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Journal of Applied Mathematics
Volume 2012, Article ID 281383, 21 pages
Research Article

On Unification of the Strong Convergence Theorems for a Finite Family of Total Asymptotically Nonexpansive Mappings in Banach Spaces

Department of Computational & Theoretical Sciences, Faculty of Sciences, International Islamic University Malaysia, P.O. Box 141, 25710, Kuantan, Malaysia

Received 5 January 2012; Revised 10 March 2012; Accepted 12 March 2012

Academic Editor: Giuseppe Marino

Copyright © 2012 Farrukh Mukhamedov and Mansoor Saburov. 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.


We unify all known iterative methods by introducing a new explicit iterative scheme for approximation of common fixed points of finite families of total asymptotically I-nonexpansive mappings. Note that such a scheme contains a particular case of the method introduced by (C. E. Chidume and E. U. Ofoedu, 2009). We construct examples of total asymptotically nonexpansive mappings which are not asymptotically nonexpansive. Note that no such kind of examples were known in the literature. We prove the strong convergence theorems for such iterative process to a common fixed point of the finite family of total asymptotically I-nonexpansive and total asymptotically nonexpansive mappings, defined on a nonempty closed-convex subset of uniformly convex Banach spaces. Moreover, our results extend and unify all known results.