Existence And Stability Of Solutions For Vector Equilibrium Problems

This thesis is devoted to studying the existence and stability of vector equilibrium problems (for short, VEPs) which provide a unified frame for several types of nonlinear problems such as vector optimization problems, vector variational inequalities, vector complementary problems, multiobjective games. VEPs belong to the crossing field of operation research, nonlinear analysis and mathematical economics, and have become a hot research topic recently. This thesis consists of seven chapters.Chapter1is an introduction. In this chapter, we briefly summarize the mathematical models, background and present situation of the research of VEPs.In chapter2, we recall some basic notions and results used in our analysis in this thesis, including theory of Baire category, topology induced by a Hausdorff metric, continuity, convexity and monotonicity of vector-valued functions with respect to a cone, continuity of set-valued mappings, KKMF lemma and several fixed point theorems.In chapter3, the solvability of VEPs is mainly considered. The existence of weak solutions for VEPs is obtained at first for the vector-valued functions without any con-tinuity and on noncompact sets. Then a series of existence theorems of the solutions for VEPs are obtained for the vector-valued functions with some weak continuity and convexity, especially, some existence theorems on noncompact subsets are established and the compactness of the solution sets are discussed. Moreover, we provide several equiv-alent theorems for our main result. As applications, we get three types of results:some generalizations for Ky Fan section theorem and Fan-Browder fixed point theorem, several theorems for solvability of vector variational inequalities, and some existence theorems of weak Pareto-Nash equilibria for multiobjective noncooperative games.In chapter4, the stability of solutions of VEPs is investigated without the assumption of linear structure or convexity. Since not all the slutions of all VEPs are stable, the generic stability is considered. In this chapter, three cases are studied. The first case is the perturbation of vector-valued functions. The second case is the perturbations of vector-valued functions and feasible sets. And the third case is the perturbation of vector quasiequilibrium problems. For three cases, we all derive the generic stability.In chapter5, we study the uniqueness of the solutions for equilibrium problems and vector Ky Fan inequalities. For each type of problems, we all consider two cases:the per- turbation of objective functions and the simultaneous perturbations of objective functions and feasible sets. Our method is that, we firstly introduced a metric in the space consist-ing of the type of problems to be considered such that it is a complete metric space, then prove the solution mapping to be a upper semicontinuous mapping with nonempty and compact values, through a further analysis of the solution mapping, we prove at last the result:in the sense of Baire category most of the type of problems have unique solution.In chapter6, the uniqueness of solutions for nonlinear problems is further studied. A unified approach to generic uniqueness of the solutions is provided. Through investigation of the sufficient and necessary conditions for a set-valued mapping to be a single-valued mapping, and using the generic continuity theorem for set valued mappings, we obtain several theorems of generic uniqueness which can be applied as a unified approach to uniqueness. As applications, the uniqueness of the solutions for lots of problems are considered such as optimization problem, saddle point problem, maximin problem, fixed point problem for nonexpansive mappings, monotone variational inequality problem, vec-tor optimization problem, vector equilibrium problem. For the problems all above, the result of generic uniqueness is derived.Chapter7is a simple summary and a working plan in future.