Convergent expansions for solutions to non-linear singular Cauchy problems

This article deals with Fuchsian type systems of the form\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\left( {t\frac{\partial }{{\partial t}} - L} \right)u = g\left( {t,y,u,\frac{{\partial u}}{{\partial y}}} \right)$$ \end{document} with g(0, … 0) = 0, Δu,v,wg(0, 0, 0, 0) = 0. Here L is a fixed endomorphism, and g is analytic in all variable, including t. It is known from Baouendi-Goulaoic that, if no eigenvalue of L is a non-negative integer, such a system has a unique analytic solution (unique precisely because the kernel contains only nonsmooth functions). The aim of this article is complementary to this result: it is to describe this kernel. The main theorem states roughly that the generalized eigenspaces associated with eigenvalues of L of positive real parts parametrize the set of all solutions. The method of proof is by constructing a formal solution, and proving convergence inductively with the aid of majorizing series.