On asymptotic behaviour of the difference equation \(x_{x+1}=\alpha + \frac {x_{n-k}}{x_n}\). (English) Zbl 1042.39001

The short paper is devoted to the difference equation in the title with \(\alpha\geq1\) and \(k=0,1,\ldots\). Using some theorems by R. DeVault, W. Kosmala et al. [Nonlinear Analysis 47, 4743–4751 (2001)] the authors obtain a few simple results on asymptotic behavior of positive solutions. In particular, if \(\alpha>1\) then for any \(k\) a unique equilibrium point \(\bar{x}=\alpha+1\) is globally asymptotically stable; if \(k\) is odd and \(\bar{x}\) is locally asymptotically stable then \(\alpha>1\); the equation has a prime period two solution iff \(\alpha=1\) and \(k\) is odd.


39A11 Stability of difference equations (MSC2000)
39A20 Multiplicative and other generalized difference equations
Full Text: DOI


[1] Amleh, A.M.; Grove, E.A.; Georgiou, D.A.; Ladas, G., On the recursive sequence xn+1=α+(xn−1/xn), J. math. anal. appl., 233, 790-798, (1999) · Zbl 0962.39004
[2] Devault, R.; Kosmala, W.; Ladas, G.; Schultz, S.W., Global behavior of yn+1=(p+yn−k)/(qyn+yn−k), Nonlinear analysis, 47, 4743-4751, (2001) · Zbl 1042.39523
[3] Kocic, V.L.; Ladas, G., Global behavior of nonlinear difference equations of higher order with applications, (1993), Kluwer Academic Publishers Dordrecht · Zbl 0787.39001
[4] Kuruklis, S.A., The asymptotic stability of xn+1−axn+bxn−k=0, J. math. anal. appl., 188, 719-731, (1994) · Zbl 0842.39004
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