= Frequency-space inverse
Use exactly the supplied inverse <Fourier transform> convention $e^{-i\omega\tau}$, with $\tau=t-t'$. Then $\partial_t\mapsto-i\omega$ and $\partial_t^3\mapsto i\omega^3$. The transformed equation is
$$
(1-\omega^2-i\varepsilon\omega^3)x(\omega)=f(\omega),\qquad
\boxed{G(\omega)=\frac1{1-\omega^2-i\varepsilon\omega^3}}.
$$
This algebraic inverse still needs a <pole> prescription to select the time-domain solution, particularly when a <pole> is on the real integration contour.
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