zbMATH — the first resource for mathematics

Towards viscosity approximations of hierarchical fixed-point problems. (English) Zbl 1143.47305
Summary: We introduce methods which seem to be a new and promising tool in hierarchical fixed-point problems. The goal of this note is to analyze the convergence properties of these new types of approximating methods for fixed-point problems. The limit attained by these curves is the solution of the general variational inequality \(0\in (I - Q)x_{\infty }+N_{\text{Fix}\,P}(x_{\infty })\), where \(N_{\text{Fix\,}P}\) denotes the normal cone to the set of fixed point of the original nonexpansive mapping \(P\) and \(Q\) a suitable nonexpansive mapping criterion. The link with other approximation schemes in this field is also made.

47J25 Iterative procedures involving nonlinear operators
47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47J20 Variational and other types of inequalities involving nonlinear operators (general)
Full Text: DOI EuDML
[1] Attouch H: Variational Convergence for Functions and Operators, Applicable Mathematics Series. Pitman, Massachusetts; 1984:xiv+423. · Zbl 0561.49012
[2] Brézis H: Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland Mathematics Studies, no. 5. North-Holland, Amsterdam; American Elsevier, New York; 1973:vi+183.
[3] Combettes PL, Hirstoaga SA: Approximating curves for nonexpansive and monotone operators. to appear in Journal of Convex Analysis · Zbl 1115.47042
[4] Lions, P-L, Two remarks on the convergence of convex functions and monotone operators, Nonlinear Analysis. Theory, Methods and Applications, 2, 553-562, (1978) · Zbl 0383.47033
[5] Marino, G; Xu, H-K, A general iterative method for nonexpansive mappings in Hilbert spaces, Journal of Mathematical Analysis and Applications, 318, 43-52, (2006) · Zbl 1095.47038
[6] Moudafi, A, Viscosity approximation methods for fixed-points problems, Journal of Mathematical Analysis and Applications, 241, 46-55, (2000) · Zbl 0957.47039
[7] Rockafellar RT, Wets R: Variational Analysis. Springer, Berlin; 1988. · Zbl 0888.49001
[8] Xu, H-K, Viscosity approximation methods for nonexpansive mappings, Journal of Mathematical Analysis and Applications, 298, 279-291, (2004) · Zbl 1061.47060
[9] Yamada, I, The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings, 473-504, (2001), New York · Zbl 1013.49005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.