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

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## 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 where

*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$$
x_{n + 1} = \alpha _n f(x_n ) + (1 - \alpha _n )S_n P_C (x_n + \lambda _n Ax_n )
$$

*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## Preview

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