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  • Research Article
  • Open Access

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

  • 1,
  • 2 and
  • 1Email author
Fixed Point Theory and Applications20082008:732193

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


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.


  • Hilbert Space
  • Mathematical Programming
  • Differential Geometry
  • Convergence Theorem
  • Hybrid Method

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Authors’ Affiliations

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


© Kittikorn Nakprasit et al. 2008

This article is published under license to BioMed Central Ltd. 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.