Gronwall's lemma

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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 $t_{0}\leq t\leq t_{1}$ .

Synonym:

Gronwall's inequality

Type of Math Object:

Theorem

Major Section:

Reference