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

2019/12/09 by Shane Kepley, Tianhao Zhang, Kepley, Shane +1 · 2 citations
Mathematics · #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical and Theoretical Analysis #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1912.03836

openalex publication_date 2019/12/09 · openalex created_date 2019/12/13 · openalex updated_date 2026/07/28

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.

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