An analytic proof of the Malgrange-Sibuya theorem on the convergence of formal solutions of an ODE

TL;DR

This paper provides an analytic proof of the Malgrange-Sibuya theorem, establishing conditions under which formal solutions to ODEs converge, using the majorant method to estimate their radius of convergence.

Contribution

It offers a new analytic proof of the Malgrange-Sibuya theorem and introduces a method to estimate the convergence radius of formal solutions to ODEs.

Findings

Proof based on the majorant method

Estimates the radius of convergence for formal solutions

Provides sufficient conditions for convergence

Abstract

We propose an analytic proof of the Malgrange-Sibuya theorem concerning a sufficient condition of the convergence of a formal power series satisfying an ordinary differential equation. The proof is based on the majorant method and allows to estimate the radius of convergence of such a series.