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Viscosity approximation methods for nonexpansive mappings. (English) Zbl 1061.47060
The author extends Moudafi’s result in Hilbert spaces [{\it 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: $\leq \alpha t\Vert x-y\Vert + (1-t)\Vert x-y\Vert = (1-t(1-\alpha ))\Vert x-y\Vert .$ (ii) Under Remark on p. 290 (line 17), it should read: For the iteration process (19) (or (38)).

MSC:
 47J25 Iterative procedures (nonlinear operator equations) 47H09 Mappings defined by “shrinking” properties 47H10 Fixed-point theorems for nonlinear operators on topological linear spaces
Full Text:
References:
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