I have to solve a finite difference equation, and I decided to attempt a solution through Fourier transform. What I obtain is an equation of the form (where I have to solve for $\hat G$ and $\hat \Delta$ is known)
$$ \hat G(p,a)\mathrm e^{2\mathrm i p a}=\hat \Delta(p,a)*_p\hat G(p,a). $$ Here, $\star_p$ is the convolution product with respect to the variable $p$: $$ \hat \Delta(p,a)*_p\hat G(p,a)=\int_{-\infty}^{+\infty}\hat \Delta(p-q,a)\hat G(q,a)\mathrm dq. $$ I have hints that a solution exists at the level of the finite difference equation (up to a multiplicative function of $a$ that I don't really need), but I have no idea about how to solve equation 1.
Is there some literature about this kind of equation? Does a solution (up to multiplicative factors) exist, and is it possible to write it in a closed form?
I am working at a purely formal level, meaning that I usually neglect all details about existence of the things I'm dealing with, convergence, existence of Fourier transforms et cetera (at least, until something breaks down!).
Thank you all!
This looks like a Fredholm equation of the second kind: https://en.wikipedia.org/wiki/Fredholm_integral_equation. Use the notation in the page, you could put $k(p,q) = e^{-2ipa}\Delta(q-p,a)$ and have that $$(KG) (p) = \int_{\mathbb{R}} k(p,q)G(q)dq$$
If $$||K||_2^2 \leq\int_{\mathbb{R}^2} | e^{-2ipa}\Delta(p-q,a)|^2 dpdq < 1 $$ then there does exist a unique solution to the problem $$(I-K)G(p) = 0$$ But since $I-K$ is linear, then a unique solution to $(I-K)G(p) = 0$ must be $G(p,a) =0$. So for nontrivial solutions, you will need $||K||\geq 1$.