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On the behaviour of the solutions of a second-order difference equation. (English) Zbl 1180.39002
Summary: We study the difference equation \(x_{n+1}= \alpha-x_n/x_{n-1}\), \(n\in\mathbb N_0\), where \(\alpha\in\mathbb R\) and where \(x_{-1}\) and \(x_0\) are so chosen that the corresponding solution \((x_n)\) of the equation is defined for every \(n\in\mathbb N\). We prove that when \(\alpha=3\) the equilibrium \(x=2\) of the equation is not stable, which corrects a result due to X. X. Yan, W. T. Li, and Z. Zhao [J. Appl. Math. Comput. 17, No. 1–2, 269–282 (2005; Zbl 1068.39030)]. For the case \(\alpha=1\), we show that there is a strictly monotone solution of the equation, and we also find its asymptotics. An explicit formula for the solutions of the equation are given for the case \(\alpha=0\).

MSC:
39A10 Additive difference equations
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