Regularized and inertial algorithms for common fixed points of nonlinear operators

This paper deals with a general fixed point iteration for computing a point in some nonempty closed and convex solution set included in the common fixed point set of a sequence of mappings on a real Hilbert space. The proposed method combines two strategies: viscosity approximations (regularization)...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2008-08, Vol.344 (2), p.876-887
Main Author: MAINGE, Paul-Emile
Format: Article
Language:English
Subjects:
Convex and discrete geometry
Demicontractive
Exact sciences and technology
Fixed point
Functional analysis
Geometry
Global analysis, analysis on manifolds
Inertial type extrapolation
Mathematical analysis
Mathematics
Operator theory
Optimization and Control
Proximal method
Pseudocontractive
Quasi-nonexpansive
Sciences and techniques of general use
Strongly convergent method
Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
Viscosity method
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Summary:This paper deals with a general fixed point iteration for computing a point in some nonempty closed and convex solution set included in the common fixed point set of a sequence of mappings on a real Hilbert space. The proposed method combines two strategies: viscosity approximations (regularization) and inertial type extrapolation. The first strategy is known to ensure the strong convergence of some successive approximation methods, while the second one is intended to speed up the convergence process. Under classical conditions on the operators and the parameters, we prove that the sequence of iterates generated by our scheme converges strongly to the element of minimal norm in the solution set. This algorithm works, for instance, for approximating common fixed points of infinite families of demicontractive mappings, including the classes of quasi-nonexpansive operators and strictly pseudocontractive ones.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2008.03.028