Stability, critical points and similar properties of solutions of nonlinear Volterra integral equationsI have a system of nonlinear Volterra integral equations of form x ( t ) = x 0 + ∫ 0 t K ( t , s ) F ( x ( s ) ) d s 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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Stability, critical points and similar properties of solutions of nonlinear Volterra integral equationsI have a system of nonlinear Volterra integral equations of form   x  (  t  )  =      x    0    +      ∫    0    t    K  (  t  ,  s  )  F  (  x  (  s  )  )  d  s and I am interested on the critical points of x(t), I mean maximum, minimum, increasing and decreasing intervals, nonnegativity etc.I imagine it&#039;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.

Clarissa Adkins · Accepted Answer

Assuming the hypotheses of the Leibniz integral rule apply,      x    ′    (  t  )  =  K  (  t  ,  t  )  F  (  x  (  t  )  )  +      ∫    0    t        ∂    1    (  K  (  t  ,  s  )  )  F  (  x  (  s  )  )        d    s  .So, can you find the zeroes of K and F? Can you find the signs of K and F on various intervals? You don&#039;t constrain K or F at all in your statement, so how could anyone possibly make more specific statements?

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.

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