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
with initial condition
where the function ƒ is defined on a rectangular domain of the form
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
where H denotes the Heaviside function defined by
It makes sense to consider the ramp function
as a solution of the differential equation. Strictly speaking though, it does not satisfy the differential equation at , 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 with initial condition
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
with defined on the rectangular domain
. If the function
satisfies the following three conditions:
is continuous in
for each fixed
,
is measurable in
for each fixed
,
- there is a Lebesgue-integrable function
,
, such that
for all
,
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
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