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Difference between revisions of "Carathéodory conditions"

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If one wants to relax the continuity assumption on a function $f$ while preserving the natural equivalence between the [[Cauchy problem|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 [[#References|[a1]]] and make the following definition.
 
If one wants to relax the continuity assumption on a function $f$ while preserving the natural equivalence between the [[Cauchy problem|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 [[#References|[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
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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|Measurable function]]);
 
1) $f (. , x ) : J \rightarrow {\bf R} ^ { m }$ is measurable for every $x \in G$ (cf. also [[Measurable function|Measurable function]]);
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====References====
 
====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  C. Carathéodory,  "Vorlesungen über reelle Funktionen" , Dover, reprint  (1948)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  E. Coddington,  N. Levinson,  "The theory of ordinary differential equations" , McGraw-Hill  (1955)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  M.A. Krasnoselskij,  "Topological methods in the theory of nonlinear integral equations" , Pergamon  (1964)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  J. Kurzweil,  "Ordinary differential equations" , Elsevier  (1986)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  A.F. Filippov,  "Differential equations with discontinuous right hand sides" , Kluwer Acad. Publ.  (1988)</td></tr></table>
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<table>
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<tr><td valign="top">[a1]</td> <td valign="top">  C. Carathéodory,  "Vorlesungen über reelle Funktionen" , Dover, reprint  (1948)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  E. Coddington,  N. Levinson,  "The theory of ordinary differential equations" , McGraw-Hill  (1955)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  M.A. Krasnoselskij,  "Topological methods in the theory of nonlinear integral equations" , Pergamon  (1964)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  J. Kurzweil,  "Ordinary differential equations" , Elsevier  (1986)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  A.F. Filippov,  "Differential equations with discontinuous right hand sides" , Kluwer Acad. Publ.  (1988)</td></tr>
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</table>

Latest revision as of 07:22, 13 February 2024

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)
How to Cite This Entry:
Carathéodory conditions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Carath%C3%A9odory_conditions&oldid=55450
This article was adapted from an original article by I. Rachůnková (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article