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.^[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\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.^[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 .

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