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Viscosity approximation methods for nonexpansive mappings. (English) Zbl 1061.47060
The author extends Moudafi’s result in Hilbert spaces [A. Moudafi, J. Math. Anal. Appl. 241, 46–55 (2000; Zbl 0957.47039)] and further proves the strong convergence of a continuous scheme of Theorem 1.4. Further the hypotheses (H1)-(H3) for the iteration process (19) (or (38)) are refinements of those of Moudafi. The author provides a remarkable proof for the iterative scheme \(x_{n+1} = \alpha _{n}f(x_{n}) + (1-\alpha _{n})Tx_{n}\) in uniformly smooth Banach spaces. These two results are of independent interest in both the theories of nonlinear operator equations and optimization.
Reviewer’s remarks: (i) There is an omission of a vital term on pp 280 (line 13). The inequality should read: \(\leq \alpha t\| x-y\| + (1-t)\| x-y\| = (1-t(1-\alpha ))\| x-y\| .\) (ii) Under Remark on p. 290 (line 17), it should read: For the iteration process (19) (or (38)).

MSC:
47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
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