A constructive proof of the Cauchy-Kovalevskaya theorem for ordinary differential equations

TL;DR

This paper presents a constructive, explicit method for proving the existence and uniqueness of analytic solutions to initial value problems in ODEs, based on a fixed point approach and radii polynomial technique.

Contribution

It introduces a novel constructive proof of the Cauchy-Kovalevskaya theorem for ODEs using a fixed point formulation and radii polynomial method, providing explicit bounds and solutions.

Findings

Provides an explicit recipe for constructing the fixed point problem

Uses radii polynomial approach for rigorous solution bounds

Ensures existence and uniqueness of analytic solutions

Abstract

We give a constructive proof of the classical Cauchy-Kovalevskaya theorem in the ODE setting which provides a sufficient condition for an initial value problem to have a unique analytic solution. Our proof is inspired by a modern functional analytic technique for rigorously solving nonlinear problems known as the radii polynomial approach. The main idea is to recast the existence and uniqueness of analytic solutions as a fixed point problem for an appropriately chosen Banach space. A key aspect of this method is the usage of an approximate solution which plays a crucial role in the theoretical proof. Our proof is constructive in the sense that we provide an explicit recipe for constructing the fixed point problem, an approximate solution, and the bounds necessary to prove the existence of the fixed point.