Sharp uniqueness conditions for one-dimensional, autonomous ordinary differential equations

TL;DR

This paper establishes precise necessary and sufficient conditions for the uniqueness of Filippov solutions in one-dimensional autonomous ODEs with discontinuous velocity fields, and proposes a selection criterion when only one condition holds.

Contribution

It introduces two exact conditions for solution uniqueness and a natural selection criterion for cases where only one condition is met.

Findings

Two conditions are necessary and sufficient for uniqueness.

A natural selection criterion guarantees uniqueness when only one condition is satisfied.

Provides a complete characterization of solution uniqueness for discontinuous scalar ODEs.

Abstract

We give two conditions that are necessary and sufficient for the uniqueness of Filippov solutions of scalar, autonomous ordinary differential equations with discontinuous velocity fields. When only one of the two conditions is satisfied, we give a natural selection criterion that guarantees uniqueness of the solution.