Let us consider the equation $$ f(t) = 1 - \int_0^t \frac{\sin(\omega s) f(s)}{\sqrt{t-s}} ds. \qquad (*) $$ Here, we assumed $\omega > 0$, and we look for solutions $f:[0, \infty) \to \mathbb{R}$.
This is a so-called Abel--Volterra integral equation of the second kind. I am trying to get a formula for the solution to this equation, and I am trying to understand the behaviour of the solution as $\omega$ grows to $+\infty$.
I tried taking the Laplace transform of both sides, but this introduces the problematic term $\mathcal{L}_t(\sin (\omega t) f(t))$, where $\mathcal{L}_t$ is the Laplace transform, and it is not well-behaved with respect to products.
I have furthermore tried integrating $(*)$ in time from $0$ to $t_1$ so as to obtain an equation of the form (Volterra equation of the first kind) $$ \int_0^{t_1} M(s,t_1) g(s) d s = H(t_1), $$ where $M$ and $H$ are known functions. But that also does not seem to help.
Does anyone have an idea whether a representation formula for solutions of such equation is possible? Any help would be much appreciated.