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.

MSC:

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

References:

 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.