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Zbl 1143.47305
Moudafi, A.; Maingé, P.-E.
Towards viscosity approximations of hierarchical fixed-point problems. (English)
[J] Fixed Point Theory Appl. 2006, Article ID 95453, 10 p. (2006). ISSN 1687-1812/e

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.

MSC 2000:: *47J25 Methods for solving nonlinear operator equations (general)
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
47H09 Mappings defined by "shrinking" properties
47J20 Inequalities involving nonlinear operators

Keywords: fixed point iteration; weak convergence; strong convergence; viscosity approximation scheme; hierarchical fixed-point problems; nonexpansive mapping

Cited in: Zbl 1168.49005

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