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Strong convergence of composite iterative methods for equilibrium problems and fixed point problems. (English) Zbl 1175.65068
Summary: We introduce a new composite iterative scheme by viscosity approximation method for finding a common point of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. It is proved that the sequence generated by the iterative scheme converges strongly to a common point of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping. Our results substantially improve the corresponding results of A. Takahashi and W. Takahashi [Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331, No. 1, 506–515 (2007; Zbl 1122.47056)]. Essentially a new approach for finding solutions of equilibrium problems and the fixed points of nonexpansive mappings is provided.

##### MSC:
 65J15 Numerical solutions to equations with nonlinear operators 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47H10 Fixed-point theorems 47J20 Variational and other types of inequalities involving nonlinear operators (general) 65K15 Numerical methods for variational inequalities and related problems
Zbl 1122.47056
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