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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.
MSC:
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