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Let $V$ be a $continuous$ vector-field defined on a domain $U$ in $R^n$ . Let $x_0$ be a point in $U$. Is it true that the differential equation $\frac{dx}{dt}$ = $V(x)$ admits a solution $\phi(t)$ satisfying initial condition $\phi(0)$=$x_0$ defined on some open interval containing $0$.

By Arnold's Book Existence theorem I know that unique solution exists in case of smooth vector field.But here vector field is only $continuous$.Can we have counterexample in this case?

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marked as duplicate by Hans Lundmark, user147263, timur, Hagen von Eitzen, Lost1 Oct 9 '14 at 18:53

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    $\begingroup$ Funny. This was asked yesterday too. $\endgroup$ – Hans Lundmark Oct 9 '14 at 16:41
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Yes.It is called Peano's existence theorem.

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  • $\begingroup$ does Peano's existence extend to higher dimensions? $\endgroup$ – Kayoken Oct 9 '14 at 16:30
  • $\begingroup$ Yes. However, you loose uniqueness (in all dimensions). $\endgroup$ – Julián Aguirre Oct 9 '14 at 16:31
  • $\begingroup$ Oh i see..i wonder why in arnold's book smooth vector fields are only considered $\endgroup$ – Kayoken Oct 9 '14 at 16:34
  • $\begingroup$ I guess it is because in most physical applications you want uniqueness of solution. $\endgroup$ – Julián Aguirre Oct 9 '14 at 16:46

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