The problem has a simple physical background. In the solid-state physics, it is possible to determine the real and imaginary parts of the dielectric permittivity \epsilon through the polarization function, depending on the frequency omega and a certain parameter q. In fact and roughly speaking, the integral is the same polarization operator, which is defined through the law of dispersion .
Usually, if one knows the dispersion law (e.g., the most common example is quadratic one), one can predict the frequency behavior of the dielectric permittivity function. I want to solve the inverse problem.
Let us assume that we know the form of the dielectric permittivity. Let us imagine that it has the form of a slowly varying function (https://en.wikipedia.org/wiki/Slowly_varying_function). The task is to determine the corresponding dispersion law.
I must confess that I chose the integration limits almost arbitrarily. In principle, there are no physical restrictions on other values of integration limits. I wanted to understand for a start how it is technically possible to numerically solve this equation, which can be called as "delay-integral equation".