Grönwall's inequality

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In mathematics, Grönwall's lemma states the following. If, for t_0\leq t\leq t_1, \phi(t)\geq 0 and \psi(t)\geq 0 are continuous functions such that the inequality

\phi(t)\leq K+L\int_{t_0}^t \psi(s)\phi(s)\,ds

holds on t_0\leq t\leq t_1, with K and L positive constants, then

\phi(t)\leq K\exp\left(L\int_{t_0}^t \psi(s)\,ds\right)

on t_0\leq t\leq t_1.

It is named for Thomas Hakon Grönwall (1877-1932).

Grönwall's lemma is an important tool used for obtaining various estimates in ordinary differential equations. In particular, it is used to prove uniqueness of a solution to the initial value problem, see the Picard-Lindelöf theorem.

This article incorporates material from Gronwall's lemma on PlanetMath, which is licensed under the GFDL.

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