Research On The Existence And Stability Of Solutions To Ky Fan's Inequality And Related Problems

In this thesis,the existence and stability of sets of solutions for Ky Fan's inequality and its related problems are studied by means of set-valued analysis.The research con-tents mainly include the strongly stability analysis of sets of solutions for Ky Fan's section problem,Ky Fan's point set and vector-valued Ky Fan's point set,and the strongly stabil-ity analysis of set of corresponding Nash equilibria for two class of related games problems,and the generic stability analysis of set of solutions for vector-valued quasi-variational inequality problems and a class of classical stochastic control problems,respectively.It is divided into six chapters and is organized as follows:In chapter 1,we first briefly summarize the research background,the current situa-tion and the significance of the study on the Ky Fan's inequality and its related problems.Moreover,we discuss the developments and current researches on essential component,generic stability,and stochastic control problems.Finally,we clarify the main research work,innovation and the basic framework on this thesis.In chapter 2,some basic notions,properties and some important conclusions are re-called which are used in following contents.These mainly include the Hausdorff distance and continuity of set-valued mapping,the continuity and convexity of vector-valued func-tions,the related concepts of stochastic process,and solution of stochastic differential equations and its existence theorem,and their related properties.In chapter 3,we mainly study the existence of the strongly essential set and strongly essential component of the solutions of Ky Fan's section problems,the strong stability of Ky Fan's points set and vector-valued Ky Fan's points set,the strongly stability of Nash equilibria of n-person non-cooperative game and Pareto-Nash equilibria of multi-objective game.Firstly,a new stronger perturbation of Ky Fan's section problems is proposed by means of the Hausdorff upper semi-metric.Based on this perturbation,a class of strongly essential set and strongly essential component of its solutions set are introduced,and the existence of the strongly essential set and strongly essential component of solutions set of Ky Fan's section problems are proposed.Moreover,for two kinds of problems of Ky Fan inequality and vector-valued Ky Fan inequality problems,these problems are transformed into the Ky Fan's section problem by using the equivalence between the Ky Fan's point and the solution of the Ky Fan's section problems.So some stronger perturbations of inequality functions and set-valued mappings are introduced by means of the Hausdorff upper semi-metric,which not only include the perturbations defined by the sup-norm of inequality function and the maximum Hausdorff metric of section mapping,but also include the case of set perturbations.And more importantly,these strong perturbations break the symmetry structure of two common perturbations,only need to considering the inclusion relationship,and extend the way of perturbations and the scope of application.Based on these perturbations,the strongly essential sets and strongly essential component of these two problems are proposed,and the existence of the strongly essential components of Ky Fan's points and vector-valued Ky Fan's points problems are proved,respectively.Finally,as applications,by using the equivalences of Nash equilibrium of n-person non-cooperative game(or weakly Pareto-Nash equilibrium of multi-objective game)with Ky Fan's points(or vector-valued Ky Fan's points),the existence of the strongly essential component of equilibria of the twokinds of games are deduced,respectively.In chapter 4,the generic stability of solution set for vector-valued quasi-variational inequality problems are studied by means of the research methods of generic property.Firstly,in the sense of graph topology,based on the graphic metric of constrain mapping,a new metric pH that is weaker than the usual uniform metric is introduced for this problem.Then,some results of generic stability of its solution set are given under this weaker metric.It is shown that,in the sense of Baire category,most of vector-valued quasi-variational inequality problems are essential with respect to pH.In chapter 5,the existence and generic stability of solutions(or optimal controls)for a kind of classical stochastic control problems are investigated mainly.Firstly,the classical Riesz-Kolmogorov compactness theorem in LP-space is generalized to the cor-responding criterion of compactness in a stochastic process space LFP([s,T];Rk),and the compactness of admissible control set U[s,T]is obtained.Secondly,we analysis the con-tinuous dependence of the solutions of the controlled system equation on the parameters,especially the continuous dependence of the solutions on the system coefficients b and a is new.Thirdly,A new existence result of optimal controls of stochastic control problems is proved by using set-valued analysis method under some assumptions without convexity of the admissible controls set and positive definiteness of the diffusion coefficient a.Fi-nally,by introducing the essential solution to the stochastic control problems,we obtain the generic stability results that in the sense of Baire category most of the stochastic control problems in our constructed problem model are essential.In chapter 6,we briefly summarize the results of this thesis and look forward to some future research prospect.