Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Carathéodory's existence theorem
Summary
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.
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. The absolute continuity of y implies that its derivative exists almost everywhere.
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.
A mapping is said to satisfy the Carathéodory conditions on if it fulfills the condition of the theorem.
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.