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On the solutions of systems of difference equations. (English) Zbl 1146.39023

Summary: We show that every solution of the following system of difference equations
\[ x_{n+1}^{(1)}=x_n^{(2)}/(x_n^{(2)}-1),\;x_{n+1}^{(2)}= x_n^{(3)}/(x_n^{(3)-1}),\dots,x_{n+1}^{(k)}=x_n^{(1)}/(x_n^{(1)}-1) \] as well as of the system
\[ x_{n+1}^{(1)}=x_n^{(k)}/(x_n^{(k)}-1), x_{n+1}^{(2)}= x_n^{(1)}/(x_n^{(1)}-1),\dots,x_{n+1}(k)=x_n(k-1)/(x_n^{(k-1)}-1) \]
is periodic with period \(2_k\) if \(k\neq 0 \pmod 2\), and with period \(k\) if \(k=0 \pmod2\) where the initial values are nonzero real numbers for \(x_0^{(1)},x_0^{(2)},\dots,x_0^{(k)}\neq 1\).

MSC:

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