Determine the stability of a system without using the $\mathcal Z$-transform (described by a difference equation) For example, let's say a causal LTI System is described by the following equation:

$$y[n] - ay[n-1] = x[n] - bx[n-1],\quad n \in Z$$

Is there a way to determine (in this case) the stability of the system without using the $\mathcal Z$-transform?

(Solving the problem using the $\mathcal Z$-transform would be a classical easy problem - verify that ROC of $H(z)$ contains the unit circle, I know how to do that)
It boils down to the same thing, with or without $\mathcal{Z}$-transform. You determine the roots of the characteristic equation, which in your case is

$$\lambda=a\tag{1}$$

For stability, the magnitude of the roots of the characteristic equation must be smaller than $1$. So for your example you get the condition $|a|&lt;1$ for stability.