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.