Rates of metastability for iterations on the unit interval

TL;DR

This paper employs proof mining techniques to derive explicit, uniform rates of metastability for iterative processes on the unit interval, extending previous convergence results to a computable framework.

Contribution

It introduces a novel application of proof mining to obtain explicit metastability rates for continuous and Lipschitz functions on the unit interval.

Findings

Derived computable rates of metastability for continuous functions.

Extended results to Lipschitz functions using Borwein and Borwein's argument.

Provided uniform bounds applicable to iterative processes.

Abstract

We use techniques of proof mining to extract computable and uniform rates of metastability (in the sense of Tao) for iterations of continuous functions on the unit interval, firstly (following earlier work of Gaspar) out of convergence proofs due to Franks, Marzec, Rhoades and Hillam and then out of an argument due to Borwein and Borwein that pertains only to Lipschitz functions.