I have an integral equation of the following form:
$$\lambda f(x) = -g(x)f(x) + \int K(x,x')f(x')\mathrm dx'$$
where $g(x)$ and $K(x,x')$ are known functions. The goal is to determine $f(x)$ and the eigenvalues $\lambda$ for which the solution $f(x)$ is non-trivial. This is obviously linear in $f(x)$. Setting $g(x)=0$ reduces this equation to the typical homogeneous Fredholm equation of the 2nd kind. In the current form (with non-trivial $g(x)$) however I do not know of any theory to solve it. Can anyone give some pointers?
Let $T$ be the integral operator $T f(x) = \int K(x,x') f(x') \; dx'$ and $G$ the multiplication operator $G f(x) = g(x) f(x)$. Then your equation is $$ (T - G) f = \lambda f$$
i.e. $\lambda$ is an eigenvalue of $T - G$.
Note that there may be no such $\lambda$. The simplest case is where $T = 0$, so you're just looking for eigenvalues of the multiplication operator $G$. Those will only exist if $g$ is constant on a set of positive measure.