An elementary proof of the existence and uniqueness of solutions to an initial value problem

2024/11/09 by Luca Tanganelli Castrillón, Castrillón, Luca Tanganelli
Engineering · Mathematics · #34A12 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Numerical methods for differential equations #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2411.06149

openalex publication_date 2024/11/09 · openalex created_date 2024/11/15 · openalex updated_date 2026/07/28

Abstract

In this note, we show a classical result on the local existence and uniqueness of a solution to an initial value problem subject to a Lipschitz condition. We use only elementary tools from mathematical analysis, without involving any integration. We proceed by showing that the Cauchy iterates converge on a dense subset of the interval and subsequently proving that the extension of this limit function to the whole interval is a solution to the Cauchy problem.

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