 # 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. Graham Beasley 2022-07-19 Answered
Stability, critical points and similar properties of solutions of nonlinear Volterra integral equations
I have a system of nonlinear Volterra integral equations of form $x\left(t\right)={x}_{0}+{\int }_{0}^{t}K\left(t,s\right)F\left(x\left(s\right)\right)ds$ 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.
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Assuming the hypotheses of the Leibniz integral rule apply,
${x}^{\prime }\left(t\right)=K\left(t,t\right)F\left(x\left(t\right)\right)+{\int }_{0}^{t}{\mathrm{\partial }}_{1}\left(K\left(t,s\right)\right)F\left(x\left(s\right)\right)\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}s\text{.}$
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?

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