Abstract and Applied Analysis
Volume 2013 (2013), Article ID 202095, 8 pages
http://dx.doi.org/10.1155/2013/202095
Research Article

Strong Convergence Theorems for Semigroups of Asymptotically Nonexpansive Mappings in Banach Spaces

D. R. Sahu,1 Ngai-Ching Wong,2,3 and Jen-Chih Yao3,4

1Department of Mathematics, Banaras Hindu University, Varanasi 221005, India
2Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 804, Taiwan
3Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 807, Taiwan
4Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 4 February 2013; Accepted 4 April 2013

Academic Editor: Qamrul Hasan Ansari

Copyright © 2013 D. R. Sahu et al. 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

Let be a real reflexive Banach space with a weakly continuous duality mapping . Let be a nonempty weakly closed star-shaped (with respect to ) subset of . Let  =  be a uniformly continuous semigroup of asymptotically nonexpansive self-mappings of , which is uniformly continuous at zero. We will show that the implicit iteration scheme: , for all , converges strongly to a common fixed point of the semigroup for some suitably chosen parameters and . Our results extend and improve corresponding ones of Suzuki (2002), Xu (2005), and Zegeye and Shahzad (2009).