Asking for results on critical points and similar properties of solutions of nonlinear Volterra integral equations - Physically coherent solutions

Question

I have a system of nonlinear Volterra integral equations of form

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

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

I have interest when $K$ is nonnegative (but the signal of $F$ depends on signal of $x$).

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.

EDIT

I am mainly interested in nonnegativity, since we need it for physically coherent solutions, and also on asymptotic equilibrium when $\int_0^\infty K(t)dt=\infty$ (but $K(t)\to 0$).

OBS.: This is a repost of this question: MathExchange.