of general equilibrium problems, a type of parametric vector equilibrium problem is obtained on the basis of some proper parametric set. The existence theorem and continuity properties of its solution mapping are proved in the same way. As its applications, the continuity of solution mappings for a class of parametric vector optimization problem and parametric vector variational inequality is obtained.3. A new type of parametric vector equilibrium problem with moving cones is introduced on the basis of some convexity and the existence theorem and continuity properties of its solution mapping are proved according to the existence results of parametric vector equilibrium problems. As its applications, the continuity of solution mappings for a class of generalized parametric vector optimization problems and parametric vector variational inequalilies is obtained.