Picard's Existence Theorem

If f is a continuous function that satisfies the Lipschitz condition


in a surrounding of (x_0,t_0) in Omega subset R^n×R={(x,t):|x-x_0|<b,|t-t_0|<a}, then the differential equation


has a unique solution x(t) in the interval |t-t_0|<d, where d=min(a,b/B), min denotes the minimum, B=sup|f(t,x)|, and sup denotes the supremum.

See also

Lipschitz Condition, Ordinary Differential Equation, Picard's Great Theorem

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Cite this as:

Weisstein, Eric W. "Picard's Existence Theorem." From MathWorld--A Wolfram Web Resource.

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