Study On Several Classes Of Generalized Vector Quasi-equilibrium Problems

By an equilibrium problem we understand the problem of finding x∈X such that where X is a given set and f :X×X→R is a given function. This problem contains as special cases optimization problems, complementarity problems, fixed point problems, and variational inequalities; it unifies these problems in a convenient way, and many of the results obtained for one of these problems can be extended, with suitable modifications, to general equilibrium problems, thus obtaining wider applicability. At the same time, equilibrium problems have been extended and generalized in many directions. Inspired and motivated by the research and activities going in this fascinating area, in this paper, we introduce and consider several new classes of generalized vector quasi-equilibrium problems. The whole paper is divided into three chapters.The first chapter introduces concisely the main contents of the paper. Basic concepts and relative results which will be used in the sequel appear in the second part of the first chapter, including the definitions of the continuity of set-valued mappings, the properties of lower semi-continuity and upper semi-continuity, several fixed points theorems etc.The first part of chapter 2 describes the process of equilibrium problem from general equilibrium to vector equilibrium, while the second part describes the five special situations of equilibrium problems, including Convex Minimization Problem, Fixed Point Problem, Complementarity Problem, Variational Inequality Problem and Vector Minimization Problem.Several new classes of generalized vector quasi-equilibrium problems are introduced in the final chapter. It is then shown that many existed problems can be seen as special situations of the new ones. The existence of solutions of these problems which are the significant contents of the paper are also given.