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Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces. (English) Zbl 0888.47034

Summary: We study the convergence of the sequence defined by \[ x_0\in C,\qquad x_{n+1} = \alpha_{n}x + (1-\alpha_{n})Tx_n,\qquad n=0,1,2, \ldots, \] where \(0 \leq \alpha_n \leq 1\) and \(T\) is a nonexpansive mapping from a closed convex subset of a Banach space into itself.

MSC:

47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47J25 Iterative procedures involving nonlinear operators
49M05 Numerical methods based on necessary conditions
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References:

[1] S. Banach, Théorie des opérations linéaires, Monografie Mat., PWN, Warszawa, 1932. · JFM 58.0420.01
[2] Benjamin Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc. 73 (1967), 957 – 961. · Zbl 0177.19101
[3] G. G. Lorentz, A contribution to the theory of divergent series, Acta Math. 80 (1948), 167-190. · Zbl 0031.29501
[4] Simeon Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl. 75 (1980), no. 1, 287 – 292. · Zbl 0437.47047
[5] Simeon Reich, Some problems and results in fixed point theory, Topological methods in nonlinear functional analysis (Toronto, Ont., 1982) Contemp. Math., vol. 21, Amer. Math. Soc., Providence, RI, 1983, pp. 179 – 187. · Zbl 0531.47048
[6] Wataru Takahashi and Yoichi Ueda, On Reich’s strong convergence theorems for resolvents of accretive operators, J. Math. Anal. Appl. 104 (1984), no. 2, 546 – 553. · Zbl 0599.47084
[7] Rainer Wittmann, Approximation of fixed points of nonexpansive mappings, Arch. Math. (Basel) 58 (1992), no. 5, 486 – 491. · Zbl 0797.47036
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