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Iterative schemes for computing fixed points of nonexpansive mappings in Banach spaces. (English) Zbl 1166.65026
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.

65J15Equations with nonlinear operators (numerical methods)
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
Full Text: DOI
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