I'm solving the integral equation with unknown parameter in the kernel,
$g(t)=\int K(t,s,\sigma) f(s) ds$,
where $f(s)$ and $\sigma$ have to be defined. Without the parameter, it is the Fredholm problem of the first kind. But I could not find the name of the problem with the parameter. Thanks!
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.
