Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Luca Tanganelli Castrillón

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.

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

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.

Paper Structure

This paper contains 1 section, 11 theorems, 27 equations.

Table of Contents

  1. The initial value problem

Key Result

Lemma 1

For all $c,d\in\mathbf{D}$, if $m\in\mathbb{N}$ is such that $x_c^m$ and $x_d^m$ are well defined, then

Theorems & Definitions (15)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Proposition 1
  • Definition 3
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Definition 4
  • Lemma 5
  • ...and 5 more