KKM Type Theorems And Applications In Various Abstract Spaces

The KKM theory was originally to study the KKM maps and their applications. nowadays, it would be better to regard as the study of applications of various equivalent formulations of the KKM principle and their generalizations. KKM theory has been widely used in variational inequalities and complementarity problems, fixed point theory and the fields of nonlinear analysis(such as game theory, mathematical programming, optimization and control, operations research and transportation, etc.). So KKM theory has become an important part in modern nonlinear analysis theory, and it is also a marginal subjects with strong cross-cutting.In recent years, many authors study various generalized KKM type theorems in topological spaces. Chang and Yen proposed KKM(X,Y) with KKM property in this context. The convexity assumptions play a crucial role which strictly restrict the applicable area of KKM principle. How to weaken convexity is the important subject of KKM principle.The thesis is composed of two parts. Some new KKM type theorems for the maps with KKM property are investigated. Furtherly, corresponding applications are gived too.Specifically, the history of KKM theory is first introduced. Then the new concept of finitely FC-closed subset is introduced in FC-space and then, the generalized KKM theorem is proved for the class of KKM(X,Y) under some conditions. As an application, a minimax inequality is proved in FC-spaces.L-spaces are more general than FC-spaces. Some new L-KKM type theorems for generalized L-KKM mappings are established and matching theorem for open covers and closed covers are proved.Correspondly locally L-uniform space generalizes locally FC-uniform space. Some almost fixed point theorems and fixed point theorems in locally L-uniform space are established for upper semicontinuous maping.More generalized spaces do not need continuous mapping (otherwise call FC-spaces) or lower semi-continuous mapping (otherwise call L-spaces). A new coincidence theorem and some KKM type theorems are studied in noncompact abstract C-convex space. And the existence theorems of vector equilibrium problems are established as applications. A KKM type theorm for the family KKM(X,Y) is obtained in abstractΓ-convex spaces. As applications, some new section theorems and coincidence theorem, fixed point theorem, maximal element theorem and generalized equilibrium problems are proved.