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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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