Theorem of Cauchy-Peano-Arzelà

Statement

Are $E$ a Espace of Banach of finished size, $H\; \backslash \; subset\; E$ a convex open part of $E$. Either $I=$ an interval of $\backslash \; mathbb\; \{R\}$ ($t\_0\; \backslash \; in\; \backslash \; mathbb\; \{R\},\; a>0$), or $f$ a continuous and limited function $I\; \backslash \; times\; H$ in $E$. That is to say $M=\; \backslash \; sup\_\; \{(T,\; X)\; \backslash \; in\; I\; \backslash \; times\; H\}\; \backslash \; mid\; \backslash \; mid\; F\; (T,\; X)\; \backslash \; mid\; \backslash \; mid$.
Are $x\_0\; \backslash \; in\; H$ and $r>0$ such as $B=B\; (x\_0,\; R)\; \backslash \; subset\; H$.
Then, there exists a solution with the problem:
$x\text{'}=f\; (T,\; X)$
$x\; (t\_0)\; =x\_0$
defined on the interval $$ where $c=\; \backslash \; inf\; (has,\; \backslash \; frac\; \{R\}\; \{M\})$, and in values in $B$.

N.B.: As opposed to what makes it possible to conclude the Théorème from Cauchy-Lipschitz under more restrictive assumptions, it does not have unicity there here.

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