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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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[1] C. Ahmed, O. S, Regularisation des problemes de points fixes, Thèse, Université Montpellier II, 1998.
[2] Y. Albert, and, S. Guerre-Delabriere, Problems of Fixed Point Theory in Hilbert and Banach Spaces, Lecture delivered, Jan. 2, 1995, Technion University, Haifa.
[3] Attouch, H., Viscosity approximation methods for minimization problems, SIAM J. optim., 6, 769-806, (1996) · Zbl 0859.65065
[4] Attouch, H.; Cominetti, R., A dynamical approach to convex minimization coupling approximation with steepest method, J. differential equations, 128, 519-540, (1996) · Zbl 0886.49024
[5] Brézis, H., Opérateurs maximaux monotones et semi-groupes de contractions dans LES espaces de Hilbert, (1974), North-Holland Amsterdam
[6] Brézis, H.; Lions, P.L., Produits infinis de resolvantes, Israel J. math., 29, 329-345, (1978) · Zbl 0387.47038
[7] Cominetti, R.; San Martin, S., Asymptotical analysis of the exponential penalty trajectory in linear programming, Math. programming, 67, 169-187, (1994) · Zbl 0833.90081
[8] Lions, P.L., Two remarks on the convergence of the convex functions and monotone operators, Nonlinear anal. theory methods appl., 2, 553-562, (1978) · Zbl 0383.47033
[9] Moreau, J.J., Rafle par un convexe variable, (1971), Séminaire d’Analyse Convexe Montpellier · Zbl 0343.49019
[10] Mosco, U., Convergence of convex set and of solution of variational inequalities, Adv. math., 3, 510-585, (1969) · Zbl 0192.49101
[11] Vasil’ev, F.V., Numerical methods for solving extremum problems, (1988), Nauka Moscow
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