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A hybrid approximation method for equilibrium and fixed point problems for a monotone mapping and a nonexpansive mapping. (English) Zbl 1163.49003
Summary: The purpose of this paper is to present an iterative scheme by a hybrid method for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of an equilibrium problem and the set of solutions of the variational inequality for $\alpha$-inverse-strongly monotone mappings in the framework of a Hilbert space. We show that the iterative sequence converges strongly to a common element of the above three sets under appropriate conditions. Additionally, the idea of our results are applied to find a zero of a maximal monotone operator and a strictly pseudocontractive mapping in a real Hilbert space.

##### MSC:
 49J40 Variational methods including variational inequalities 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 47H05 Monotone operators (with respect to duality) and generalizations 49M30 Other numerical methods in calculus of variations 47J20 Inequalities involving nonlinear operators