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Picard's Existence Theorem

If $f$ is a continuous function that satisfies the Lipschitz Condition

\vert f(x,t)-f(y,t)\vert\leq L\vert x-y\vert

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 $x(t)$ 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

© 1996-9 Eric W. Weisstein