On a solvable system of difference equations of higher-order with period two coefficients. (English) Zbl 1492.39010
Summary: We show that the next difference equations system
\[
x_{n+1}= \frac{a_n x_{n-k+1}y_{n-k}}{y_n-\alpha_n}+\beta_{n+1},\quad y_{n+1}=\frac{b_n y_{n-k+1}x_{n-k}}{x_n-\beta_n}+ \alpha_{n+1},\quad n\in\mathbb{N}_0,
\]
where \(\mathbb{N}_0=\mathbb{N}\cup\{0\}\), the sequences \((a_n)_{n\in \mathbb{N}_0}\), \((b_n)_{n\in \mathbb{N}_0}\), \((\alpha_n)_{n\in \mathbb{N}_0}\), \((\beta_n)_{n\in\mathbb{N}_0}\) are two periodic and the initial conditions \(x_{-i}\), \(y_{-i}\), \(i\in\{0, 1,\dots, k\}\), are non-zero real numbers, can be solved. Also, we investigate the behavior of solutions of above-mentioned system when \(a_0=b_1\) and \(a_1= b_0\).
MSC:
| 39A23 | Periodic solutions of difference equations |
| 39A20 | Multiplicative and other generalized difference equations |
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