Identify whether to have unique output in this ARMA system
As I mentioned in my comment, I think there is a typographical error in the equation. I think it is supposed to be
\begin{equation}
y[k] = \sum_{i=1}^{M}a_iy[k-i] + \sum_{j=1}^{N}b_jx[k-j].
\end{equation}
This is an example of a linear constant-coefficient difference equation. These are introduced very early in most textbooks about digital signal processing. For example, they are introduced in Section 2.5 (page 33) in the 1989 edition of Discrete-Time Signal Processing by Oppenheim and Schafer.
Given an input signal $x$, there will not be a unique solution signal $y$. This is because a specific solution requires specifying the input signal as well as initial conditions for $y$. Different initial conditions lead to different solutions.
You may gain some insight into the subject from Oppenheim himself in lecture 3 of a DSP course recorded in 1975 and now posted online.