A Sufficient And Necessary Conditions For The Iteration Convergence Of Generalized Asymptotically Quasi-nonexpansive Type Non-self Mappings

Since the Banach contraction mapping principle was proved by Banach in 1921, 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. If T is non-self mappings, the Tx may not belong to the domain of T , so defined by the xn +1 iteration may not be meaningful. So this paper main discuss the non-self mapping, in order to address the limitations of self-image in the application of fixed point theory.Based on the real Banach space,in the appropriate constraints on the parameters,we discuss iterative convergence the problem of asymptotically quasi-nonexpansive non-self mappings and generalized asymptotically quasi-nonexpansive type non-self mappings .Chapter I, we introduce three contents of this paper, the significance of fixed points, including: theoretical and practical significance, and the present research situation of domestic and foreign country.Chapter II, first, we introduce the concept of generalized asymptotically quasi-nonexpansive type non-self mappings, then give a example of asymptotically quasi-nonexpansive non-self mappings. The example indicates that asymptotically quasi-nonexpansive non-self mapping is more generally than asymptotically quasi-nonexpansive self-mappings, thus it shows generalized asymptotically quasi-nonexpansive type non-self mappings is more generally than asymptotically quasi-nonexpansive type mappings. And the modified Ishikawa iterative sequence is used as follows: (?),Where x 0∈D,{αn} , {βn} ? [ 0,1]. In real Banach space, by appropriate restrictions on the parameters we prove that a sufficient and necessary conditions for the iteration convergence of asymptotically quasi-nonexpansive non-self mappings and generalized asymptotically quasi-nonexpansive type non-self mappingsChapter III, we discuss the N-asymptotically quasi-nonexpansive non-self mappings and N-type generalized asymptotically quasi-nonexpansive non-self mappings, prove the N-step iterative sequence strongly converges to common fixed points, in real Banach space.