Picard's Existence Theorem

If is a continuous function that satisfies the Lipschitz Condition

$\begin{displaymath} \vert f(x,t)-f(y,t)\vert\leq L\vert x-y\vert \end{displaymath}$

in a surrounding of $(x_0,t_0)\in\Omega\subset\Bbb{R}\times\Bbb{R}^n=\{(x,t):\vert x-x_0\vert<b, \vert t-t_0\vert<a\}$ , then the differential equation

$\displaystyle {df\over dx}$	$\textstyle =$	$\displaystyle f(x,t)$
$\displaystyle x(t_0)$	$\textstyle =$	$\displaystyle x_0$

has a unique solution

in the interval $\vert t-t_0\vert<d$ , where $d=\min(a, b/B)$ , min denotes the Minimum, $B=\sup \vert f(t,x)\vert$ , and sup denotes the Supremum.

See also Ordinary Differential Equation