Reich-takahashi Iterative Approximation Of Fixed Points For Asymptotically Nonexpansive Nonself-mappings

Since the Banach fixed point theorem was proved by Banach in 1922, the problem of approximating of fixed points of nonlinear mappings and the studying about solution of nonlinear equation become more and more extensively. In 1972, Goebel and Kirk brought in the asymptotically nonexpansive mappings, after that, people have been studying to approximate to fixed points of asymptotically nonexpansive mappings by using different iterative sequences such as modified Mann iterative and Ishikawa iterative 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 Banach space E.In this paper, in real Banach space, under appropration paramaters restriction, we discuss the modified Reich-Takahashi iterative sequence approximating to fixed points of asymptotically nonexpansive nonself mappings.In the first paper, we introduce the significance of fixed point research, including: theory and practical significance, the present research situation of domestic and foreign country.In the second chapter, firstly, we give the example of asymptotically nonexpansive nonself-mapping, it shows that the asymptotically nonexpansive nonself-mapping is the real promotion of the asymptotically nonexpansive self-mapping. Adopt the following adment iterative sequence: Where x 0∈D,{αn} , {βn} ? [ 0,1]. In a real Banach spece, we prove that the iterative sequence above converges strongly to fixed points of asymptotically nonexpansive nonself mappings under some restrictive conditions on the parmaters.Then, we discuss the iterative sequence with errors, under some restrictive conditions on the parmaters, we prove the the iterative sequence strongly convergence to the fixed points.In the third chapter, in real Banach space , for finite asymptotically nonexpansive mappings T i ( i = 1,2, ???, m), Under some restrictive conditions of the parmeters, the iterative sequence strongly convergence to the common fixed points of T 1, T 2, ???, Tm.