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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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##### References:
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