## 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).

### MSC:

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