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I have an Integro-Differential Equation (IDE) of the following form: $$ x'(t) = f(t,x(t)) + \int_0^t K(t-s, x(s), x(t)) ds, $$ I have found this classical reference, but the IDEs considered therein have $K$ as a function of $(t,s,x(s))$, whilst for me it is essential to also consider the dependence on $x(t)$. Here the domain of $x$ is a function from $[0,\infty)$ to $\mathbb{R}$, $f$ and $K$ are both smooth functions.

I am simply looking for criteria under which the IDE has a (unique) solution.

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  • $\begingroup$ Are you just asking for a good reference or you have a real trouble establishing some desired solvability properties in your particular setting? $\endgroup$
    – fedja
    Commented May 16, 2018 at 22:46
  • $\begingroup$ just a reference $\endgroup$
    – Darkwizie
    Commented May 17, 2018 at 5:57
  • $\begingroup$ I have been able to show that if, as a boundary condition, we have $x_1(0) \leq x_2(0)$ that the associated solutions to the IDE satisfy $x_1(s) \leq x_2(s)$ for all $s$. $\endgroup$
    – Darkwizie
    Commented May 17, 2018 at 7:20

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