On the Nagumo uniqueness theorem

TL;DR

This paper improves the Nagumo uniqueness theorem by establishing a more flexible criterion for the uniqueness of solutions to nonlinear ODEs without Lipschitz conditions, using a reparametrisation of integral curves.

Contribution

It introduces a new approach through reparametrisation to extend the Nagumo theorem to broader classes of nonlinear ODEs without Lipschitz-like nonlinearities.

Findings

Established a more general uniqueness criterion for nonlinear ODEs.

Extended the applicability of Nagumo's theorem beyond Lipschitz conditions.

Provided a simplified proof technique via reparametrisation.

Abstract

By a convenient reparametrisation of the integral curves of a nonlinear ordinary differential equation (ODE), we are able to improve the conclusions of the recent contribution [A. Constantin, Proc. Japan Acad. {\bf 86(A)} (2010), 41--44]. In this way, we establish a flexible uniqueness criterion for ODEs without Lipschitz-like nonlinearities.