How do I solve the following integral equation for $f(x)$ given $k(x,y)$ and $g(x)$ $$ f(x) = \int_0^x f(x-y)k(x,y)dy+g(x). $$ Any suggestion or reference would be appreciated.
What I have tried: I know how to solve the special case where $k(x,y)$ is only a function of $y$. In that case,
$$
f(x) = \int_0^x f(x-y)k(y)dy+g(x)=(f*k)(x)+g(x)\\\implies F(s) = F(s)K(s)+G(s)\\\implies F(s) = \frac{G(s)}{1-K(s)},
$$
where $F(s)$, $K(s)$, and $G(s)$ are Laplace transforms of $f(x)$, $k(y)$, and $g(x)$. Then we have
$$
f(x) = \mathcal L^{-1}\left\{\frac{G(s)}{1-K(s)}\right\}(x).
$$
I have no, idea what to do with the gerenal case.
Edit: Assume $k(x,y)$ and $g(x)$ satisfy any necessary condition for the solution to exist.