Solving a linear functional equation

Question

Working with Green functions, I have found to solve the following equation $$ -\omega^2G(\omega)-m^2G(\omega)+\kappa\sum_{n=-\infty}^\infty b_nG(\omega-n\omega_0)=1 $$ where $m$, $\kappa$ and $\omega_0$ are arbitrary constants, $b_n$ are coefficients of a Fourier series and so $$ \sum_{n=-\infty}^\infty|b_n|^2<\infty. $$ Is there any approach I can attempt to solve it? Is there anything in literature to approach this kind of problems?