Rates of asymptotic regularity for Halpern iterations of nonexpansive mappings

TL;DR

This paper uses proof-theoretic methods to derive effective rates of asymptotic regularity for Halpern iterations of nonexpansive mappings in normed spaces, advancing the proof mining project.

Contribution

It provides new explicit bounds for the convergence behavior of Halpern iterations, applying proof mining techniques to nonexpansive mappings.

Findings

Derived effective rates of asymptotic regularity

Applied proof-theoretic methods to iteration analysis

Contributed to the proof mining project in nonlinear analysis

Abstract

In this paper we obtain new effective results on the Halpern iterations of nonexpansive mappings using methods from mathematical logic or, more specifically, proof-theoretic techniques. We give effective rates of asymptotic regularity for the Halpern iterations of nonexpansive self-mappings of nonempty convex sets in normed spaces. The paper presents another case study in the project of {\em proof mining}, which is concerned with the extraction of effective uniform bounds from (prima-facie) ineffective proofs.