Hostname: page-component-848d4c4894-xm8r8 Total loading time: 0 Render date: 2024-06-23T16:18:13.707Z Has data issue: false hasContentIssue false
  • English
  • Français

Bounds for a nonlinear ergodic theorem for Banach spaces

Part of: Nonlinear operators and their properties Proof theory and constructive mathematics

Published online by Cambridge University Press:  18 February 2022

ANTON FREUND Open the ORCID record for ANTON FREUND [Opens in a new window]  and
ULRICH KOHLENBACH Open the ORCID record for ULRICH KOHLENBACH [Opens in a new window]
ANTON FREUND*
Affiliation:
Department of Mathematics, Technical University of Darmstadt, Schlossgartenstr. 7, 64289 Darmstadt, Germany (e-mail: kohlenbach@mathematik.tu-darmstadt.de)
ULRICH KOHLENBACH
Affiliation:
Department of Mathematics, Technical University of Darmstadt, Schlossgartenstr. 7, 64289 Darmstadt, Germany (e-mail: kohlenbach@mathematik.tu-darmstadt.de)
*
e-mail: freund@mathematik.tu-darmstadt.de
Get access Rights & Permissions [Opens in a new window]

Abstract

We extract quantitative information (specifically, a rate of metastability in the sense of Terence Tao) from a proof due to Kazuo Kobayasi and Isao Miyadera, which shows strong convergence for Cesàro means of non-expansive maps on Banach spaces.

Keywords

nonlinear ergodic theoremBanach spacequantitative analysisproof miningmetastability

MSC classification

Primary: 47H10: Fixed-point theorems
Secondary: 03F10: Functionals in proof theory
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 5 , May 2023 , pp. 1570 - 1593
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Avigad, J., Gerhardy, P. and Towsner, H.. Local stability of ergodic averages. Trans. Amer. Math. Soc. 362 (2010), 261–288.CrossRefGoogle Scholar
Avigad, J. and Rute, J.. Oscillation and the mean ergodic theorem for uniformly convex Banach spaces. Ergod. Th. & Dynam. Sys. 35 (2015), 1009–1027.CrossRefGoogle Scholar
Brézis, H. and Browder, F.. Nonlinear ergodic theorems. Bull. Amer. Math. Soc. (N.S.) 82(6) (1976), 959–961.CrossRefGoogle Scholar
Bruck, R. E.. On the almost-convergence of iterates of a nonexpansive mapping in Hilbert space and the structure of the weak $\omega$ -limit set. Israel J. Math. 29 (1978), 1–16.CrossRefGoogle Scholar
Bruck, R. E.. A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces. Israel J. Math. 32 (1979), 107–116.CrossRefGoogle Scholar
Bruck, R. E.. On the convex approximation property and the asymptotic behavior of nonlinear contractions in Banach spaces. Israel J. Math. 38 (1981), 304–314.CrossRefGoogle Scholar
Clarkson, J. A.. Uniformly convex spaces. Trans. Amer. Math. Soc. 40(3) (1936), 396–414.Google Scholar
Dilworth, S. J., Howard, R. and Roberts, J. W.. On the size of approximately convex sets in normed spaces. Studia Math. 140 (2000), 213–241.CrossRefGoogle Scholar
Freund, A.. Proof lengths for instances of the Paris–Harrington principle. Ann. Pure Appl. Logic 168 (2017), 1361–1382.CrossRefGoogle Scholar
Gerhardy, P.. Proof mining in topological dynamics. Notre Dame J. Form. Log. 49(4) (2008), 431–446.CrossRefGoogle Scholar
Haagerup, U.. The best constants in the Khintchine inequality. Studia Math. 70 (1982), 231–283.CrossRefGoogle Scholar
Hanner, O.. On the uniform convexity of ${L}^p$ and ${l}^p$ . Ark. Mat. 3(3) (1956), 239–244.CrossRefGoogle Scholar
Kobayasi, K. and Miyadera, I.. On the strong convergence of the Césaro means of contractions in Banach spaces. Proc. Japan Acad. 56 (1980), 245–249.Google Scholar
Kohlenbach, U.. On the computational content of the Krasnoselski and Ishikawa fixed point theorems. Computability and Complexity in Analysis (Lecture Notes in Computer Science, 2064). Eds. Blanck, J., Brattka, V. and Hertling, P.. Springer, Berlin, 2001, pp. 119–145.CrossRefGoogle Scholar
Kohlenbach, U.. Uniform asymptotic regularity for Mann iterates. J. Math. Anal. Appl. 279(2) (2003), 531–544.CrossRefGoogle Scholar
Kohlenbach, U.. Some logical metatheorems with applications in functional analysis. Trans. Amer. Math. Soc. 357(1) (2005), 89–128.CrossRefGoogle Scholar
Kohlenbach, U.. Applied Proof Theory: Proof Interpretations and their Use in Mathematics (Springer Monographs in Mathematics). Springer, Berlin, 2008.Google Scholar
Kohlenbach, U.. On quantitative versions of theorems due to F. E. Browder and R. Wittmann. Adv. Math. 226(3) (2011), 2764–2795.Google Scholar
Kohlenbach, U.. On the asymptotic behavior of odd operators. J. Math. Anal. Appl. 382 (2011), 615–620.Google Scholar
Kohlenbach, U.. A uniform quantitative form of sequential weak compactness and Baillon’s nonlinear ergodic theorem. Commun. Contemp. Math. 14(1) (2012), 1250006.CrossRefGoogle Scholar
Kohlenbach, U.. On the quantitative asymptotic behavior of strongly nonexpansive mappings in Banach and geodesic spaces. Israel J. Math. 216 (2016), 215–246.CrossRefGoogle Scholar
Kohlenbach, U.. Quantitative results on the proximal point algorithm in uniformly convex Banach spaces. J. Convex Anal. 28(1) (2021), 11–18.Google Scholar
Kohlenbach, U. and Leuştean, L.. A quantitative mean ergodic theorem for uniformly convex Banach spaces. Ergod. Th. & Dynam. Sys. 29 (2009), 1907–1915.Google Scholar
Kohlenbach, U. and Leuştean, L.. Asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces. J. Eur. Math. Soc. (JEMS) 12 (2010), 71–92.CrossRefGoogle Scholar
Kohlenbach, U., Leuştean, L. and Nicolae, A.. Quantitative results on Fejér monotone sequences. Commun. Contemp. Math. 20(2) (2018), 1750015.CrossRefGoogle Scholar
Kohlenbach, U. and Sipoş, A.. The finitary content of sunny nonexpansive retractions. Commun. Contemp. Math. 23(1) (2021), 1950093.CrossRefGoogle Scholar
Ledoux, M. and Talagrand, M.. Probability in Banach spaces (Ergebnisse der Mathematik und ihrer Grenzgebiete, 23). Springer, Berlin, 1991.CrossRefGoogle Scholar
Pisier, G.. Sur les espaces qui ne contiennent pas de ${l}_n^1$ uniformément. Séminaire Maurey–Schwartz (1973–1974), Espaces ${L}^p$ , Applications Radonifiantes et Géométrie des Espaces de Banach. Centre de Mathématiques, École Polytechnique, Paris, 1974, Exp. No. 7.Google Scholar
Powell, T.. A new metastable convergence criterion and an application in the theory of uniformly convex Banach spaces. J. Math. Anal. Appl. 478(2) (2019), 790–805.CrossRefGoogle Scholar
Safarik, P.. A quantitative nonlinear strong ergodic theorem for Hilbert spaces. J. Math. Anal. Appl. 391(1) (2012), 26–37.CrossRefGoogle Scholar
Tao, T.. Structure and Randomness: Pages from Year One of a Mathematical Blog. American Mathematical Society, Providence, RI, 2008.Google Scholar
Wittmann, R.. Mean ergodic theorems for nonlinear operators. Proc. Amer. Math. Soc. 108(3) (1990), 781–788.CrossRefGoogle Scholar