Theorem. Picard-Lindeloef (improved) [00A8]

Let be open, and smooth. Assume that . Then there exists an open neighbourhood of such that and an , with a unique smooth map such that

1. for all "initial value"
2. for all and "solution of the OdE "
3. If for the solution can be found on with then for every we find a neighbourhood of such that the (unique) solution can be found on .
4. For each , there is a maximal open interval with on which one can define the (unique) solution of the initial value problem. By (3.) this defines a maximal open subset such that the intersection with is an interval containing on which the solution function can be defined.