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

$y^{\prime }(t)=f(t,y(t))$

with initial condition

$y(t_0)=y_0,$

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

$R=\{(t,y)\in 𝐑\times {𝐑}^n:|t-t_0|\le a,|y-y_0|\le b\}.$

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.

However, it is also possible to consider differential equations with a discontinuous right-hand side, like the equation

$y^{\prime }(t)=H(t),y(0)=0,$

where H denotes the Heaviside function defined by

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

It makes sense to consider the ramp function

$y(t)={\int }_0^{t}H(s)\mathrm{d}s=\{\begin{array}{cc}0,& \text{if }t\le 0;\\ t,& \text{if }t>0\end{array}$

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 $y^{\prime }=f(t,y)$ 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

$y^{\prime }(t)=f(t,y(t)),y(t_0)=y_0,$

with $f$ defined on the rectangular domain $R=\{(t,y)||t-t_0|\le a,|y-y_0|\le b\}$. 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 $m:[t_0-a,t_0+a]\to [0,\mathrm{\infty })$ such that $|f(t,y)|\le m(t)$ for all $(t,y)\in R$,

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

A mapping $f:R\to {𝐑}^n$ is said to satisfy the Carathéodory conditions on $R$ if it fulfills the condition of the theorem.

Uniqueness of a solution

Assume that the mapping $f$ satisfies the Carathéodory conditions on $R$ and there is a Lebesgue-integrable function $k:[t_0-a,t_0+a]\to [0,\mathrm{\infty })$, such that

$|f(t,y_1)-f(t,y_2)|\le k(t)|y_1-y_2|,$

for all $(t,y_1)\in R,(t,y_2)\in R.$ Then, there exists a unique solution $y(t)=y(t,t_0,y_0)$ to the initial value problem

$y^{\prime }(t)=f(t,y(t)),y(t_0)=y_0.$

Moreover, if the mapping $f$ is defined on the whole space $𝐑\times {𝐑}^n$ and if for any initial condition $(t_0,y_0)\in 𝐑\times {𝐑}^n$, there exists a compact rectangular domain $R_{(t_0,y_0)}\subset 𝐑\times {𝐑}^n$ such that the mapping $f$ satisfies all conditions from above on $R_{(t_0,y_0)}$. Then, the domain $E\subset {𝐑}^{2+n}$ of definition of the function $y(t,t_0,y_0)$ is open and $y(t,t_0,y_0)$ is continuous on $E$.

Example

Consider a linear initial value problem of the form

$y^{\prime }(t)=A(t)y(t)+b(t),y(t_0)=y_0.$

Here, the components of the matrix-valued mapping $A:𝐑\to {𝐑}^{n\times n}$ and of the inhomogeneity $b:𝐑\to {𝐑}^n$ 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.

See also

• Picard–Lindelöf theorem
• Cauchy–Kowalevski theorem

References

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