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

Understand

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.