On the Picard-Lindel\"of Argument and the Banach-Caccioppoli Contraction Mapping Principle

TL;DR

This paper refines the Contraction Mapping Principle to accurately determine the convergence rate in the Picard-Lindel"of Theorem, enhancing understanding of solution convergence in differential equations.

Contribution

It introduces a slight refinement of the Contraction Mapping Principle that precisely captures the convergence rate in the Picard-Lindel"of Theorem.

Findings

Provides a refined contraction principle for better convergence rate estimation

Enhances the theoretical understanding of solution iteration in differential equations

Simplifies the proof of convergence in the Picard-Lindel"of Theorem

Abstract

The aim of this note is to present the simple observation that a slight refinement of the Contraction Mapping Principle allows one to recover the precise convergence rate in the Picard-Lindel\"of Theorem.