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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.

##### MSC:
 47H10 Fixed-point theorems 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 49J40 Variational inequalities
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##### References:
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