Iterative Methods For Approximating Fixed Points For Nonlinear Operators

In this paper,we will study mainly several iterative sequences approximating fixed points of nonlinear operators:we proved four convergence theorems of contractive mapping in metric space;we proved the iterative sequence of nonexpansive mapping,accretive mapping approximating their fixed points in Banach space;we proved the iterative sequence of strictly asymptotically pseudocontractive mappings approximating their common fixed points in Hilbert space.Firstly,for a nonempty complete metric space X,T₁ and T₂ are self-mappings on X,we proved that the iterative sequence {x_n}satisfied some errors conditions, converges to a common fixed point of T₁ and T₂.This result extends and improves the corresponding ones of Meng liang,Yang Xiaoye and Liu Qihou[2,Meng Liang,Yang Xiaoye,Liu Qihou,Iterative sequence with errors for the extended contractive mappings pair in the metric space,The First International Conference on Iterative Method for Fixed Point of Nonlinear Operator and Solution of Variational Inequality,page:90-94,Tianjin,2006.].Secondly,we proved the strong convergence of nonexpansive mapping iterative sequence{x_n} by a new method in a uniformly smooth Banach space.This result extends and improves the corresponding ones of Hongkun Xu[6,J.Math. Anal.Appl.(2007),doi:10.1016/j.jmaa.2007.03.078].Subsequently,in a strictly convex Banach space X which has a uniformly Gateaux differentiable norm,for a finite family m-accretive mappings{S_r}_r∈Γ on C,and C is a nonempty closed convex subset of X,we proposed a new iteration sequence {x_n},and proved that {x_n} converges strongly to a common fixed point of a finite family m-accretive mappings.Our result extends and improves the corresponding results of H.Zegeye and N.Shahza[3,Strong convergence theorems for a common zero of a finite family of m-accretive mappings,Nonlinear Anal,Nonlinear Analysis 66(2007)1161-1169].We extended the viscosity iteration of nonexpansive mapping and considered the viscosity iteration of accretive operators,we proved the sequence{x_n}converges strongly to a common zero of m-accretive mappings in a reflexive Banach space.This result extends and improves the corresponding ones of Rudong Chen and Zhichuan Zhu[4, Fixed Point Theory and Applications.2006,Article ID 81325,10 pages,2006]and Xiaolong Qin,Yongfu Su[5,Volume 329,Issue 1,1 May 2007,Pages 415-424].Finally,for C a closed convex subset of a Hilbert space H.We proved that {x_n} strongly converges to a common fixed point of Strictly asymptotically pseudo-contractive mappings T_i.This result presented extends and improves the corresponding ones of Ta-Hwa Kim,Hong-Kun Xu[7,Nonlinear Analysis (2007),doi:10.1016/j.na.2007.02.029]and so on.