On computational properties of Cauchy problems generated by accretive operators

TL;DR

This paper develops quantitative versions of convergence results for nonlinear semigroups generated by accretive operators, using proof mining techniques to extract explicit rates of convergence based on a new notion of a convergence condition with modulus.

Contribution

It introduces a quantitative framework for convergence conditions of accretive operators, enabling explicit rates of convergence for associated nonlinear semigroups.

Findings

Derived notions of convergence condition with modulus.

Extracted explicit rates of convergence.

Provided quantitative versions of classical asymptotic results.

Abstract

In this paper, we provide quantitative versions of results on the asymptotic behavior of nonlinear semigroups generated by an accretive operator due to O. Nevanlinna and S. Reich as well as H.-K. Xu. These results themselves rely on a particular assumption on the underlying operator introduced by A. Pazy under the name of `convergence condition'. Based on logical techniques from `proof mining', a subdiscipline of mathematical logic, we derive various notions of a `convergence condition with modulus' which provide quantitative information on this condition in different ways. These techniques then also facilitate the extraction of quantitative information on the convergence results of Nevanlinna and Reich as well as Xu, in particular also in the form of rates of convergence which depend on these moduli for the convergence condition.