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 ( 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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Answers (1)

Clarissa Adkins
Answered 2022-07-20 Author has 16 answers
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't constrain K or F at all in your statement, so how could anyone possibly make more specific statements?

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Can anyone confirm if my answer is right?
Find the critical points and the intervals of the increase and decrease of the function f ( x ) = ( x + 5 ) 2 ( x 2 ) 5
The critical points are: 3 ,   5 ,   2
The four intervals -
from left to right these open intervals are ( , 5 ) ,   ( 5 , 3 ) ,   ( 3 , 2 )   and   ( 2 , )
This part I am not sure of:
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