An application of proof mining to nonlinear iterations

TL;DR

This paper employs proof mining techniques to derive a uniform effective rate of asymptotic regularity for Ishikawa iterations in convex geodesic spaces, enhancing understanding of convergence behavior.

Contribution

It introduces a novel application of proof mining to obtain explicit convergence rates for nonlinear iterations in a broad class of spaces.

Findings

Established a uniform effective rate of asymptotic regularity

Applied logical metatheorems to derive convergence guarantees

Extended results to nonexpansive mappings in convex geodesic spaces

Abstract

In this paper we apply methods of proof mining to obtain a highly uniform effective rate of asymptotic regularity for the Ishikawa iteration associated to nonexpansive self-mappings of convex subsets of a class of uniformly convex geodesic spaces. Moreover, we show that these results are guaranteed by a combination of logical metatheorems for classical and semi-intuitionistic systems.