In Wikipedia https://en.wikipedia.org/wiki/Gr%C3%B6nwall%27s_inequality#Integral_form_for_continuous_functions we can find the following statement:
Let $I$ denote an interval of the real line of the form $[a, ∞)$ or $[a, b]$ or $[a, b)$ with $a < b$. Let $\alpha$, $\beta$, and $u$ be real-valued functions defined on $I$. Assume that $\beta$ and $u$ are continuous and that the negative part of α is integrable on every closed and bounded subinterval of $I$. If $\beta$ is non-negative and $u$ satisfies \begin{equation} u(t) \leq \alpha(t) + \int_a^t \beta(s)u(s)\,ds \end{equation} for all $t\in I$, then \begin{equation} u(t) \leq \alpha(t) + \int_a^t \alpha(s)\beta(s)\exp\left(\int_s^t \beta(r)\,dr\right)\,ds \end{equation} for all $t \in I$.
Is it possible to relax the non-negativity assumption on $\beta$? Specifically, my question is: can we derive any meaningful upper bound on $u$ when $\beta$ is negative?
I searched the Internet but could not find any useful information. It seems most of the research focus on non-negative $\beta$. I guess the reason might be that the case of a negative $\beta$ is simply difficult to handle.