Grönwall's inequality

Lemma (Grönwall’s lemma in differential form)
Let \(I\) be an interval of the real line of one of the forms \([a, \infty)\), \([a, b]\), or \([a, b)\) with \(a < b\), and let \(\beta \in C(I)\) such that \(\beta(t) \geq 0\) for all \(t \in I\). If \(v \in C^1(I)\) is a nonnegative function that satisfies the inequality

\[v'(t) \leq \alpha(t) + \beta(t) v(t), \quad \text{for all } t \in I,\]

then

\[v(t) \leq v(a) \exp\left( \int_a^t \beta(s) ds \right) + \int_a^{t} \alpha(r) \exp\left(\int_{r}^{t}\beta(s) ds\right) dr, \quad \text{for all } t \in I.\]