Popularization And Application Of KKM Theorem

This paper mainly further studies several focus problems of nonlinear functional analysis in G-convex spaces,L-convex spaces and finitely continuous topological spaces (in short, FC-spaces) without any convexity and linear structure, some new KKM-type theory are obtained. These results unites and generalize some known results in recent literature.In the first chapter, we simply introduce the background of the KKM theory and its development.In the second chapter, firstly, by using transfer compactly open-valued (closed-valued), the generalized KKM-type theorems in G-convex spaces are proved. As applications, some minimax inequalities and existence of maximal elements are obtained in G-convex spaces. Finally, new classes of L-KKM mappings,generalized G-LKKM mappings and generalized G-S-LKKM mappings in a nonempty subset of a L-convex space are obtained. Some new generalized L-KKM theorems,G-LKKM theorems and generalized G-S-LKKM theorems are established under the noncompact setting of a L-convex space. By using these results, some minimax inequalities and saddle theorems are proved in L-convex space.In the third chapter, we introduce the new concepte of the class B+ for admissible set-valued mappings in FC-spaces. With this new concept, some new KKM-type theorems involving the set-valued mapping with compactly local intersection property are proved in FC-spaces. By applying the KKM-type theorems, some new coincidence theorems are obtained in FC-spaces under much weak assumptions.