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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 A. Meir and 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