#jsDisabledContent { display:none; } My Account | Register | Help

# Carathéodory's existence theorem

Article Id: WHEBN0021075081
Reproduction Date:

 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.[1]

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.[2] The absolute continuity of y implies that its derivative exists almost everywhere.[3]

## 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.[4]

## References

• .
• .
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.

Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.