Carathéodory's existence theorem

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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[edit]

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[edit]

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]

A mapping is said to satisfy the Carathéodory conditions on if it fulfills the condition of the theorem.[5]

Uniqueness of a solution[edit]

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 .[6]

Example[edit]

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.[7]

Notes[edit]

  1. ^ Coddington & Levinson (1955), Theorem 1.2 of Chapter 1
  2. ^ Coddington & Levinson (1955), page 42
  3. ^ Rudin (1987), Theorem 7.18
  4. ^ Coddington & Levinson (1955), Theorem 1.1 of Chapter 2
  5. ^ Hale (1980), p.28
  6. ^ Hale (1980), Theorem 5.3 of Chapter 1
  7. ^ Hale (1980), p.30

References[edit]