Frequency response of marginally stable LTI systems
As long as the system is linear and time-invariant, it can be described using the frequency response / transfer function $H(j\omega)$. For marginally stable systems, $H(j\omega)$ does exist, but cannot be expressed in terms of the Fourier-transformed impulse response $h(t)$.

The Form $H(j\omega) = \frac{Y(j\omega)}{X(j\omega)}$ however is still valid for systems not stable in the more narrow sense simply because they are both linear and time-invariant.