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.


47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
49J40 Variational inequalities
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