# Grönwall's inequality

(Redirected from Gronwall's lemma)

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.