In
mathematics, Carathéodory's existence theorem says that an
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
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
Constantin Carathéodory ( el, Κωνσταντίνος Καραθεοδωρή, Konstantinos Karatheodori; 13 September 1873 – 2 February 1950) was a Greek mathematician who spent most of his professional career in Germany. He made significant ...
.
Introduction
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
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** 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
The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argume ...
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
In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central ope ...
, ''y'' satisfies the differential equation
almost everywhere
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
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
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
in
for each fixed
,
*
is
measurable
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simila ...
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
.
Example
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.
[, p.30]
See also
* Picard–Lindelöf theorem
* Cauchy–Kowalevski theorem
Notes
References
* .
* .
* .
{{DEFAULTSORT:Caratheodory's existence theorem
Ordinary differential equations
Theorems in analysis