Stability, critical points and similar properties of solutions of nonlinear Volterra integral equations

Question

I've posted also on MathOverflow, but repost here once I think it's an important question and would like to get more attention. Thank you!

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 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. $K$ is nonnegative and the signal of $F$ depends on signal of $x$. Also interested on asymptotic equilibrium when $\int_0^\infty K(t)dt=\infty$ (but $K(t)\to 0$).

Answer 1

Assuming the hypotheses of the Leibniz integral rule apply, $$ x'(t) = K(t,t)F(x(t)) + \int_0^t \partial_1(K(t,s)) F(x(s)) \,\mathrm{d}s \text{.} $$ So, can you find the zeroes of $K$ and $F$? Can you find the signs of $K$ and $F$ on various intervals? You don't constrain $K$ or $F$ at all in your statement, so how could anyone possibly make more specific statements?