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Viscosity approximation methods for monotone map**s and a countable family of nonexpansive map**s

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Mathematica Slovaca

Abstract

We use viscosity approximation methods to obtain strong convergence to common fixed points of monotone map**s and a countable family of nonexpansive map**s. 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 = x ∈ C 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-map**s of a closed convex subset C of H, and A is an inverse-strongly-monotone map** 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 map**s and the set of solutions of the variational inequality for an inverse-strongly-monotone map** 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 map** in a real Hilbert space. The results of this paper extend and improve the results of Chen, Zhang and Fan.

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Authors and Affiliations

  1. Department of Mathematics Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT) Bangmod, Bangkok, 10140, Thailand

    Poom Kumam

  2. Department of Mathematics Faculty of Science, Naresuan University, Phitsanulok, 65000, Thailand

    Somyot Plubtieng

Authors
  1. Poom Kumam
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  2. Somyot Plubtieng
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Corresponding author

Correspondence to Poom Kumam.

Additional information

Communicated by L’ubica Holá

This work was completed with the support of the Thailand Research Fund and the Commission on Higher Education under grant No. MRG5380044. The first author was supported by the Higher Education Commission and the Thailand Research Fund under Grant MRG5380044.

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Kumam, P., Plubtieng, S. Viscosity approximation methods for monotone map**s and a countable family of nonexpansive map**s. Math. Slovaca 61, 257–274 (2011). https://doi.org/10.2478/s12175-011-0010-9

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  • DOI: https://doi.org/10.2478/s12175-011-0010-9

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