International Journal of Mathematics and Mathematical Sciences
Volume 27 (2001), Issue 11, Pages 653-662

Convergence theorems of the sequence of iterates for a finite family asymptotically nonexpansive mappings

Jui-Chi Huang

Department of General Education, Kuang Wu Institute of Technology, Peito, Taipei 11271, Taiwan

Received 30 March 2001

Copyright © 2001 Jui-Chi Huang. 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.


Let E be a uniformly convex Banach space, C a nonempty closed convex subset of E. In this paper, we introduce an iteration scheme with errors in the sense of Xu (1998) generated by {Tj:CC}j=1r as follows: Un(j)=an(j)I+bn(j)TjnUn(j1)+cn(j)un(j), j=1,2,,r, x1C, xn+1=an(r)xn+bn(r)TrnUn(r1)xn+cn(r)un(r), n1, where Un(0):=I, I the identity map; and {un(j)} are bounded sequences in C; and {an(j)}, {bn(j)}, and {cn(j)} are suitable sequences in [0,1]. We first consider the behaviour of iteration scheme above for a finite family of asymptotically nonexpansive mappings. Then we generalize theorems of Schu and Rhoades.