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If one wants to relax the continuity assumption on a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c1200301.png" /> while preserving the natural equivalence between the [[Cauchy problem|Cauchy problem]] for the differential equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c1200302.png" /> 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.
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c1200303.png" /> be an open set and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c1200304.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c1200305.png" />. One says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c1200306.png" /> satisfies the Carathéodory conditions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c1200307.png" />, written as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c1200308.png" />, if
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1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c1200309.png" /> is measurable for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c12003010.png" /> (cf. also [[Measurable function|Measurable function]]);
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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.
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c12003011.png" /> is continuous for almost every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c12003012.png" />;
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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
  
3) for each compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c12003013.png" /> the function
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1) $f (. , x ) : J \rightarrow {\bf R} ^ { m }$ is measurable for every $x \in G$ (cf. also [[Measurable function|Measurable function]]);
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c12003014.png" /></td> </tr></table>
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2) $f ( t , . ) : G \rightarrow \mathbf{R} ^ { m }$ is continuous for almost every $t \in J$;
  
is Lebesgue integrable (cf. also [[Lebesgue integral|Lebesgue integral]]) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c12003015.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c12003016.png" /> is the norm in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c12003017.png" />.
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3) for each compact set $K \subset G$ the function
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c12003018.png" /> is a non-compact interval, one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c12003019.png" /> satisfies the local Carathéodory conditions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c12003020.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c12003021.png" /> for every compact interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c12003022.png" />. This is written as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c12003023.png" />.
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\begin{equation*} h _ { K } ( t ) = \operatorname { sup } \{ \| f ( t , x ) \| : x \in K \} \end{equation*}
  
Note that any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c12003024.png" /> which is the composition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c12003025.png" /> and a measurable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c12003026.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c12003027.png" /> (cf. also [[Composite function|Composite function]]), is measurable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c12003028.png" />.
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is Lebesgue integrable (cf. also [[Lebesgue integral|Lebesgue integral]]) on $J$, where $\| .\|$ is the norm in $\mathbf{R} ^ { m }$.
  
To specify the space of the majorant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c12003029.png" /> more precisely, one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c12003030.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c12003032.png" />-Carathéodory, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c12003033.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c12003034.png" /> satisfies 1)–3) above with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c12003035.png" />.
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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 $f \in \operatorname { Car }  _ { \text{loc} } ( I \times G )$.
  
One can see that any function continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c12003036.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c12003037.png" />-Carathéodory for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c12003038.png" />.
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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|Composite function]]), is measurable on $I$.
  
Similarly, one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c12003039.png" /> is locally <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c12003041.png" />-Carathéodory on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c12003042.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c12003043.png" /> restricted to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c12003044.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c12003045.png" />-Carathéodory for every compact interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120030/c12003046.png" />.
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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 $h _ { K } \in L ^ { p } ( J )$.
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One can see that any function continuous on $J \times G$ is $L ^ { p }$-Carathéodory for any $p$.
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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====
 
====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=19225
This article was adapted from an original article by I. Rachůnková (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article