Journal of Inequalities and Applications
Volume 2006 (2006), Article ID 65983, 11 pages
doi:10.1155/JIA/2006/65983
Implicit predictor-corrector iteration process for finitely many
asymptotically (quasi-)nonexpansive mappings
1Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
2Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 804, Taiwan
Received 13 February 2006; Revised 3 June 2006; Accepted 5 June 2006
Copyright © 2006 L. C. Ceng 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
We study an implicit predictor-corrector iteration process for finitely many asymptotically quasi-nonexpansive self-mappings on a nonempty closed convex subset of a Banach space E. We derive a necessary and sufficient condition for the strong convergence of this iteration process to a common fixed point of these mappings. In the case E is a uniformly convex Banach space and the mappings are asymptotically nonexpansive, we verify the weak (resp., strong) convergence of this iteration process to a common fixed point of these mappings if Opial's condition is satisfied (resp., one of these mappings is semicompact). Our results improve and extend earlier and recent ones in the literature.