The Nonlinear Operator Fixed Points Of Iterative Approximation

Fixed point theorems act as very important roles in nonlinear functional analysis theory. Since Banach's fixed points theorem was proved by Banach in 1992, the iterative approximations methods of fixed points for nonlinear mapping have formed unity by contractive mappings theorems and the Mann and Ishikawa iterative sequences now.This paper is devoted to introduce a new class of mappings and to approximation to the fixed points by the modified Ishikawa iterative sequences, which is an unity and extension of large number of known results. The research is carried on from three aspects. One is to modify the Ishikawa iterative sequence and study the iterative approximation problems by the modified Ishikawa iterative sequences by Liu L S and Xu Y G's ideas. Two is to introduce a new class of generalized uniformly Lipschitzian mappings and to study the iterative approximations of fixed points to generalized uniformly Lipschitzian mappings. Three is to modify the iterative parameter and to study the iterative approximations of fixed points. Details are as follows.The background of the iterative methods of fixed points for nonlinear mappings is introduced briefly.Some notions and conclusions in this paper have been reviewed.A new class of generalized uniformly Lipschitzian mappings are introduced. The iterative approximations of fixed points involving asymptotically nonexpansive mappings and asymptotically pseudocontra-ctive mappings in Banach spaces and generalized uniformly Lipschitzian mappings in uniformly smooth Banach spaces by the modified Ishikawa iterative sequences are proved.Some convergence theorems of Ishikawa and Mann type iterative sequence with error for generalized quasi-contractive mappings in convex metric spaces are proved.A characteristic condition is given for a generalized steepest decent method to converge to the zeros of quasi-accretive and a kind of accretive mappings and pseudocontractive mappings in a real normed linear spaces.The iterative convergences strongly to fixed points of asymptotically nonexpansive mappings in uniformly convex Banach spaces by Ishikawa iteration scheme are proved.