Title: Picard-Lindelöf beyond Banach spaces
Abstract: The Picard-Lindelöf (PL) theorem asserts the existence (and uniqueness) of solutions to ODEs on Banach spaces with arbitrary initial conditions. Beyond Banach spaces, things become wild. Even for a natural class of ODEs, those with “finite loss of norms/derivatives”, existence and uniqueness is usually not satisfied.
However, it is still possible to formulate a version of the PL theorem on Fréchet spaces. The contraction property of the Picard map now depends on both the initial condition and the underlying vector field. Although the contraction property is usually quite strong, it does have some applications beyond to what is standard.
A PDE in normal (a.k.a. Cauchy-Kovalevskaya) form can be interpreted as an ODE on the space of smooth functions. This way, the PL theorem can be applied to prove existence of solutions to some types of smooth Cauchy problems, where the Cauchy-Kovalevskaya theorem fails to apply. (From “Giordano, Luperi Baglini - Picard-Lindelöf for smooth PDE”)
