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

Sornsak Thianwan

Abstract

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 quasi-nonexpansive mappings converges strongly to a common fixed point of the family in a uniformly convex Banach 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 implicit iteration process for a finite family of nonexpansive Mappings, Scientiae Mathematicae Joponicae 66 (2007), 221–229], Sun [Z.H. Sun, Strong convergence of an implicit iteration for a finite family of asymptotically quasi-nonexpansive mappings, J. Math. Anal. Appl. 286 (2003), 351–358], and other authors.