Proof mining in ${\mathbb R}$-trees and hyperbolic spaces

TL;DR

This paper develops logical theorems for hyperbolic spaces and ${\mathbb R}$-trees, enabling extraction of uniform bounds in proofs, with applications to fixed-point theory in nonlinear analysis.

Contribution

It extends proof mining techniques to hyperbolic and ${\mathbb R}$-tree spaces, providing new metatheorems and applications in metric fixed-point theory.

Findings

Established logical metatheorems for hyperbolic spaces and ${\mathbb R}$-trees.

Demonstrated the extraction of uniform bounds in nonlinear functional analysis.

Applied results to bounds on asymptotic regularity in fixed-point iterations.

Abstract

This paper is part of the general project of proof mining, developed by Kohlenbach. By "proof mining" we mean the logical analysis of mathematical proofs with the aim of extracting new numerically relevant information hidden in the proofs. We present logical metatheorems for classes of spaces from functional analysis and hyperbolic geometry, like Gromov hyperbolic spaces, ${\mathbb R}$-trees and uniformly convex hyperbolic spaces. Our theorems are adaptations to these structures of previous metatheorems of Gerhardy and Kohlenbach, and they guarantee a-priori, under very general logical conditions, the existence of uniform bounds. We give also an application in nonlinear functional analysis, more specifically in metric fixed-point theory. Thus, we show that the uniform bound on the rate of asymptotic regularity for the Krasnoselski-Mann iterations of nonexpansive mappings in uniformly convex hyperbolic spaces obtained in a previous paper is an instance of one of our metatheorems.