Gronwall’s lemma

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