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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)).

47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
Full Text: DOI
[1] Browder, F. E.: Convergence of approximants to fixed points of non-expansive maps in Banach spaces. Arch. rational mech. Anal. 24, 82-90 (1967) · Zbl 0148.13601
[2] Geobel, K.; Kirk, W. A.: Topics in metric fixed point theory. Cambridge studies in advanced mathematics 28 (1990)
[3] Geobel, K.; Reich, S.: Uniform convexity, nonexpansive mappings, and hyperbolic geometry. (1984) · Zbl 0537.46001
[4] Halpern, B.: Fixed points of nonexpanding maps. Bull. amer. Math. soc. 73, 957-961 (1967) · Zbl 0177.19101
[5] Lions, P. L.: Approximation de points fixes de contractions. C. R. Acad. sci. Sèr. A--B Paris 284, 1357-1359 (1977) · Zbl 0349.47046
[6] Moudafi, A.: Viscosity approximation methods for fixed-points problems. J. math. Anal. appl. 241, 46-55 (2000) · Zbl 0957.47039
[7] Reich, S.: Strong convergence theorems for resolvents of accretive operators in Banach spaces. J. math. Anal. appl. 75, 287-292 (1980) · Zbl 0437.47047
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[9] Xu, H. K.: An iterative approach to quadratic optimization. J. optim. Theory appl. 116, 659-678 (2003) · Zbl 1043.90063