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| \le a, |y-y_0| \le 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.^[1]

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

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.^[4]

Notes

↑ Coddington & Levinson (1955), Theorem 1.2 of Chapter 1
↑ Coddington & Levinson (1955), page 42
↑ Rudin (1987), Theorem 7.18
↑ Coddington & Levinson (1955), Theorem 1.1 of Chapter 2

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 924157 .