I have come across this set of inequalities in $f,\,g:[0,T]\to \mathbb{R}^+$: $$f(t)\le a(t)+\int_0^ta(t-s)f^3(s)ds+c_1\int_0^ta(t-s)g(s)ds$$ $$g(t)\le c_2\int_0^ta(t-s)g(s)ds+c_3\int_0^ta(t-s)f(T-s)ds$$
For some continuous function $a(t)>0$ and positive constants $c_1, \, c_2,$ and $c_3$. Any reference to such inequalities, and how they are solved or treated would be helpful.