3 | PRELIMINARY LEMMAS
Lemma | 3.1 (classical differential version
of Gronwall lemma).
We assume that \(\Theta\) ∈ C1([0, T); R), T ∈ (0,
∞), satisfies the differential inequality
\(\dot{\Theta}\left(t\right)\leq b\left(t\right)+\varphi\left(t\right)\Theta(t)\)on (0, T) (6)
for some \(\varphi\), b ∈ C (0, T),
Then, \(\Theta\) satisfies the estimate
\(\Theta\left(t\right)\leq\Theta\left(0\right)e^{\Phi\left(t\right)}+\int_{0}^{t}{b\left(\zeta\right)e^{\Phi\left(t\right)-\Phi\left(\zeta\right)}}d\zeta,\ \ \ \ \ \forall t\in\left[0,t\right.\ )\)(7)
where we have defined the primitive function \(\Phi\)(t)
:=\(\int_{0}^{t}{\varphi\left(\zeta\right)\text{dζ}}\)