Beyond Cauchy–Kowalewsky: a Picard–Lindelöf theorem for smooth PDE

We prove that Picard-Lindelof iterations for an arbitrary smooth normal Cauchy problem for PDE converge if we assume a suitable Weissinger-like sufficient condition. This condition includes both a large class of non-Gevrey PDE or initial conditions, and more classical real analytic functions. The proof is based on a Banach fixed point theorem for contractions with loss of derivatives. From the latter, we also prove an inverse function theorem for locally Lipschitz maps with loss of derivatives in arbitrary graded Fr & eacute;chet spaces, not necessarily of tame type or with smoothing operators.