This would just be a family of Fredholm equations depending on a parameter. Things like this pop up with solutions to the Helmholtz equation when using integral equations, where the Helmholtz equation has a complex parameter $s$ indicating the wavenumber. The behavior of the problem may change depending on $s$, though. For example, if you're looking at the resolvent Laplace equations (forgoing boundary conditions), you have something like
-\Delta u + su = 0
which can be shown to be coercive for $s \neq 0$. But when $s=0$ or is an eigenvalue of the Laplacian, you lose coercivity, and so need a different approach.