Some Equivalent Forms Of Fixed Point Theorems And Generalizations

Fixed point theory has become the main component of Nonlinear Ananlysis. It is applied successfully in many fields such as partial differential equation , cybernetics, economic balance theory , game theory and so on. In this thesis , firstly, we discuss some fixed point theorems and their equivalent forms. And then, we establish some new fixed point theorems and section theorems in H-space and give some applications.This thesis consists of three chapters.In chapter one ,we simply introduce some general concepts and propositions about convex analysis, topolopy space and set-valued mappings.In chapter two ,we discuss some equivalent forms of fixed point theorems.Fisrtly, we simply introduce some important generalizations of Brouwer fixed point theorem.And then, we obtain that Brouwer fixed point theorem,KKM theorem,FKKM theorem ,Ky Fan minimax inequality, Browder fixed point theorem , Ky Fan inequality I,the geometry form of Ky Fan minimax ,Ky Fan section theorem , Fan-Browder fixed point theorem and Ky Fan inequality II are all equivalent.In chapter three, firstly ,we introduce some important concepts about H-space.And then, we obtain a new fixed point theorem of Fan-Browder type in H-space and some equivalent forms.As application,these results are utilized to study the existence problems of fixed point and nearest point in H-space. These results are generalizations of Zhang(1999) in H-space. Finally,we establish a new section theorem of Fan-Ha type in locally convex H-space and some corresponding equivalent versions.As applications, some minimax theorems in H-space are obtained.The above results improve and extend the corresponding results in Wu (2000) to the H-space.