Bounds for a nonlinear ergodic theorem for Banach spaces

TL;DR

This paper derives explicit quantitative bounds, including a rate of metastability, from a proof demonstrating strong convergence of Cesàro means of nonexpansive maps in Banach spaces.

Contribution

It provides the first explicit rate of metastability for the convergence of Cesàro means in Banach spaces, enhancing the quantitative understanding of ergodic theorems.

Findings

Established a rate of metastability for Cesàro means

Quantified convergence in nonexpansive map iterations

Enhanced the ergodic theorem with explicit bounds

Abstract

We extract quantitative information (specifically, a rate of metastability in the sense of Terence Tao) from a proof due to Kazuo Kobayasi and Isao Miyadera, which shows strong convergence for Ces\`aro means of nonexpansive maps on Banach spaces.