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According to Wikipedia, A version of Grönwall's inequality for the integral of continuous functions is the following:

Let $I$ denote an interval of the real line of the form $[a,\infty)$ or $[a,b]$ or [a,b) with $a<b.$ Let $\alpha,\beta$ and $u$ be real-valued continuous functions defined on $I.$ If $\color{blue}{\beta\;is \;nonnegative}$ and if $u$ satisfies the integral inequality$$u(t)\leq\alpha(t)+\int_{a}^{t}\beta(s)u(s)ds,\hspace{20 pt}\forall t\in I,$$then $$u(t)\leq\alpha(t)+\int_{a}^{t}\alpha(s)\beta(s)\text{exp}\bigg(\int_{s}^{t}\beta(r)dr\bigg)ds,\hspace{20 pt}t\in I.$$

A problem I have asks to derive the same conclusion given everything except the condition of nonnegativity of $\beta$. But I was not sure if the problem was correct or not. So please let me know if the result could still be obtained without the nonnegativity of $\beta$.

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  • $\begingroup$ Dear Daum Yoon, I believe the result can still be obtained without the non-negativity of $ \beta $. I have what I consider to be the proof of the case where $ \beta $ is any constant in $ \mathbb{R} $. $\endgroup$ – scjorge Jan 12 '16 at 10:53

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