The Stability Of Equilibrium Points Of Set-valued Mappings

In this paper, we mainly discuss the generic stability and the existence of essential connected components to equilibrium points of set-valued mappings. As a generalization, we study the generic stability of equilibrium points of set-valued mappings under graph topology. As applications, we obtain that any set-valued mapping fixed point set as well as any coincidence point set exists at least one essential connected component. Finally, we study the existence of essential connected components of the set of solutions of differential inclusion problems and that of the set of maximum likelihood estimates in linear models by the unified approach in Yu et al (2004).It consists of three chapters.In Chapter one, we introduce some basic notions and results including compact sets, connected sets, completed distance space, Hausdorff distance, Baire space, generic stability, convex sets, convex functions and semi-continuity of set-valued mappings.In Chapter two, we study the stability of equilibrium points of set-valued mappings. We firstly introduce the generic stability of equilibrium points of set-valued mappings under uniform distance topology. As a generalization, we study the generic stability of equilibrium points of set-valued mappings under graph topology. Then we rededuce the existence of essential connected components of the set of the equilibrium points of set-valued mapping by the unified approach in Yu et al. (2004). Finally, as applications, we obtain that any set-valued mapping fixed point set as well as any coincidence point set exists at least one essential connected component.In Chapter three, we further study the stability of solutions of differential inclusion problems and that of maximum likelihood estimates in linear models. We prove that both the set of solutions of every differential inclusion problem and the set of maximum likelihood estimates in every linear model exist at least one essential connected component.