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Viscosity approximation methods for fixed-points problems. (English) Zbl 0957.47039
The main result of the paper is: Let $C$ be a closed convex set of a real Hilbert space $X$ and $P:C\rightarrow C$ a nonexpansive operator so that its fixed point set $S$ is nonempty. Given a sequence of positive real numbers $\varepsilon_{n}\rightarrow 0$ and a contraction $\pi:C\rightarrow C,$ the sequence $x_{n}\in C$ given by the unique fixed point in $C$ of the contraction $\frac{1}{1+\varepsilon_{n}}P+\frac{\varepsilon_{n}} {1+\varepsilon_{n}}\pi$ strongly converges to the unique fixed point in $S$ of the operator proj$_{S}\circ\pi.$ Applications are given to viscosity principles for optimization problems and inclusions for monotone operators.

47H10Fixed-point theorems for nonlinear operators on topological linear spaces
47H09Mappings defined by “shrinking” properties
49J40Variational methods including variational inequalities
Full Text: DOI
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