I know about existence and uniqueness of solutions to differential equations, but when it comes to difference equations, I am struggling to find a reference.
I am looking for conditions under which, for all $k >0$, the solutions of
$x(k+1) = f(k,x(k)), \quad x(0) \in \mathbb{R}^{n},$
exist and are unique, where $f: \mathbb{Z}^{+} \cup \{0 \} \times \mathbb{R}^{n} \to \mathbb{R}^{n}$.
If there is an book that has theory on difference equations and has something about this topic, I would really like to know about it.
In case you are curious, this came up while I was looking for references on discrete time Lyapunov stability theorems.