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Mathematica Slovaca

, Volume 61, Issue 2, pp 257–274 | Cite as

Viscosity approximation methods for monotone mappings and a countable family of nonexpansive mappings

  • Poom Kumam
  • Somyot Plubtieng
Article
  • 52 Downloads

Abstract

We use viscosity approximation methods to obtain strong convergence to common fixed points of monotone mappings and a countable family of nonexpansive mappings. Let C be a nonempty closed convex subset of a Hilbert space H and P C is a metric projection. We consider the iteration process {x n } of C defined by x 1 = xC is arbitrary and
$$ x_{n + 1} = \alpha _n f(x_n ) + (1 - \alpha _n )S_n P_C (x_n + \lambda _n Ax_n ) $$
where f is a contraction on C, {S n } is a sequence of nonexpansive self-mappings of a closed convex subset C of H, and A is an inverse-strongly-monotone mapping of C into H. It is shown that {x n } converges strongly to a common element of the set of common fixed points of a countable family of nonexpansive mappings and the set of solutions of the variational inequality for an inverse-strongly-monotone mapping which solves some variational inequality. Finally, the ideas of our results are applied to find a common element of the set of equilibrium problems and the set of solutions of the variational inequality problem, a zero of a maximal monotone operator and a strictly pseudocontractive mapping in a real Hilbert space. The results of this paper extend and improve the results of Chen, Zhang and Fan.

Keywords

nonexpansive mapping monotone mapping equilibrium problem variational inequality accretive operator 

2010 Mathematics Subject Classification

Primary 46C05, 47D03, 47H09 Secondary 47H10, 47H20 

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Copyright information

© © Versita Warsaw and Springer-Verlag Wien 2011

Authors and Affiliations

  1. 1.Department of Mathematics Faculty of ScienceKing Mongkut’s University of Technology Thonburi (KMUTT) BangmodBangkokThailand
  2. 2.Department of Mathematics Faculty of ScienceNaresuan UniversityPhitsanulokThailand

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