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I am looking for a proof or refrence for the following Gronwall-type inequality:


Let $ \varphi (t,s) $ is a continous function for $0 \leq s < t \leq T$. If the following inequality holds: $$ \varphi (t,s) \leq A + B \int_s^t ( t - \sigma)^{\alpha -1} \varphi ( \sigma , s) d \sigma $$ for some $ \alpha > 0$ on $0 \leq s < t \leq T$, then there exists $C$ such that $ \varphi (t,s) \leq C$.


Any help very welcome.

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