A Lipschitz condition along a transversal foliation implies local uniqueness for ODEs

TL;DR

This paper establishes that a Lipschitz condition along a transversal foliation guarantees local uniqueness of solutions to ODEs, extending classical uniqueness results to more general geometric settings.

Contribution

It introduces a new criterion for local uniqueness of ODE solutions based on Lipschitz continuity restricted to foliated hypersurfaces, with illustrative examples and conditions.

Findings

Lipschitz condition along foliation implies local uniqueness

Provides sufficient conditions for applying the main theorem

Includes illustrative examples demonstrating the result

Abstract

We prove the following result: if a continuous vector field $F$ is Lipschitz when restricted to the hypersurfaces determined by a suitable foliation and a transversal condition is satisfied at the initial condition, then $F$ determines a locally unique integral curve. We also present some illustrative examples and sufficient conditions in order to apply our main result.