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Iterative algorithms for hierarchical fixed points problems and variational inequalities. (English) Zbl 1205.65192
Summary: This paper deals with a method for approximating a solution of the fixed point problem: find $\stackrel{˜}{x}\in H$; $\stackrel{˜}{x}=\left({\mathrm{proj}}_{F\left(t\right)}S\right)\stackrel{˜}{x}$, where $H$ is a Hilbert space, $S$ is some nonlinear operator and $T$ is a nonexpansive mapping on a closed convex subset $C$ and ${\mathrm{proj}}_{F\left(t\right)}$ denotes the metric projection on the set of fixed points of $T$. First, we prove a strong convergence theorem by using a projection method which solves some variational inequality. As a special case, this projection method also solves some minimization problems. Secondly, under different restrictions on parameters, we obtain another strong convergence result which solves the above fixed point problem.
##### MSC:
 65J15 Equations with nonlinear operators (numerical methods) 65K15 Numerical methods for variational inequalities and related problems 49J40 Variational methods including variational inequalities