If I have $y(n) = - \sum_{k=1}^N a_k \cdot y(n-k) + \sum_{k=0}^M b_k \cdot x(n-k)$ with
$|x(n)|\leq L\\ y(n-k) = 0 \text{ for all } n-k \lt 0 \\ x(n-k) = 0 \text{ for all } n-k \lt 0 \\ n \in \mathbb{Z} \\ k \in \mathbb{N} \\ a_{k} \text{ and } \ b_{k} \text{ are constants }\in \mathbb{R} $
How can I get the maximum $y(n)$ value of this difference equation? Can you suggest me some book?
In this generality, little can be said. Anything can happen. For example, $$y(n)=2y(n-1)+x(n-1)$$ is a particular instance of this problem. Suppose $x(n)=1$ for all $n\in\mathbb N$ (or even just for one), then $y$ grows exponentially, and there is no maximum at all.
To bring the problem into focus, you should impose some conditions that will prevent $y$ from growing indefinitely. For example, $\sum_{k=1}^N |a_k|<1$.