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.
Consider the differential equation
with initial condition
where the function ƒ is defined on a rectangular domain of the form
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.
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.
A mapping is said to satisfy the Carathéodory conditions on if it fulfills the condition of the theorem.
Uniqueness of a solution
Assume that the mapping satisfies the Carathéodory conditions on and there is a Lebesgue-integrable function , , such that
for all 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 such that the mapping satisfies all conditions from above on . Then, the domain of definition of the function is open and is continuous on .
Consider a linear initial value problem of the form
Here, the components of the matrix-valued mapping and of the inhomogeneity 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.
- 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
- Hale (1980), p.28
- Hale (1980), Theorem 5.3 of Chapter 1
- Hale (1980), p.30
- Coddington, Earl A.; Levinson, Norman (1955), Theory of Ordinary Differential Equations, New York: McGraw-Hill.
- Hale, Jack K. (1980), Ordinary Differential Equations (2nd ed.), Malabar: Robert E. Krieger Publishing Company, ISBN 0-89874-011-8.
- Rudin, Walter (1987), Real and complex analysis (3rd ed.), New York: McGraw-Hill, ISBN 978-0-07-054234-1, MR 924157.