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Picard's Existence Theorem

Picard's existence theorem states that if f is a continuous function that satisfies the Lipschitz condition

|f(x,t)-f(y,t)|<=L|x-y|
(1)

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

(dx)/(dt)=f(x,t)
(2)
x(t_0)=x_0
(3)

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 Resource. https://mathworld.wolfram.com/PicardsExistenceTheorem.html

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