Open Access

Towards viscosity approximations of hierarchical fixed-point problems

Fixed Point Theory and Applications20062006:95453

DOI: 10.1155/FPTA/2006/95453

Received: 10 February 2006

Accepted: 18 September 2006

Published: 27 November 2006


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, , where denotes the normal cone to the set of fixed point of the original nonexpansive mapping and a suitable nonexpansive mapping criterion. The link with other approximation schemes in this field is also made.


Authors’ Affiliations

GRIMAAG, Département Scientifique interfacultaires, Université Antilles Guyane, Schelcher


  1. Attouch H: Variational Convergence for Functions and Operators, Applicable Mathematics Series. Pitman, Massachusetts; 1984:xiv+423.Google Scholar
  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.Google Scholar
  3. Combettes PL, Hirstoaga SA: Approximating curves for nonexpansive and monotone operators. to appear in Journal of Convex AnalysisGoogle Scholar
  4. Lions P-L: Two remarks on the convergence of convex functions and monotone operators. Nonlinear Analysis. Theory, Methods and Applications 1978,2(5):553–562. 10.1016/0362-546X(78)90003-2View ArticleMathSciNetMATHGoogle Scholar
  5. Marino G, Xu H-K: A general iterative method for nonexpansive mappings in Hilbert spaces. Journal of Mathematical Analysis and Applications 2006,318(1):43–52. 10.1016/j.jmaa.2005.05.028MathSciNetView ArticleMATHGoogle Scholar
  6. Moudafi A: Viscosity approximation methods for fixed-points problems. Journal of Mathematical Analysis and Applications 2000,241(1):46–55. 10.1006/jmaa.1999.6615MathSciNetView ArticleMATHGoogle Scholar
  7. Rockafellar RT, Wets R: Variational Analysis. Springer, Berlin; 1988.MATHGoogle Scholar
  8. Xu H-K: Viscosity approximation methods for nonexpansive mappings. Journal of Mathematical Analysis and Applications 2004,298(1):279–291. 10.1016/j.jmaa.2004.04.059MathSciNetView ArticleMATHGoogle Scholar
  9. Yamada I: The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. In Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications. Elsevier, New York; 2001:473–504.View ArticleGoogle Scholar


© A. Moudafi and P.-E. Maingé. 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.