Does Not Have The Kkm Theory And Its Application In Any Convex Structure In The Topological Space

This paper is aimed to study the KKM theory and its applications in general topological spaces without any convexity structure.In the first chapter, we simply introduce the background of the KKM theory and its development.In the second chapter, by using generalized R-KKM mappings, we obtain a new coincidence theorem for admissible set-valued mappings in topological spaces without any convexity structure. As applications, an abstract variational inequality, a KKM type theorem and some fixed point theorems are obtained.In the third chapter, by using generalized R-KKM mappings, we generalize the concept of Φ-mappings from G-convex spaces to topological spaces without any convexity structure. We obtain a new continuous selection theorem for Φ-mappings in topological spaces without any convexity structure. As applications, we give some fixed point theorems, coincidence theorems and a nonempty intersection theorem.In the forth chapter, by using generalized R-KKM mappings, we obtain a new coincidence theorem for admissible set-valued mappings in topological spaces without any convexity structure. As applications, we prove some new minimax inequalities, section theorem and best approximation theorem. Using the obtained new minimax inequalities, some existence theorems of weighted Nash equilibria and Pareto equilibria for multiobjective games are given.In the fifth chapter, we introduce the concepts of weakly R-KKM mappings, R-convex and R-β-quasiconvex in general topological spaces without any convexity structure. We generalize Fan's matching theorem to general topological space without any convexity structure, namely that Lemma 5.1.2 in this paper. By using Lemma 5.1.2, two intersection theorems are proved in topological spaces without any convexity structure. By using intersection theorems, some minimax inequalities of Ky Fan type are also proved.In the sixth chapter, we introduce the new concepts of better admissible class B_FC(Y,X), Φ_FC mappings and Φ_FC-spaces in FC-spaces. On this basis, firstly, we obtain a continuous selection theorem for Φ_FC-mappings in FC-spaces. Secondly, using the continuous selection theorem, we proof a fixed point theorem involving B_FC ,Φ_FC-spaces, which is closely related to the well-known Schauder conjecture. Finally, as applications of our fixed point theorem, a quasi-equilibrium theorem and a generalized quasi-equilibrium theorem involving B_FC(F, X) are proved.