Journal of Inequalities and Applications
Volume 2007 (2007), Article ID 68616, 10 pages
doi:10.1155/2007/68616
Research Article

Convergece Theorems for Finite Families of Asymptotically Quasi-Nonexpansive Mappings

C. E. Chidume1 and Bashir Ali2

1Mathematics Section, The Abdus Salam International Centre for Theoretical Physics, Trieste 34014, Italy
2Department of Mathematical Sciences, Bayero University, Kano, Nigeria

Received 20 October 2006; Revised 30 January 2007; Accepted 31 January 2007

Academic Editor: Donal O'Regan

Copyright © 2007 C. E. Chidume and Bashir Ali. 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 E be a real Banach space, K a closed convex nonempty subset of E, and T1,T2,…,Tm:K→K asymptotically quasi-nonexpansive mappings with sequences (resp.) {kin}n=1∞ satisfying kin→1 as n→∞, and ∑n=1∞(kin−1)<∞, i=1,2,…,m. Let {αn}n=1∞ be a sequence in [ε, 1−ε],  ε∈(0,1). Define a sequence {xn} by x1∈K, xn+1=(1−αn)xn+αnT1nyn+m−2, yn+m−2=(1−αn)xn+αnT2nyn+m-3, …, yn=(1−αn)xn+αnTmnxn, n≥1,  m≥2. Let ⋂i=1mF(Ti)≠∅. Necessary and sufficient conditions for a strong convergence of the sequence {xn} to a common fixed point of the family {Ti}i=1m are proved. Under some appropriate conditions, strong and weak convergence theorems are also proved.