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On some solvable systems of difference equations. (English) Zbl 1253.39011
The author studies the following system of difference equations \[ x_{n+1}=\frac{u_n}{1+v_n}, \qquad y_{n+1}=\frac {w_n}{1+s_n}, \qquad n\in \mathbb{N}_0, \] where \(u_n\), \(v_n\), \(w_n\), \(s_n\) are some of the sequences \(x_n\) or \(y_n\), with real initial values \(x_0\) and \(y_0\). For fourteen of the sixteen possible cases, the author obtains the general solutions. The unsolved cases are \[ x_{n+1}=\frac{y_n}{1+y_n}, \qquad y_{n+1}=\frac{y_n}{1+x_n}, \qquad n\in \mathbb{N}_0 \] and the symmetric counterpart \[ x_{n+1}=\frac {x_n}{1+y_n}, \qquad y_{n+1}=\frac{x_n}{1+x_n}, \qquad n\in \mathbb{N}_0. \]

MSC:
39A20 Multiplicative and other generalized difference equations
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