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
y
′
(
t
)
=
f
(
t
,
y
(
t
)
)
with initial condition
y
(
t
0
)
=
y
0
,
where the function ƒ is defined on a rectangular domain of the form
R
=
{
(
t
,
y
)
∈
R
×
R
n
:
|
t
−
t
0
|
≤
a
,
|
y
−
y
0
|
≤
b
}
.
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
y
′
(
t
)
=
H
(
t
)
,
y
(
0
)
=
0
,
where H denotes the Heaviside function defined by
H
(
t
)
=
{
0
,
if
t
≤
0
;
1
,
if
t
>
0.
It makes sense to consider the ramp function
y
(
t
)
=
∫
0
t
H
(
s
)
d
s
=
{
0
,
if
t
≤
0
;
t
,
if
t
>
0
as a solution of the differential equation. Strictly speaking though, it does not satisfy the differential equation at
t
=
0
, 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
y
′
=
f
(
t
,
y
)
with initial condition
y
(
t
0
)
=
y
0
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
y
′
(
t
)
=
f
(
t
,
y
(
t
)
)
,
y
(
t
0
)
=
y
0
,
with
f
defined on the rectangular domain
R
=
{
(
t
,
y
)
|
|
t
−
t
0
|
≤
a
,
|
y
−
y
0
|
≤
b
}
. If the function
f
satisfies the following three conditions:
f
(
t
,
y
)
is continuous in
y
for each fixed
t
,
f
(
t
,
y
)
is measurable in
t
for each fixed
y
,
there is a Lebesgue-integrable function
m
(
t
)
,
|
t
−
t
0
|
≤
a
, such that
|
f
(
t
,
y
)
|
≤
m
(
t
)
for all
(
t
,
y
)
∈
R
,
then the differential equation has a solution in the extended sense in a neighborhood of the initial condition.
A mapping
f
:
R
→
R
n
is said to satisfy the Carathéodory conditions on
R
if it fulfills the condition of the theorem.
Assume that the mapping
f
satisfies the Carathéodory conditions on
R
and there is a Lebesgue-integrable function
k
(
t
)
,
|
t
−
t
0
|
≤
a
, such that
|
f
(
t
,
y
1
)
−
f
(
t
,
y
2
)
|
≤
k
(
t
)
|
y
1
−
y
2
|
,
for all
(
t
,
y
1
)
∈
R
,
(
t
,
y
2
)
∈
R
.
Then, there exists a unique solution
y
(
t
)
=
y
(
t
,
t
0
,
y
0
)
to the initial value problem
y
′
(
t
)
=
f
(
t
,
y
(
t
)
)
,
y
(
t
0
)
=
y
0
.
Moreover, if the mapping
f
is defined on the whole space
R
×
R
n
and if for any initial condition
(
t
0
,
y
0
)
∈
R
×
R
n
, there exists a compact rectangular domain
R
(
t
0
,
y
0
)
⊂
R
×
R
n
such that the mapping
f
satisfies all conditions from above on
R
(
t
0
,
y
0
)
. Then, the domain
E
⊂
R
2
+
n
of definition of the function
y
(
t
,
t
0
,
y
0
)
is open and
y
(
t
,
t
0
,
y
0
)
is continuous on
E
.
Consider a linear initial value problem of the form
y
′
(
t
)
=
A
(
t
)
y
(
t
)
+
b
(
t
)
,
y
(
t
0
)
=
y
0
.
Here, the components of the matrix-valued mapping
A
:
R
→
R
n
×
n
and of the inhomogeneity
b
:
R
→
R
n
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.
