Open Access Research

Almost stability of the Mann type iteration method with error term involving strictly hemicontractive mappings in smooth Banach spaces

Nawab Hussain1, Arif Rafiq2, Ljubomir B Ciric3 and Saleh Al-Mezel1*

Author Affiliations

1 Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

2 Hajvery University, 43-52 Industrial Area, Gulberg-III, Lahore, Pakistan

3 Faculty of Mechanical Engineering, University of Belgrade, Al. Rudara 12-35, Belgrade, 11 070, Serbia

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Journal of Inequalities and Applications 2012, 2012:207 doi:10.1186/1029-242X-2012-207

Published: 24 September 2012

Abstract

Let K be a nonempty closed bounded convex subset of an arbitrary smooth Banach space X and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/207/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/207/mathml/M1">View MathML</a> be a continuous strictly hemicontractive mapping. Under some conditions, we obtain that the Mann iteration method with error term converges strongly to a unique fixed point of T and is almost T-stable on K. As an application of our results, we establish strong convergence of a multi-step iteration process.

Keywords:
Mann iteration method with error term; strictly hemicontractive operators; strongly pseudocontractive operators; local strongly pseudocontractive operators; continuous mappings; Lipschitz mappings; smooth Banach spaces