Oscillatory solution of a difference equation

Remarks

Example: $x_n = (-1)^n$ solves $x_{n+1} = -x_n$. For every $n$ one has $x_n x_{n+1} = -1 \le 0$, so the solution is oscillatory.

Non-example: $x_n = 2^{-n}$ solves $x_{n+1} = x_n / 2$. Here $x_n x_{n+1} > 0$ for all $n$, so the condition fails already for $N = n_0$, and the solution is nonoscillatory.

One also speaks of oscillation about a value $x^*$, often an equilibrium, meaning that $(x_n - x^*)$ is oscillatory. For the linear equation $x_{n+1} = a x_n$ with $a \ne 0$, every nonzero solution is oscillatory exactly when $a < 0$.