Shioji, Naoki; Takahashi, Wataru Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces. (English) Zbl 0888.47034 Proc. Am. Math. Soc. 125, No. 12, 3641-3645 (1997). 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. Cited in 5 ReviewsCited in 204 Documents MSC: 47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. 47J25 Iterative procedures involving nonlinear operators 49M05 Numerical methods based on necessary conditions Keywords:nonexpansive mappings; strong convergence; Banach limits; iteration PDFBibTeX XMLCite \textit{N. Shioji} and \textit{W. Takahashi}, Proc. Am. Math. Soc. 125, No. 12, 3641--3645 (1997; Zbl 0888.47034) Full Text: DOI 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.