Grönwall's inequality

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In mathematics, Grönwall's lemma allows one to estimate a function that is known to satisfy a certain differential inequality. There are two forms of the lemma, an integral form and a differential form.

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.

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

[edit] Integral form

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) \, \mathrm{d} s

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)\, \mathrm{d} s\right)

on t_0\leq t\leq t_1.

[edit] Differential form

The same conclusion holds if the inequality

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

is replaced by the corresponding differential inequality

\frac{\mathrm{d} \phi}{\mathrm{d} t} (t) \leq L \psi(t) \phi (t).

In this case one concludes that

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

for t_0 \leq t \leq t_1.


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

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