A Necessary And Sufficient Conditions For The Iterative Sequence Convergence To The Uniformly L-Lipschitz Asymptotically Pseudo-contractive Nonself-mappings

Since the Banach contraction mapping theorem was proved in 1921 by Banach, the problem of approximating to fixed points of nonlinear mappings and the studying about solution of nonlinear equation become more and more extensively. After that, people have been studying to approximate to the fixed points of asymptotically pseudo-contractive mappings by using different iterative sequences such as modified Mann iterative and Ishikawa iterative,modified implicit iteration etc in different spaces, The results have been more mature. However, all the results they discussed require that the mapping T is a self-mapping in the subset of the Banach space E. They also have studied about the asymptotically pseudo-contractive nonself-mappings. In this paper,We continue to study the convergence theorem of uniformly L-Lipschitz asymptotically pseudo-contractive nonself-mappings under some restrictive conditions . The results presented in this paper improve and extend recent corresponding results. This paper includes four chapters.In chapter 1,we discuss the significance of this paper. Furthermore,we present some research situation about the problem of fixed points of asymptotically pseudo-contractive mappings of domestic and foreign country and we also give a summary of this work.In chapter 2,we mainly study about the necessary and sufficient conditions for strong convergence of uniformly L-Lipschitz asymptotically pseudo-contractive nonself-mappings.In chapter 3 ,We study the convergence theorem of a finite family of uniformly L-Lipschitz asymptotically pseudo-contractive nonself-mappings. .We introduce modified implicit iterative sequence, under some restrictive conditions, we gain the necessary and sufficient conditions for strong convergence of a finite family uniformly L-Lipschitz asymptotically pseudo-contractive nonself-mappings.In the last chapter,we give the example of the asymptotically pseudo-contractive nonself-mapping and illustrated that asymptotically pseudo-contractive nonself-Mapping is the general of asymptotically pseudo-contractive self-mappings in real Banach space E.