Grönwall's inequality

From Exampleproblems

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.

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Categories: Inequalities | Lemmas