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# Carathéodory's existence theorem

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 Title: Carathéodory's existence theorem Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Carathéodory's existence theorem

In mathematics, Carathéodory's existence theorem says that an ordinary differential equation has a solution under relatively mild conditions. It is a generalization of Peano's existence theorem. Peano's theorem requires that the right-hand side of the differential equation is continuous, while Carathéodory's theorem shows existence of solutions (in a more general sense) for some discontinuous equations. The theorem is named after Constantin Carathéodory.

## Introduction

Consider the differential equation

$y\text{'}\left(t\right) = f\left(t,y\left(t\right)\right) \,$

with initial condition

$y\left(t_0\right) = y_0, \,$

where the function ƒ is defined on a rectangular domain of the form

$R = \\left\{ \left(t,y\right) \in \mathbf\left\{R\right\}\times\mathbf\left\{R\right\}^n \,:\, |t-t_0| \le a, |y-y_0| \le b \\right\}.$

Peano's existence theorem states that if ƒ is continuous, then the differential equation has at least one solution in a neighbourhood of the initial condition.

However, it is also possible to consider differential equations with a discontinuous right-hand side, like the equation

$y\text{'}\left(t\right) = H\left(t\right), \quad y\left(0\right) = 0,$

where H denotes the Heaviside function defined by

$H\left(t\right) = \begin\left\{cases\right\} 0, & \text\left\{if \right\} t \le 0; \\ 1, & \text\left\{if \right\} t > 0. \end\left\{cases\right\}$

It makes sense to consider the ramp function

$y\left(t\right) = \int_0^t H\left(s\right) \,\mathrm\left\{d\right\}s = \begin\left\{cases\right\} 0, & \text\left\{if \right\} t \le 0; \\ t, & \text\left\{if \right\} t > 0 \end\left\{cases\right\}$

as a solution of the differential equation. Strictly speaking though, it does not satisfy the differential equation at $t=0$, because the function is not differentiable there. This suggests that the idea of a solution be extended to allow for solutions that are not everywhere differentiable, thus motivating the following definition.

A function y is called a solution in the extended sense of the differential equation $y\text{'} = f\left(t,y\right)$ with initial condition $y\left(t_0\right)=y_0$ if y is absolutely continuous, y satisfies the differential equation almost everywhere and y satisfies the initial condition. The absolute continuity of y implies that its derivative exists almost everywhere.

## Statement of the theorem

Consider the differential equation

$y\text{'}\left(t\right) = f\left(t,y\left(t\right)\right), \quad y\left(t_0\right) = y_0, \,$

with $f$ defined on the rectangular domain $R=\\left\{\left(t,y\right) \, | \, |t - t_0 | \leq a, |y - y_0| \leq b\\right\}$. If the function $f$ satisfies the following three conditions:

• $f\left(t,y\right)$ is continuous in $y$ for each fixed $t$,
• $f\left(t,y\right)$ is measurable in $t$ for each fixed $y$,
• there is a Lebesgue-integrable function $m\left(t\right)$, $|t - t_0| \leq a$, such that $|f\left(t,y\right)| \leq m\left(t\right)$ for all $\left(t, y\right) \in R$,

then the differential equation has a solution in the extended sense in a neighborhood of the initial condition.

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