渐近拟伪压缩型非自映象的收敛性定理

Convergence Theorem for Asymptotically Quasi Pseudo-Contractive Type Nonself-mappings

Chidume首次提出渐近非扩张非自映象、一致L-Lipschitz非自映象的定义,并证明了所引入的迭代序列强收敛于渐进非扩张非自映象的不动点。本文引入渐近拟伪压缩型非自映象的概念。设E是实Banach空间,K是E的收缩核,P是从E到K上的非扩张收缩映象,T是一致L-Lipschitz的渐近拟伪压缩型非自映象,在对参数的一些限制条件下,给出了带误差修改的Ishikawa迭代序列强收敛于T的不动点的充要条件,即存在[0,+∞)上的严格增加函数φ(s),φ(0)=0,使得lim supn→∞j(xn+1-x*)inf∈J(xn+1-x*)[〈T(PT)n-1 xn+1-x*,j(xn+1-x*)〉-kn‖xn+1-x*‖2+φ(‖xn+1-x*‖)]≤0。目的是把对渐近拟伪压缩型自映象的迭代结果推广到渐近拟伪压缩型非自映象,从而推广了以前的结果。

Abstract ： Chidume first introduced the definition of asymptotically nonexpansive nonself-mappings and uniformly L-Lipschitzan nonself- mappings. Furthermore he proved that the iterative sequence converged strongly to fixed points of asymptotically nonexpansive nonself- mappings. In this paper,the definition of asymptotically pseudo-contractive type nonself-mappings, asymptotically quasi pseudo-contrac- tive type nonself-mappings is introduced. Suppose E is a real Banach space ,let K be a retract of E ,P be a nonexpansive retraction from E to K, T is L-Lipschitzan asymptotically quasi pseudo-contractive type nonself-mappings, under some restricted conditions on the pa- rameters. An necessary and sufficient condition is given for the modified Ishikawa iterative sequence with enor to converge strongly to afixed point of T, Suppose there exist a strictly increasing function Ф： [0, ＋ ∞ ） → [0, ＋ ∞）, Ф （0） = 0, such that lim sup n→∞ j（xn＋1^-1＊）∈J（xn＋1^-x＊）[〈T（PT）^n-1xn＋1-X＊,J（xn＋1-x＊）〉-kn｜｜xn＋1-x＊｜｜^2＋Ф（｜｜xn＋1-x＊｜｜）]≤0.The objective of this article is to extend the asymptotically quasi pseudo-contractive type mappings to asymptotically quasi pseudo-contractive type nonself-mappings. Therefore, the results presented in this paper extended the previous work.