Carathéodory conditions

If one wants to relax the continuity assumption on a function $f$ while preserving the natural equivalence between the Cauchy problem for the differential equation $x ^ { \prime } = f ( t , x )$ and the integral equation which can be obtained by integrating the Cauchy problem, one can follow ideas of C. Carathéodory [a1] and make the following definition.

Let $G \subset {\bf R} ^ { n }$ be an open set and $J = [ a, b ] \subset \mathbf{R}$, $a < b$. One says that $f : J \times G \rightarrow \mathbf{R} ^ { m }$ satisfies the Carathéodory conditions on $J \times G$, written as $f \in \operatorname { Car } ( J \times G )$, if

1) $f (. , x ) : J \rightarrow {\bf R} ^ { m }$ is measurable for every $x \in G$ (cf. also Measurable function);

2) $f ( t , . ) : G \rightarrow \mathbf{R} ^ { m }$ is continuous for almost every $t \in J$;

3) for each compact set $K \subset G$ the function

\begin{equation*} h _ { K } ( t ) = \operatorname { sup } \{ \| f ( t , x ) \| : x \in K \} \end{equation*}

is Lebesgue integrable (cf. also Lebesgue integral) on $J$, where $\| .\|$ is the norm in $\mathbf{R} ^ { m }$.

If $I \subset \mathbf{R}$ is a non-compact interval, one says that $f : I \times G \rightarrow \mathbf{R} ^ { m }$ satisfies the local Carathéodory conditions on $I \times G$ if $f \in \operatorname { Car } ( J \times G )$ for every compact interval $J \subset I$. This is written as $f \in \operatorname { Car } _ { \text{loc} } ( I \times G )$.

Note that any function $g : I \rightarrow {\bf R} ^ { m }$ which is the composition of $f \in \operatorname { Car } _ { \text{loc} } ( I \times G )$ and a measurable function $u : I \rightarrow G$, i.e. $g ( t ) = f ( t , u ( t ) )$ (cf. also Composite function), is measurable on $I$.

To specify the space of the majorant $h _ { K }$ more precisely, one says that $f$ is $L ^ { p }$-Carathéodory, $1 \leq p \leq \infty$, if $f$ satisfies 1)–3) above with $h _ { K } \in L ^ { p } ( J )$.

One can see that any function continuous on $J \times G$ is $L ^ { p }$-Carathéodory for any $p$.

Similarly, one says that $f$ is locally $L ^ { p }$-Carathéodory on $I \times G$ if $f$ restricted to $J \times G$ is $L ^ { p }$-Carathéodory for every compact interval $J \subset I$.

References

[a1]	C. Carathéodory, "Vorlesungen über reelle Funktionen" , Dover, reprint (1948)
[a2]	E. Coddington, N. Levinson, "The theory of ordinary differential equations" , McGraw-Hill (1955)
[a3]	M.A. Krasnoselskij, "Topological methods in the theory of nonlinear integral equations" , Pergamon (1964)
[a4]	J. Kurzweil, "Ordinary differential equations" , Elsevier (1986)
[a5]	A.F. Filippov, "Differential equations with discontinuous right hand sides" , Kluwer Acad. Publ. (1988)