Copyright © 2010 Tae-Hwa Kim and Kui-Yeon Kim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Motivated and inspired by the recent works on implicit iteration methods, we prove necessary and sufficient conditions for strong convergence of the eventually implicit iteration methods with errors to a common fixed point of a discrete family which is continuous total asymptotically nonexpansive (in brief, TAN) on q-uniformly smooth Banach spaces with a perturbed mapping F, 1<q≤2, under some suitable control conditions of parameters. Some applications to viscosity approximation methods or to the eventually implicit algorithms with errors for a finite family of TAN self-mappings in real Banach spaces are also added.