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.
A maximal solution of a locally Lipschitz ordinary differential equation on a finite-dimensional space can end at a finite time only by leaving every compact subset of the vector field's domain. In particular, a bounded solution for an everywhere-defined vector field extends globally.
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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.
