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
:$y'(t) = f(t,y(t))$
with initial condition
:$y(t_0) = y_0,$
where the function ƒ is defined on a rectangular domain of the form
:$R = \.$
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
:$y'(t) = H(t), \quad y(0) = 0,$
where ''H'' denotes the Heaviside function defined by
:$H(t) = \begin 0, & \text t \le 0; \\ 1, & \text t > 0. \end$
It makes sense to consider the ramp function
:$y(t) = \int_0^t H(s) \,\mathrms = \begin 0, & \text t \le 0; \\ t, & \text t > 0 \end$
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. The absolute continuity of ''y'' implies that its derivative exists almost everywhere.

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=\$. 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 : _0 - a, t_0 + a\to [0, \infty) 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.

A mapping $f \colon R \to \mathbf^n$ 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 : _0 - a, t_0 + a\to [0, \infty), such that
:$, f(t,y_1) - f(t,y_2), \leq k(t) , y_1 - y_2, ,$
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
:$y'(t) = f(t,y(t)), \quad y(t_0) = y_0.$

Moreover, if the mapping $f$ is defined on the whole space $\mathbf \times \mathbf^n$
and if for any initial condition $(t_0,y_0) \in \mathbf \times \mathbf^n$, there exists a compact rectangular domain $R_ \subset \mathbf \times \mathbf^n$ such that the mapping $f$ satisfies all conditions from above on $R_$. Then, the domain $E \subset \mathbf^$ 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
:$y'(t) = A(t)y(t) + b(t), \quad y(t_0) = y_0.$
Here, the components of the matrix-valued mapping $A \colon \mathbf \to \mathbf^$ and of the inhomogeneity $b \colon \mathbf \to \mathbf^$ 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., p.30

See also

* Picard–Lindelöf theorem
* Cauchy–Kowalevski theorem

Notes

References

* .
* .
* .

{{DEFAULTSORT:Caratheodory's existence theorem
Ordinary differential equations
Theorems in analysis