Some systems of nonlinear difference equations of higher order with periodic solutions. (English) Zbl 1098.39003

The paper deals with the two systems (1) \(x^{(m)}_{n+1} x^{(m+2)}_{n-1}=1+x_n^{(m+1)}=1 + x^{(m+1)}_n\) and (2) \(x^{(m)}_{n+1}x^{(m+3)}_{n-2}=1+x^{(m+1)}_n+x^{(m+2)}_{n-1}\). The solutions of both systems are assumed to be \(k\)-periodic in \(m\) with a fixed \(k\in\mathbb N\). It is shown that all solutions are \(p\)-periodic in \(n\) with \(p=5k\) when \(5\nmid k\) and \(p = k\) else in the case (1), whereas \(p = 8k\) when \(k = 2^jq\) \((0\leq j\leq 2\)), \(2 \nmid q\) and \(p = k\) else in the case (2).


39A11 Stability of difference equations (MSC2000)
39A20 Multiplicative and other generalized difference equations