A quadratic rate of asymptotic regularity for CAT(0)-spaces

By Laurentiu Leustean. 2005

252 citations recorded by the source corpus.

Abstract In this paper we obtain a quadratic bound on the rate of asymptotic regularity for the Krasnoselski–Mann iterations of nonexpansive mappings in CAT(0)-spaces, whereas previous results guarantee only exponential bounds. The method we use is to extend to the more general setting of uniformly convex hyperbolic spaces a quantitative version of a strengthening of Groetsch's theorem obtained by Kohlenbach using methods from mathematical logic (so-called “proof mining”).

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/ea8ae94477dac931865b378adfd1f2359aac08cb

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Ulrich Kohlenbach
2. Laurentiu Leustean
3. Genaro López-Acedo
4. David Ariza-Ruiz
5. Hafiz Fukhar-ud-din
6. Mohamed Amine Khamsi
7. Abdul Rahim Khan

In Computational Theory and Mathematics

1. Laurentiu Leustean
2. Genaro López-Acedo
3. David Ariza-Ruiz
4. Ulrich Kohlenbach
5. Mohamed Amine Khamsi
6. Abdul Rahim Khan
7. Hafiz Fukhar-ud-din
8. Paulo Oliva

Citations in the corpus

• Firmly nonexpansive mappings in classes of geodesic spaces (2012, methodology): “The methods of proof are inspired by those used by Kohlenbach [23] and the second author [29] for computing rates of asymptotic regularity for the Krasnoselski-Mann iterations of nonexpansive mappings in uniformly convex Banach spaces and UCW -hyperbolic spaces.”
• Asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces (2007, background): “…the proof even holds for asymptotically weakly-quasi nonexpansive functions for which one can show that no uniform effective rate does exist), general logical metatheorems from [13, 4] guarantee (see also section 5 below) effective uniform bound on the so-called no-counterexample…”
• Strong convergence of a general iteration scheme in C A T ( 0 ) spaces (2011, background): “CAT (0) spaces provide wonderful examples of uniformly convex metric spaces (see [22]).”
• General logical metatheorems for functional analysis (2005, unknown): “[21, 22, 23, 24, 26, 31 , 32] and the discussion in section 8 below).”