I want to derive a Gronwall-type inequality from the inequality below. Here all the functions are nonnegative, continuous and if you need some assumptions you may use that.
$$ f^2(t) \leqslant g^2(t) + \int\_0^t (f(s) +c) f(s) ds \;\;\;\; (t \in [0,T]) $$
So please help!



A Gronwall-type inequality.
Consider the function 
$$h(t):=\int\_0^t f(s)e^{t-s}ds\,,$$
which solves the ODE $h'=h+f$ with $h(0)=0$, so $$h(t):=\int\_0^t \Big(h(s)+f(s)\Big)ds\, .$$ 
Adding the term $-c\, h(t)$ to both sides, your inequality takes a more familiar form of a Gronwall inequality:
$$f(t)^2-c\;h(t)\le g(t)^2 + \int\_0^t \Big(f(s)^2 -c\;h(s) \Big)ds $$
relative to the function $f(t)^2-c\; h(t)$.


