Determine the stability of a system without using the $\mathcal Z$-transform (described by a difference equation)
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.