I am dealing with the following equation: $$ f(x) = g(x) + \intop_{X} dt K(x-t)f(t) \;,\qquad \left\lbrace \begin{array}{c}f(x)>0\;,\;x\in X \\ f(x)<0\;,\;x\notin X \end{array}\right.$$ where $X$ is a compact, not necessarily connected, subset of $\mathbb R$ and $$ g(x) = -\cosh x \;,$$ and $K(x)$ is a function in $L^2(\mathbb R)$. The subset $X$ is to be treated as an unknown, to be determined self-consistently by the requirements on the positivity of the function $f(x)$. Hence, by solution to the above equation I mean a function $f(x)$ together with a subset $X$.
A specific case is $X = [-A,A]$ and $$ K(x) = \frac{1}{2\pi}\sum_{\sigma,\sigma'=\pm1}\frac{1}{\cosh\left(x+\sigma\theta+i\sigma'\gamma\right)}\;,$$ but I would like to keep things more general.
My question consists of two parts:
there exist known results on existence and uniqueness of solutions?
is there a known procedure to solve this equation for $f(x)$ and $X$? I tried to extend the method of Wiener and Hopf, but I do not seem to be getting anywhere useful.
Thanks,
Stefano
