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Zbl 1166.65026
Chancelier, Jean-Philippe
Iterative schemes for computing fixed points of nonexpansive mappings in Banach spaces.
(English)
[J] J. Math. Anal. Appl. 353, No. 1, 141-153 (2009). ISSN 0022-247X

The author discusses the convergence of an explicit iterative schemes involving a sequence of nonexpansive mappings $\{T_n\}$ on a real Banach space (with some properties), and also a contraction $f$. A general framework is developed to prove the strong convergence of the iterative schemes to the fixed point of a nonexpansive mapping $T$, related to the sequence $\{T_n\}$. By specifying the sequence $\{T_n\}$, the author recovers and extends some known convergence theorems. In the iterative schemes, the contraction $f$ can be replaced by the Meir-Keeler contraction [see {\it A. Meir} and {\it E. Keeler}, J. Math. Anal. Appl. 28, 326--329 (1969; Zbl 0194.44904)]. Seven examples of iterative schemes from the literature which are generalized by the iterative schemes proposed by the author, are analyzed in detail.
[Iulian Coroian (Baia Mare)]
MSC 2000:
*65J15 Equations with nonlinear operators (numerical methods)
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
47J25 Methods for solving nonlinear operator equations (general)
47H09 Mappings defined by "shrinking" properties

Keywords: nonexpansive mappings; fixed point; viscosity approximation; Meir-Keeler contraction; numerical examples; convergence; iterative schemes; Banach space

Citations: Zbl 0194.44904

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