Ivan, Daniel; Leuştean, Laurenţiu A rate of asymptotic regularity for the Mann iteration of \(\kappa\)-strict pseudo-contractions. (English) Zbl 1334.47063 Numer. Funct. Anal. Optim. 36, No. 6, 792-798 (2015). Let \(H\) be a Hilbert space, \(C\subset H\) be a nonexpty convex subset and \(T:C\to C\) be a \(\kappa\)-strict pseudocontraction, where \(\kappa\in[0,1)\), i.e., \(\| Tx-Ty\|^2\leq\| x-y\|^2+\kappa\| x-Tx-(y-Ty)\|^2\) for all \(x,y\in C\). G. Marino and H.-K. Xu [J. Math. Anal. Appl. 329, No. 1, 336–346 (2007; Zbl 1116.47053)] showed that, if \(T\) has a fixed point, then the Mann iteration defined by \(x_{n+1}:=(1-\lambda_n)x_n+\lambda_nTx_n\) is asymptotically regular (i.e., \(\| Tx_n-x_n\|\to0\)) for all starting points \(x_0\in C\) and all sequences \((\lambda_n)\subset(0,1)\) that satisfy (1) \(\kappa<\lambda_n<1\) for all nonnegative integers \(n\) and (2) \(\sum_{n=0}^{\infty}(\lambda_n-\kappa)(1-\lambda_n)=\infty\).By applying general proof-theoretic methods, the paper under review improves this result twofold. First of all, the authors show that the mapping \(T\) need not have a real fixed point, but only arbitrarily good approximate fixed points, i.e., for all \(\varepsilon>0\), there exists an \(x_{\varepsilon}\in C\) such that \(\| x_{\varepsilon}-Tx_{\varepsilon}\|<\varepsilon\).Moreover, if \(\theta:\mathbb{N}\to\mathbb{N}\) is a rate of divergence for \(\sum_{n=0}^{\infty}(\lambda_n-\kappa)(1-\lambda_n)\) and \(x\in C\) is a point such that \(\| Tx-x\|\leq b\) and \(T\) has approximate fixed points in a \(b\)-neighorhood of \(x\), they provide a rate of convergence for \(\| Tx_n -x_n\|\to0\). As a corollary, for the Krasnoselskii iteration, which is obtained by setting \(\lambda_n=\lambda\in(\kappa,1)\), this translates into a quadratic rate of convergence. Reviewer: Daniel Körnlein (Darmstadt) Cited in 6 Documents MSC: 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. 03F10 Functionals in proof theory Keywords:asymptotic regularity; effective bounds; Mann iteration; proof mining; strict pseudocontractions Citations:Zbl 1116.47053 PDFBibTeX XMLCite \textit{D. Ivan} and \textit{L. Leuştean}, Numer. Funct. Anal. Optim. 36, No. 6, 792--798 (2015; Zbl 1334.47063) Full Text: DOI arXiv References: [1] DOI: 10.1090/S0002-9904-1966-11544-6 · Zbl 0138.08202 [2] DOI: 10.1016/0022-247X(67)90085-6 · Zbl 0153.45701 [3] DOI: 10.1016/0022-247X(72)90056-X · Zbl 0244.47042 [4] DOI: 10.1016/S0022-247X(03)00028-3 · Zbl 1043.47045 [5] DOI: 10.1090/S0002-9947-04-03515-9 · Zbl 1079.03046 [6] Krasnoselskii M. A., Usp. Math. Nauk 10 pp 123– (1955) [7] DOI: 10.1016/j.jmaa.2006.01.081 · Zbl 1103.03057 [8] DOI: 10.1090/S0002-9939-1953-0054846-3 [9] DOI: 10.1016/j.jmaa.2006.06.055 · Zbl 1116.47053 [10] DOI: 10.1016/0022-247X(79)90024-6 · Zbl 0423.47026 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.