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Viscosity approximation methods for strongly positive and monotone operators. (English) Zbl 1196.47044

The purpose of this paper is to suggest and analyze both explicit and implicit iterative schemes for two strongly positive operators and a nonexpansive mapping S on a Hilbert space, and to study explicit and implicit versions of iterative schemes for an inverse-strongly monotone mapping T and S by an extragradient-like approximation method. The viscosity approximation methods are employed to establish strong convergence of the iterative schemes to a common element of the set of fixed points of S and the set of solutions of the variational inequality for T. Further, some applications of finding a common fixed point of a nonexpansive mapping and a strictly pseudocontractive mapping which solves some variational inequalities are given.

The results presented in this paper improve and unify various results about viscosity approximation methods for fixed-point problems and variational inequality problems.

47J25Iterative procedures (nonlinear operator equations)
47J20Inequalities involving nonlinear operators
49J40Variational methods including variational inequalities
90C31Sensitivity, stability, parametric optimization