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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 Ω 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)Ω 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.
47J25Iterative procedures (nonlinear operator equations)
47J20Inequalities involving nonlinear operators
47H09Mappings defined by “shrinking” properties
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