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

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.

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.