WEAK AND STRONG CONVERGENCE OF MANN'S-TYPE ITERATIONS FOR A COUNTABLE FAMILY OF NONEXPANSIVE MAPPINGS

Abstract

Let K be a nonempty closed convex subset of a Banach space E. Suppose <TEX>$\{T_{n}\}$</TEX> (n = 1,2,...) is a uniformly asymptotically regular sequence of nonexpansive mappings from K to K such that <TEX>${\cap}_{n=1}^{\infty}$</TEX> F<TEX>$\(T_n){\neq}{\phi}$</TEX>. For <TEX>$x_0{\in}K$</TEX>, define <TEX>$x_{n+1}={\lambda}_{n+1}x_{n}+(1-{\lambda}_{n+1})T_{n+1}x_{n},n{\geq}0$</TEX>. If <TEX>${\lambda}_n{\subset}[0,1]$</TEX> satisfies <TEX>$lim_{n{\rightarrow}{\infty}}{\lambda}_n=0$</TEX>, we proved that <TEX>$\{x_n\}$</TEX> weakly converges to some <TEX>$z{\in}F\;as\;n{\rightarrow}{\infty}$</TEX> in the framework of reflexive Banach space E which satisfies the Opial's condition or has <TEX>$Fr{\acute{e}}chet$</TEX> differentiable norm or its dual <TEX>$E^*$</TEX> has the Kadec-Klee property. We also obtain that <TEX>$\{x_n\}$</TEX> strongly converges to some <TEX>$z{\in}F$</TEX> in Banach space E if K is a compact subset of E or there exists one map <TEX>$T{\in}\{T_{n};n=1,2,...\}$</TEX> satisfy some compact conditions such as T is semi compact or satisfy Condition A or <TEX>$lim_{n{\rightarrow}{\infty}}d(x_{n},F(T))=0$</TEX> and so on.