  We prove that Picard-Lindel\"of 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-analytic 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\'echet spaces.
