Sipoş, Andrei Effective results on a fixed point algorithm for families of nonlinear mappings. (English) Zbl 1422.03123 Ann. Pure Appl. Logic 168, No. 1, 112-128 (2017). Summary: We use proof mining techniques to obtain a uniform rate of asymptotic regularity for the instance of the parallel algorithm used by G. L. Acedo and H.-K. Xu [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 67, No. 7, 2258–2271 (2007; Zbl 1133.47050)] to find common fixed points of finite families of \(k\)-strict pseudocontractive self-mappings of convex subsets of Hilbert spaces. We show that these results are guaranteed by a number of logical metatheorems for classical and semi-intuitionistic systems. Cited in 2 Documents MSC: 03F10 Functionals in proof theory 03F60 Constructive and recursive analysis 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. Keywords:proof mining; effective bounds; asymptotic regularity; strict pseudocontractions; parallel algorithm Citations:Zbl 1133.47050 PDFBibTeX XMLCite \textit{A. Sipoş}, Ann. Pure Appl. Logic 168, No. 1, 112--128 (2017; Zbl 1422.03123) Full Text: DOI arXiv References: [1] Browder, F. E.; Petryshyn, W. V., Construction of fixed points of nonlinear mappings in Hilbert spaces, J. Math. Anal. Appl., 20, 197-228 (1967) · Zbl 0153.45701 [2] Gerhardy, P.; Kohlenbach, U., Strongly uniform bounds from semi-constructive proofs, Ann. Pure Appl. Logic, 141, 89-107 (2006) · Zbl 1099.03048 [3] Gerhardy, P.; Kohlenbach, U., General logical metatheorems for functional analysis, Trans. Amer. Math. Soc., 360, 2615-2660 (2008) · Zbl 1130.03036 [4] Ivan, D.; Leuştean, L., A rate of asymptotic regularity for the Mann iteration of \(κ\)-strict pseudo-contractions, Numer. Funct. Anal. Optim., 36, 792-798 (2015) · Zbl 1334.47063 [5] Khan, M. A.A.; Kohlenbach, U., Bounds on Kuhfittig’s iteration schema in uniformly convex hyperbolic spaces, J. Math. Anal. Appl., 403, 633-642 (2013) · Zbl 1307.47078 [6] Kohlenbach, U., Analysing proofs in analysis, (Hodges, W.; Hyland, M.; Steinhorn, C.; Truss, J., Logic: From Foundations to Applications, European Logic Colloquium. Logic: From Foundations to Applications, European Logic Colloquium, Keele, 1993 (1996), Oxford University Press), 225-260 · Zbl 0881.03032 [7] Kohlenbach, U., Relative constructivity, J. Symbolic Logic, 63, 1218-1238 (1998) · Zbl 0928.03065 [8] Kohlenbach, U., Some logical metatheorems with applications in functional analysis, Trans. Amer. Math. Soc., 357, 1, 89-128 (2005) · Zbl 1079.03046 [9] Kohlenbach, U., Applied Proof Theory: Proof Interpretations and Their Use in Mathematics, Springer Monogr. Math. (2008), Springer-Verlag: Springer-Verlag Berlin, Heidelberg · Zbl 1158.03002 [11] Kreisel, G., Interpretation of analysis by means of constructive functionals of finite types, (Heyting, A., Constructivity in Mathematics (1959), North-Holland: North-Holland Amsterdam), 101-128 [12] Leuştean, L., A quadratic rate of asymptotic regularity for CAT(0)-spaces, J. Math. Anal. Appl., 325, 1, 386-399 (2007) · Zbl 1103.03057 [13] Leuştean, L., An application of proof mining to nonlinear iterations, Ann. Pure Appl. Logic, 165, 1484-1500 (2014) · Zbl 1388.03052 [14] López-Acedo, G.; Xu, H.-K., Iterative methods for strict pseudo-contractions in Hilbert spaces, Nonlinear Anal., 67, 7, 2258-2271 (2007) · Zbl 1133.47050 [15] Marino, G.; Xu, H.-K., Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl., 329, 1, 336-346 (2007) · Zbl 1116.47053 [16] Oliva, P., Hybrid functional interpretations of linear and intuitionistic logic, J. Logic Comput., 22, 305-328 (2012) · Zbl 1242.03086 [17] Sipoş, A., A note on the Mann iteration for \(k\)-strict pseudocontractions in Banach spaces, Numer. Funct. Anal. Optim. (2016), in press, preprint [18] Spector, C., Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles formulated in current intuitionistic mathematics, (Dekker, J. C.E., Proc. Sympos. Pure Math., vol. 5 (1962), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 1-27 · Zbl 0143.25502 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.