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

L. C. Ceng,1 N. C. Wong,2 and J. C. Yao2

 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

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.