existence and uniqueness for $dot x(t)=f(t,x(t))$, with $f$ measurable in $t$.
$begingroup$
Consider the following scalar ODE
$$dot x(t) = f(t,x(t))$$
defined on a compact interval $[0,T]$.
Assume that $f$ is Lipschitz continuous in $x$, but it is measurable in $t$.
What can we say about the existence and uniqueness?
The standard Picard theorem assumes that $f$ is continuous in $t$.
real-analysis ordinary-differential-equations
asked Dec 25 '18 at 21:59
M.AM.A
1599
$endgroup$
add a comment |
$begingroup$
Consider the following scalar ODE
$$dot x(t) = f(t,x(t))$$
defined on a compact interval $[0,T]$.
Assume that $f$ is Lipschitz continuous in $x$, but it is measurable in $t$.
What can we say about the existence and uniqueness?
The standard Picard theorem assumes that $f$ is continuous in $t$.
real-analysis ordinary-differential-equations
asked Dec 25 '18 at 21:59
M.AM.A
1599
$endgroup$
add a comment |
$begingroup$
Consider the following scalar ODE
$$dot x(t) = f(t,x(t))$$
defined on a compact interval $[0,T]$.
Assume that $f$ is Lipschitz continuous in $x$, but it is measurable in $t$.
What can we say about the existence and uniqueness?
The standard Picard theorem assumes that $f$ is continuous in $t$.
real-analysis ordinary-differential-equations
asked Dec 25 '18 at 21:59
M.AM.A
1599
$endgroup$
Consider the following scalar ODE
$$dot x(t) = f(t,x(t))$$
defined on a compact interval $[0,T]$.
Assume that $f$ is Lipschitz continuous in $x$, but it is measurable in $t$.
What can we say about the existence and uniqueness?
The standard Picard theorem assumes that $f$ is continuous in $t$.
real-analysis ordinary-differential-equations
real-analysis ordinary-differential-equations
asked Dec 25 '18 at 21:59
M.AM.A
1599
asked Dec 25 '18 at 21:59
M.AM.A
1599
asked Dec 25 '18 at 21:59
M.AM.A
1599
asked Dec 25 '18 at 21:59
M.AM.A
1599
asked Dec 25 '18 at 21:59
M.AM.A
1599
1599
add a comment |
add a comment |
1 Answer
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$begingroup$
The answer is almost provided by Carathéodory’s existence theorem.
answered Dec 25 '18 at 22:21
mathcounterexamples.netmathcounterexamples.net
26.9k22158
$endgroup$
add a comment |
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$begingroup$
The answer is almost provided by Carathéodory’s existence theorem.
answered Dec 25 '18 at 22:21
mathcounterexamples.netmathcounterexamples.net
26.9k22158
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add a comment |
$begingroup$
The answer is almost provided by Carathéodory’s existence theorem.
answered Dec 25 '18 at 22:21
mathcounterexamples.netmathcounterexamples.net
26.9k22158
$endgroup$
add a comment |
$begingroup$
The answer is almost provided by Carathéodory’s existence theorem.
answered Dec 25 '18 at 22:21
mathcounterexamples.netmathcounterexamples.net
26.9k22158
$endgroup$
The answer is almost provided by Carathéodory’s existence theorem.
answered Dec 25 '18 at 22:21
mathcounterexamples.netmathcounterexamples.net
26.9k22158
answered Dec 25 '18 at 22:21
mathcounterexamples.netmathcounterexamples.net
26.9k22158
answered Dec 25 '18 at 22:21
mathcounterexamples.netmathcounterexamples.net
26.9k22158
answered Dec 25 '18 at 22:21
mathcounterexamples.netmathcounterexamples.net
26.9k22158
26.9k22158
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