I knew the following Gronwall's inequality (Integral form)
If $\alpha$ is non-negative and $H(t)$ satisfies the integral inequality
\begin{align*} H(t) \leq c+ \int_0^t \alpha(s) H(s)ds \quad \text{(c: constant)}, \end{align*}
then
\begin{align*} H(t) \leq c \exp \left( \int_0^t \alpha(s) ds \right). \end{align*}
But, I wonder whether there are some Gronwall-type inequalities below :
New Version If $H(t)$ satisfies the integral inequality \begin{align*} H(t) \leq c + \int_0^t \color{red}{\alpha(t,s)} H(s) ds, \end{align*} then it holds \begin{align*} H(t) \leq c \exp \left( \int_0^t \alpha(t,s) ds\right). \end{align*}
(Specifically, in my case, $\alpha(t,s) = (1+(t-s)^{-1/2}) e^{-s}$.)
I'm curious whether Gronwall type inequality mentioned above can be established, and if not, whether it can be applicable in my case. I would appreciate it if you could provide references or assistance.