Parametric Type Of KKM Theorems And Iterative Processes For Totally Asymptotically Quasi-nonexpansive Mappings

KKM theorems and the corresponding applications in different spaces are proved in the first two chapters of this paper. The third chapter independently studies It-erative Processes for Totally Asymptotically Quasi-nonexpansive Mappings. The main content of the paper is demonstrated as following:Firstly, the background and current state of KKM theory and variational in-equality theory are discussed.Secondly, by introducing the abstract convex space,using the connectedness of set and a characteristic property in abstract convex spaces, a parametric type of KKM theorem is established in noncompact abstract convex space. The result is applied to get noncompact minimax inequalities, saddle point theorem and section theorem.Thirdly, introducing a new space that is GFC-space, a parametric type of KKM theorem is proved. The result is applied to get noncompact minimax inequalities, saddle point theorem.Finally, by extended the Ishikawa iteration with errors to obtain a new se-quence, based on this sequence a new inequality is proved and some sufficient and necessary conditions on the strong convergence of a new iterative process for totally asymptotically quasi-nonexpansive mappings in convex metric spaces is obtained.