Eigenvalue Problem for Fredholm (Generalised?) Integro-Differential Equations

Question

Consider the following problem: For $\Omega\subset\mathbb{R}^2$ a bounded domain, find $(\lambda, f(x))\in\mathbb{R}\times L^2(\Omega)$ such that \begin{align*} Lf(x) & = \lambda Kf(x) && \textrm{ in $\Omega$} \\ Bf(x) & = 0 && \textrm{ on $\partial\Omega$} \end{align*} where $L$ is a partial differential operator, $\lambda$ the eigenvalue, $K$ the integral operator with kernel $G(x,y)$ and $Bf(x)$ is the "boundary condition operator", for example $Bf(x) = f(x)$ for Dirichlet boundary condition. Are there any known theoretical results/references about the solutions to this eigenvalue problem? Note that this is a Fredholm integro-differential equation and $x = (x_1,x_2)$.