For instance, in this article of Wikipédia a statement is made for Differential form of the Grönwall's Lema for a $\beta, u:I:=[a,b]\subset \mathbb{R} \longrightarrow \mathbb{R}$ continuous functions and $u$ differentiable in the interior $I^{\circ}$ of $I$. That is, $ u $ and $ \beta $ are not necessarily non-negatives. Where, $$u'(t) \leq \beta(t)u(t) , \; \forall \; t \in I^{\circ}\Rightarrow u(t) \leq u(a)e^{\int_{a}^{t}\beta(s)}ds, \; \forall \; t \in I.$$

The versions of the Grönwall's Lema that I know of require that the functions be non-negative. Is that what's on Wikipedia right? If so, is there a reference that contains this result?

In the statement of the article, quoted above, an observation is made: There are no assumptions on the signs of the functions $\beta$ and $u$.

I did not see error in the proof presented there.


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