Open Access

Implicit Iteration Methods with Errors for Discrete Non-Lipschitzian Families with Perturbed Mappings

Journal of Inequalities and Applications20102010:820535

https://doi.org/10.1155/2010/820535

Received: 31 October 2009

Accepted: 3 March 2010

Published: 25 March 2010

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 -uniformly smooth Banach spaces with a perturbed mapping , , 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.

Keywords

Iteration MethodFull ArticlePublisher NoteImplicit IterationPerturbed Mapping

Publisher note

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Authors’ Affiliations

(1)
Division of Mathematical Sciences, Pukyong National University, Busan, South Korea

Copyright

© T.-H. Kim and K.-Y. Kim. 2010

This article is published under license to BioMed Central Ltd. 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.