Let $p(t), a(t)$ be non-negative, continuous functions on $[0,T]$. Suppose that we have: $$p(t) \leq a(t) + C \int_0^t du e^{-\kappa(t-u)}p(u) \int_0^u ds e^{-\kappa(u-s)} p(s),$$ where $\kappa, C > 0$ are constants.

Can we derive a Gronwall type inequality for $p(t)$, under possibly additional assumptions on $p(t)$ and/or $a(t)$?


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