Convergence in norm of modified Krasnoselski–Mann iterations for fixed points of demicontractive mappings

This paper deals with fixed points methods related to the general class of demicontractive mappings (including the well-known classes of nonexpansive and quasi-nonexpansive mappings) in Hilbert spaces. Specifically, we point out some historical aspects concerning the concept of demicontactivity and...

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Bibliographic Details
Published in:Applied mathematics and computation 2011-08, Vol.217 (24), p.9864-9874
Main Authors: MAINGE, Paul-Emile, MARUSTER, Stefan
Format: Article
Language:English
Subjects:
Algorithms
Computation
Convergence
Convex feasibility problem
Demicontractive operator
Exact sciences and technology
Feasibility
Fixed point problem
Fixed points (mathematics)
Functional analysis
Global analysis, analysis on manifolds
Iterative methods
Krasnoselski–Mann iteration
Mapping
Mathematical analysis
Mathematical models
Mathematics
Norms
Numerical analysis
Numerical analysis. Scientific computation
Optimization and Control
Sciences and techniques of general use
Strong convergence
Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
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Summary:This paper deals with fixed points methods related to the general class of demicontractive mappings (including the well-known classes of nonexpansive and quasi-nonexpansive mappings) in Hilbert spaces. Specifically, we point out some historical aspects concerning the concept of demicontactivity and we investigate a regularized variant of the Krasnoselski-Mann iteration that can be alternatively regarded as a simplified form of the inertial iteration (P-E. Maingé, J. Math. Anal. Appl. 344 (2008) 876–887) with non-constant relaxation factors. These two methods ensure the strong convergence of the generated sequence towards the least norm element of the set of fixed-points of demicontractive mappings. However, for convergence, our method does not require anymore the knowledge of some constant related to the involved demicontractive operator. A new and simpler proof is also proposed for its convergence even when involving non-constant relaxation factors. We point out the simplicity of this algorithm (at least from computational point of view) in comparison with other existing methods. We also present some numerical experiments concerning a convex feasibility problem, experiments that emphasize the characteristics of the considered algorithm comparing with a classical cyclic projection-type iteration.
ISSN:0096-3003
1873-5649
DOI:10.1016/j.amc.2011.04.068