I have a system of Volterra integral equations of form

$$x(t)=x_0+\int_0^t K(x,t)F(x(t))$$

and I am interested on the critical points of $x(t)$, I mean maximum, minimum, increasing and decreasing intervals, nonnegativity etc.

I imagine it's impossible to get complete informations about that, but here I am asking for theorems and general results to help me to study these aspects, once is impossible know the true solution.

Thank you.