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Logical aspects of rates of convergence in metric spaces

Published online by Cambridge University Press:  12 March 2014

Eyvind Martol Briseid
Eyvind Martol Briseid*
Affiliation:
Technische Universität Darmstadt, Department of Mathematics, Schlossgartenstrasse 7, 64289 Darmstadt., Germany, E-mail: briseid@mathematik.tu-darmstadt.de
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Abstract

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. In recent case studies we were able to find such explicit rates of convergence in two concrete cases. 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.

Keywords

Metric fixed point theoryrates of convergenceproof mining
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 74 , Issue 4 , December 2009 , pp. 1401 - 1428
Copyright
Copyright © Association for Symbolic Logic 2009

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