In the process of solving a problem about diffusion from a sphere, we want to find the Green's function for our problem. It is complicated by the fact that we have two media with different diffusion coefficient and attenuation time constants inside and outside the radius $R$.
At this stage, my goal is to arrive at a Green's function $G(r, t)$ which we currently only have defined by its fourier transform $\tilde{G}(q, \omega)$, where $q$ is the Fourier transform of $r$ and $\omega$ is the Fourier transform of $t$, which is given like this
\begin{equation} \tilde{G}(q, \omega) = \tilde{G}_0(q,\omega) \left[ \lambda\int_0^R \tilde{G}(r,\omega)e^{-i q r} dr +1 \right] , \end{equation}
where $\tilde{G}(r,\omega)$ is the Fourier transform of $G(r,t)$ done only over $t$.
In this equation, $\tilde{G}_0$ is the solution if we assumed a homogeneous medium with the properties of the medium at $r>R$, and the rest can be seen as a perturbation to $\tilde{G}_0$ to include the effects of the medium inside $R$. (Equivalently, a similar equation can be written with the integral over $\int_R^\infty$ and $\tilde{G}_0$ being the inside solution with the perturbation due to "outside" effects.)
The problem is that this equation does not have the form of anything I can find anywhere. If $\tilde{G}(q,\omega)$ had not been a factor on the integral, then it would essentially be a Fredholm integral, which we could expand in a series to approximate $\tilde{G}(q,\omega)$ by setting $\tilde{G}(r,\omega)$ equal to the inverse FT of $\tilde{G}_0(q,\omega)$ over $q$ and so on, but as it is now, I'm at a loss.
We have been suggested to attack this problem with perturbation theory, but I am not experienced with that and cannot see how to apply what I find online to this problem.