# 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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## References

- [1]AOYAMA, K.—KIMURA, Y.—TAKAHASHI, W.—TOYODA M.:
*Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space*, Nonlinear Anal.**67**(2007), 2350–2360.CrossRefMATHMathSciNetGoogle Scholar - [2]BLUM, E.—OETTLI, W.:
*From optimization and variational inequalities to equilibrium problems*, Math. Student**63**(1994), 123–145.MATHMathSciNetGoogle Scholar - [3]CHEN, R.—ZHU, Z.:
*Viscosity approximation method for accretive operator in Banach space*, Nonlinear Anal.**69**(2008), 1356–1363.CrossRefMATHMathSciNetGoogle Scholar - [4]CHEN, J.—ZHANG, L.—FAN, T.:
*Viscosity approximation methods for nonexpansive mappings and monotone mappings*, J. Math. Anal. Appl.**334**(2007), 1450–1461.CrossRefMATHMathSciNetGoogle Scholar - [5]COMBETTES, P. L.—HIRSTOAGA, S. A.:
*Equilibrium programming in Hilbert spaces*, J. Nonlinear Convex Anal.**6**(2005), 117–136.MATHMathSciNetGoogle Scholar - [6]FLAM, S. D.—ANTIPIN, A. S.:
*Equilibrium progamming using proximal-link algolithms*, Math. Program.**78**(1997), 29–41.CrossRefMATHMathSciNetGoogle Scholar - [7]GOEBEL, K.—KIRK, W. A.:
*Topics in Metric Fixed Point Theory*, Cambridge University Press, Cambridge, 1990.CrossRefMATHGoogle Scholar - [8]IIDUKA, H.—TAKAHASHI, W.:
*Strong convergence theorems for nonexpansive mapping and inverse-strong monotone mappings*, Nonlinear Anal.**61**(2005), 341–350.CrossRefMATHMathSciNetGoogle Scholar - [9]MOUDAFI, A.:
*Viscosity approximation methods for fixed-points problems*, J. Math. Anal. Appl.**241**(2000), 46–55.CrossRefMATHMathSciNetGoogle Scholar - [10]MOUDAFI, A.—THERA, M.:
*Proximal and dynamical approaches to equilibrium problems*. In: Lecture Notes in Econom. and Math. Systems 477, Springer-Verlag, New York, 1999, pp. 187–201.Google Scholar - [11]NILSRAKOO, W.—SAEJUNG, S.:
*Weak and strong convergence theorems for countable Lipschitzian mappings and its applications*, Nonlinear Anal.**69**(2008), 2695–2708.CrossRefMATHMathSciNetGoogle Scholar - [12]OPIAL, Z.:
*Weak convergence of successive approximations for nonexpansive mappings*, Bull. Amer. Math. Soc.**73**(1967), 591–597.CrossRefMATHMathSciNetGoogle Scholar - [13]PLUBTIENG, S.—PUNPAENG, R.:
*Fixed-point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces*, Math. Comput. Modelling**48**(2008), 279–286.CrossRefMATHMathSciNetGoogle Scholar - [14]PLUBTIENG, S.—KUMAM, P.:
*Weak convergence theorem for monotone mappings and a countable family of nonexpansive mappings*, J. Comput. Appl. Math.**224**(2009), 614–621.CrossRefMATHMathSciNetGoogle Scholar - [15]TAKAHASHI, W.:
*Nonlinear Functional Analysis*, Yokohama Publishers, Yokohama, 2000.MATHGoogle Scholar - [16]TAKAHASHI, S.—TAKAHASHI, W.:
*Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces*, J. Math. Anal. Appl.**331**(2007), 506–515.CrossRefMATHMathSciNetGoogle Scholar - [17]TAKAHASHI, W.—TOYODA, M.:
*Weak convergence theorems for nonexpansive mappings and monotone mappings*, J. Optim. Theory Appl.**118**(2003), 417–428.CrossRefMATHMathSciNetGoogle Scholar - [18]XU, H. K.:
*Iterative algorithms for nonlinear operators*, J. London Math. Soc. (2)**66**(2002), 240–256.CrossRefMATHMathSciNetGoogle Scholar - [19]XU, H. K.:
*Viscosity approximation methods for nonexpansive mappings*, J. Math. Anal. Appl.**298**(2004), 279–291.CrossRefMATHMathSciNetGoogle Scholar

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