Carathéodory's existence theorem

Statement of the theorem

Consider the differential equation

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

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

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

Uniqueness of a solution

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

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

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

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

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 ${\displaystyle E}$.

Example

Consider a linear initial value problem of the form

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

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