A Picard-Lindel\"of theorem for smooth PDE

TL;DR

This paper extends the Picard-Lindel"of theorem to smooth PDEs by establishing convergence conditions for iterative solutions, including non-analytic cases, using a fixed point approach in graded Fréchet spaces.

Contribution

It introduces a Weissinger-like condition ensuring convergence of Picard iterations for smooth PDEs, broadening applicability beyond analytic functions.

Findings

Proves convergence of Picard iterations under new conditions

Establishes an inverse function theorem in graded Fréchet spaces

Includes non-analytic PDEs and initial conditions

Abstract

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.