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.

Introduction

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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

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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

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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

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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]

See also

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Notes

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  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

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  • 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 0924157.