Fixed Point Theory and Applications
Volume 2008 (2008), Article ID 732193, 18 pages
Research Article

Weak and Strong Convergence Theorems of an Implicit Iteration Process for a Countable Family of Nonexpansive Mappings

Kittikorn Nakprasit,1 Weerayuth Nilsrakoo,2 and Satit Saejung1

1Department of Mathematics, Khon Kaen University, Khon Kaen 40002, Thailand
2Department of Mathematics, Statistics and Computer, Ubon Rajathanee University, Ubon Ratchathani 34190, Thailand

Received 22 July 2008; Accepted 18 November 2008

Academic Editor: Anthony Lau

Copyright © 2008 Kittikorn Nakprasit 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.


Using the implicit iteration and the hybrid method in mathematical programming, we prove weak and strong convergence theorems for finding common fixed points of a countable family of nonexpansive mappings in a real Hilbert space. Our results include many convergence theorems by Xu and Ori (2001) and Zhang and Su (2007) as special cases. We also apply our method to find a common element to the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem. Finally, we propose an iteration to obtain convergence theorems for a continuous monotone mapping.