# Singular Gronwall type inequality

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.