Journal of Symbolic Logic
- J. Symbolic Logic
- Volume 74, Issue 4 (2009), 1401-1428.
Logical aspects of rates of convergence in metric spaces
In this paper we develop a method for finding, under general conditions, explicit and highly uniform rates of convergence for the Picard iteration sequences for selfmaps on bounded metric spaces from ineffective proofs of convergence to a unique fixed point. We are able to extract full rates of convergence by extending the use of a logical metatheorem recently proved by Kohlenbach. %This metatheorem could earlier be used to %extract such computable rates of convergence only in cases where the %selfmappings are also %nonexpansive. In recent case studies we were able to find such explicit rates of convergence in two concrete cases. %without assuming the selfmappings in %question to be nonexpansive. Our novel method now provides an explanation in logical terms for these findings. This amounts, loosely speaking, to general conditions under which we in this specific setting can transform a ∀ ∃ ∀-sentence into a ∀ ∃-sentence via an argument involving product spaces. This reduction in logical complexity allows us to use the existing machinery to extract quantitative bounds of the sort we need.
J. Symbolic Logic, Volume 74, Issue 4 (2009), 1401-1428.
First available in Project Euclid: 5 October 2009
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 03F10: Functionals in proof theory 03F35: Second- and higher-order arithmetic and fragments [See also 03B30] 47H10: Fixed-point theorems [See also 37C25, 54H25, 55M20, 58C30] 47H09: Contraction-type mappings, nonexpansive mappings, A-proper mappings, etc.
Briseid, Eyvind Martol. Logical aspects of rates of convergence in metric spaces. J. Symbolic Logic 74 (2009), no. 4, 1401--1428. doi:10.2178/jsl/1254748697. https://projecteuclid.org/euclid.jsl/1254748697