Open Access Research Article

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

Kittikorn Nakprasit1, Weerayuth Nilsrakoo2 and Satit Saejung1*

Author Affiliations

1 Department of Mathematics, Khon Kaen University, Khon Kaen 40002, Thailand

2 Department of Mathematics, Statistics and Computer, Ubon Rajathanee University, Ubon Ratchathani 34190, Thailand

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Fixed Point Theory and Applications 2008, 2008:732193  doi:10.1155/2008/732193


The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2008/1/732193


Received: 22 July 2008
Accepted: 18 November 2008
Published: 26 November 2008

© 2008 The Author(s).

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

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.

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