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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.

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up vote 0 down vote accepted

Existence and uniqueness is clear once $x(0)$ is chosen: $$ \begin{align*} x(1)&=f(0,x(0))\\ x(2)&=f(1,x(1))\\ x(3)&=f(2,x(2))\\ \dots \end{align*} $$ You can compute the solution iteratively.

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