Weak and Strong Convergence of an Implicit Iteration Process for a Finite Family of Asymptotically Quasi-Nonexpansive Mappings

In this paper, a new implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings is introduced and studied. We prove that the implicit iteration sequence for a finite family of asymptotically quasinonexpansive mappings converges strongly to a common fixed point of the family in a uniformly convex Baliach space, requiring; one member T in the family which is either semi-compact or satisfies condition (C). More precisely, weak convergence theorems are established for the implicit, iteration process in a uniformly convex Banach space which satisfies Opial's condition. Our results generalize and extend the recent ones announced by Thianwan and Suantai [S. Thianwan and S. Suantai, Weak and strong convergence of an i itiplicit iteration process for a finite family of nonexpansive Mappings, Scientiae Matlientaticae Joponicae 66 (2007), 221-229J, Sun Z.H.Sun, Strong convergence of an implicit iteration for a finite family of asymptotically quasi-nonexpansive mappings,.1. Matti. A ppl. 286 (2003), 351-358, and other authors.