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.
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.
defined on the rectangular domain
satisfies the following three conditions: then the differential equation has a solution in the extended sense in a neighborhood of the initial condition.
is said to satisfy the Carathéodory conditions on
if it fulfills the condition of the theorem.
[5] Assume that the mapping
satisfies the Carathéodory conditions on
and there is a Lebesgue-integrable function
Then, there exists a unique solution
to the initial value problem Moreover, if the mapping
is defined on the whole space
and if for any initial condition
, there exists a compact rectangular domain
satisfies all conditions from above on
of definition of the function
[6] Consider a linear initial value problem of the form Here, the components of the matrix-valued mapping
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.