Iteration Approximation Scheme Of Asymptotically Quasi-nonexpansive Mappings In Banach Space

The main purpose of this paper to study the necessary and sufficient conditions offixed points and strong convergence theorems for nonself asymptotically quasi-nonexpansive mappings in real uniformly convex Banach spaces, a modified Mann andIshikawa iterations with errors is established for a family of nonself asymptotically quasi-nonexpansive mappings, extends two step nonself asymptotically quasi-nonexpansivemappings to two finite families, and set-valued asymptotically quasi-nonexpansivemappings is introduced.The following is our main work which consists of four parts.In the first chapter, we narrate the development of the fixed point theory, thesignificance, the current research lever and the tendency of this topic home and abroad inthe preface. And simply introduce why we choose the topic.In the second chapter, we introduce convergence theorems for fixed points ofset-valued asymptotically quasi-nonexpansive mappings in uniformly convex Banachspaces.In the third chapter, we introduce approximating common fixed points of a family ofnonself asymptotically quasi-nonexpansive mappings. A necessary and sufficient conditionfor the convergence of the projection type Ishikawa iteration process is obtained. Threestrong convergence theorems for such iteration process are established.In the forth chapter, we introduce a finite-step projection type iteration process withtwo finite families of asymptotically quasi-nonexpansive nonself-mappings,theconvergence theorems for the iteration process are established.