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

@inproceedings{Cid2018ALC,
  title={A Lipschitz condition along a transversal foliation implies local uniqueness for ODEs},
  author={Jos'e 'Angel Cid and F. Adri'an F. Tojo},
  year={2018}
}
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.