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

$y'(t)=f(t,y(t))\,$

with initial condition

$y(t_{0})=y_{0},\,$

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

$R=\{(t,y)\in \mathbf {R} \times \mathbf {R} ^{n}\,:\,|t-t_{0}|\leq a,|y-y_{0}|\leq b\}.$

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

$y'(t)=H(t),\quad y(0)=0,$

where H denotes the Heaviside function defined by

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

It makes sense to consider the ramp function

$y(t)=\int _{0}^{t}H(s)\,\mathrm {d} s={\begin{cases}0,&{\text{if }}t\leq 0;\\t,&{\text{if }}t>0\end{cases}}$

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 $y'=f(t,y)$ with initial condition $y(t_{0})=y_{0}$ 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

$y'(t)=f(t,y(t)),\quad y(t_{0})=y_{0},\,$

with $f$ defined on the rectangular domain $R=\{(t,y)\,|\,|t-t_{0}|\leq a,|y-y_{0}|\leq b\}$. If the function $f$ satisfies the following three conditions:

• $f(t,y)$ is continuous in $y$ for each fixed $t$,
• $f(t,y)$ is measurable in $t$ for each fixed $y$,
• there is a Lebesgue-integrable function $m(t)$, $|t-t_{0}|\leq a$, such that $|f(t,y)|\leq m(t)$ for all $(t,y)\in R$,

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

References

• Coddington, Earl A.; Levinson, Norman (1955), Theory of Ordinary Differential Equations, New York: McGraw-Hill.
• Rudin, Walter (1987), Real and complex analysis (3rd ed.), New York: McGraw-Hill, ISBN 978-0-07-054234-1, MR 0924157.