Gronwall’s lemma

If, for $t_0\le t\le t_1$, $\varphi (t)\ge 0$ and $\psi (t)\ge 0$ are continuous functions such that the inequality

$$\varphi (t)\le K+L{\int }_{t_0}^t\psi (s)\varphi (s)𝑑s$$

holds on $t_0\le t\le t_1$, with $K$ and $L$ positive constants, then

$$\varphi (t)\le K\exp \left(L{\int }_{t_0}^t\psi (s)𝑑s\right)$$

on $t_0\le t\le t_1$.