We're stuck on the following question in a problem relevant to a physics paper on AdS/CFT that we are working on. Given a Fredholm equation of second kind with the form $f$+$\int_D K f\,dx = 0$, where $\int_D K f\,dx$ is not a contraction and $D$ is compact, what are some of the general cases in which $f=0$ is the unique solution?

Alternatively, what restrictions on $K$ does one have to put to obtain a unique solution?