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

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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Correspondence to Satit Saejung.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Nakprasit, K., Nilsrakoo, W. & Saejung, S. Weak and Strong Convergence Theorems of an Implicit Iteration Process for a Countable Family of Nonexpansive Mappings. Fixed Point Theory Appl 2008, 732193 (2008). https://doi.org/10.1155/2008/732193

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  • DOI: https://doi.org/10.1155/2008/732193

Keywords

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