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On the difference equation \(X_{n+1} = \alpha + \frac{x_{n-1}}{x_n}\). (English) Zbl 1155.39305

Summary: We study the behavior of the solutions of the difference equation \[ x_{n+1} = \alpha + \frac{x_{n-1}}{x_n}\quad n=0,1,\dots \] where \(\alpha \) is a negative number. Included are results which considerably improve and correct those in the recently published paper [A.E. Hamza, J. Math. Anal. Appl. 322, No. 2, 668–674 (2006; Zbl 1105.39008)]. We also refute Conjecture 2 in [G. Ladas et al., J. Difference Equ. Appl. 7, No. 3, 477–482 (2001; Zbl 1081.39503)].

MSC:

39A23 Periodic solutions of difference equations
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