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][1], 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 $x(t)$.

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


  [1]: https://www.amazon.com/Theory-Integro-Differential-Equations-Stability-Control/dp/2884490000