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 $x(t)$.

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

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.