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 be 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

${\displaystyle y'(t)=f(t,y(t))\,}$

with initial condition

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

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

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

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

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

where H denotes the Heaviside function defined by

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

It makes sense to consider the ramp function

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

as a solution of the differential equation. Strictly speaking though, it does not satisfy the differential equation at ${\displaystyle t=0}$, 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 ${\displaystyle y'=f(t,y)}$ with initial condition ${\displaystyle y(t_{0})=y_{0}}$ 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

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

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

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

then the differential equation has a solution in the extended sense in a neighborhood of the initial condition.[4]

A mapping ${\displaystyle f\colon R\to \mathbf {R} ^{n}}$ is said to satisfy the Carathéodory conditions on ${\displaystyle R}$ if it fulfills the condition of the theorem.[5]

Uniqueness of a solution

Assume that the mapping ${\displaystyle f}$ satisfies the Carathéodory conditions on ${\displaystyle R}$ and there is a Lebesgue-integrable function ${\displaystyle k(t)}$, ${\displaystyle |t-t_{0}|\leq a}$, such that

${\displaystyle |f(t,y_{1})-f(t,y_{2})|\leq k(t)|y_{1}-y_{2}|,}$

for all ${\displaystyle (t,y_{1})\in R,(t,y_{2})\in R.}$ Then, there exists a unique solution ${\displaystyle y(t)=y(t,t_{0},y_{0})}$ to the initial value problem

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

Moreover, if the mapping ${\displaystyle f}$ is defined on the whole space ${\displaystyle \mathbf {R} \times \mathbf {R} ^{n}}$ and if for any initial condition ${\displaystyle (t_{0},y_{0})\in \mathbf {R} \times \mathbf {R} ^{n}}$, there exists a compact rectangular domain ${\displaystyle R_{(t_{0},y_{0})}\subset \mathbf {R} \times \mathbf {R} ^{n}}$ such that the mapping ${\displaystyle f}$ satisfies all conditions from above on ${\displaystyle R_{(t_{0},y_{0})}}$. Then, the domain ${\displaystyle E\subset \mathbf {R} ^{2+n}}$ of definition of the function ${\displaystyle y(t,t_{0},y_{0})}$ is open and ${\displaystyle y(t,t_{0},y_{0})}$ is continuous on ${\displaystyle E}$.[6]

Example

Consider a linear initial value problem of the form

${\displaystyle y'(t)=A(t)y(t)+b(t),\quad y(t_{0})=y_{0}.}$

Here, the components of the matrix-valued mapping ${\displaystyle A\colon \mathbf {R} \to \mathbf {R} ^{n\times n}}$ and of the inhomogeneity ${\displaystyle b\colon \mathbf {R} \to \mathbf {R} ^{n}}$ are assumed to be integrable on every finite interval. Then, the right hand side of the differential equation satisfies the Carathéodory conditions and there exists a unique solution to the initial value problem.[7]

Notes

1. ^ Coddington & Levinson (1955), Theorem 1.2 of Chapter 1
2. ^ Coddington & Levinson (1955), page 42
3. ^ Rudin (1987), Theorem 7.18
4. ^ Coddington & Levinson (1955), Theorem 1.1 of Chapter 2
5. ^ Hale (1980), p.28
6. ^ Hale (1980), Theorem 5.3 of Chapter 1
7. ^ Hale (1980), p.30