Stability Of Vector Quasi-equilibrium Problems

In chapter 1, we introduce the background, including the history and recent development of equilibrium problem. The equilibrium problem is important in applied nonlinear analysis. Many problems can be transformed to equilibrium problem, such as optimization problem, Nash equilibrium problem, complementarity problem, fixed point problem, saddle point problem, variational inequality and so on. In this chapter, we introduce the original and generalized types of equilibrium problem. We also introduce new types which appeare in the recent years, namely , the implicit vector equilibrium problem and the dual vector equilibrium problem. Due to the importance of stability in theory and application, we also introduce some classical stability concept, such as the generic stability and essential component.In chapter 2, we give some preliminaries, namely, the vector cone and its property, the continuity and convexity of set-valued mapping, some basic concept and lemma in functional analysis and topology and so on. We also introduce the mutual relation of some concept.In chapter 3, we discuss the generic stability of the solution of the vector quasi-equilibrium problem. First, we establish a metric on a set M consisting of some vector equilibrium problem which satisfy some continuity and convexity. We prove the completeness of M, and prove the solution mapping is an usco-mapping. We gain that the solution of most vector quasi-equilibrium problem are stable. We give an example and show that the solution of some vector quasi-equilibrium problem were unstable. Indeed, all solutions of some vector quasi-equilibrium problem are unstable.In chapter 4, we discuss the set-wise stability of the solution set to vector quasi-equilibrium problem, namely, the essential set and essential component. We prove that there exists at least one minimal essential set in the solution set of the vector quasi-equilibrium problem, in which the partial order is the inclusion of sets. And each minimal essential set is connected under some stronger conditions. Our results include some results in the literature as special cases.