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A hybrid extragradient-viscosity method for monotone operators and fixed point problems. (English) Zbl 1178.90273
The author proposes an iterative method, in a real Hilbert space for computing a common element of the set of fixed points of a demicontractive operator (possibly quasi-nonexpansive or strictly pseudocontractive) and the set of solutions of a variational inequality problem for a monotone, Lipschitz continuous mapping. Strongly convergent algorithms are of fundamental importance for solving problems in infinite dimensional spaces. The considered algorithm can be regarded as a combination of a variation of the hybrid steepest descent method and the so-called extragradient method. Under classical conditions, the author proves the strong convergence of the sequences of iterates given by the considered scheme. The main convergence theorem is also applied to some specific examples.
90C25Convex programming
49M37Methods of nonlinear programming type in calculus of variations
65C20Models (numerical methods)