I have an integral equation which is not exactly an eigenvalue type equation, but similar:

$$\int d^3 y\, K(x,y,\lambda) f(y) = f(x)$$

Here $\lambda$ can be thought of as an eigenvalue, so it expected that only for a discrete set of $\lambda$ we have a solution. $K(x,y,\lambda)$ is some fairly simple expression involving ratios of polynomials and also $\lambda$ appears in a rational fashion.

What would be the typical numerical methods to find the lowest eigenvalue $\lambda$ and eigenfunction $f(x)$?