PlanetMath: Gronwall's lemma

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Gronwall's lemma

(Theorem)

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

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

holds on $t_0\leq t\leq t_1$ , with

and

positive constants, then

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

on $t_0\leq t\leq t_1$ .

"Gronwall's lemma" is owned by jarino.

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Other names:

Gronwall's inequality

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Cross-references: positive, inequality, continuous functions
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This is version 1 of Gronwall's lemma, born on 2003-01-18.
Object id is 3901, canonical name is GronwallsLemma.
Accessed 22147 times total.

Classification:

26D10 (Real functions :: Inequalities :: Inequalities involving derivatives and differential and integral operators)

Pending Errata and Addenda

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