 Picard-Lindelöf theorem

A vector field that is continuous in time and locally Lipschitz continuous in the state variable gives a unique local solution to its initial-value problem. A solution extends while it remains in a compact subset of the vector field's domain.

The Picard–Lindelöf theorem, also known as the Picard existence theorem or the Picard-Lindelöf theorem, is a fundamental result in the theory of ordinary differential equations (ODEs). It provides conditions under which a first-order ordinary differential equation has a unique solution in a specified interval.