## Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems.(English)Zbl 1198.47081

Summary: The purpose of this paper is to investigate the problem of finding a common element of the set of fixed points $$F(S)$$ of a nonexpansive mapping $$S$$ and the set of solutions $$\Omega_{A }$$ of the variational inequality for a monotone, Lipschitz continuous mapping $$A$$. We introduce a hybrid extragradient-like approximation method which is based on the well-known extragradient method and a hybrid (or outer approximation) method. The method produces three sequences which are shown to converge strongly to the same common element of $${F(S)\cap\Omega_{A}}$$. As applications, the method provides an algorithm for finding the common fixed point of a nonexpansive mapping and a pseudocontractive mapping, or a common zero of a monotone Lipschitz continuous mapping and a maximal monotone mapping.

### MSC:

 47J25 Iterative procedures involving nonlinear operators 47J20 Variational and other types of inequalities involving nonlinear operators (general) 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.
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