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On asymptotic behaviour of the difference equation ${x}_{n+1}=\alpha +\frac{{x}_{n-1}^{p}}{{x}_{n}^{p}}$. (English) Zbl 1052.39005

The authors investigate the oscillation with respect to the equilibrium, and the asymptotic behaviour of the positive solutions to the difference equation

${x}_{n+1}=\alpha +{\left({x}_{n-1}/{x}_{n}\right)}^{p},\phantom{\rule{0.166667em}{0ex}}n=0,1,\cdots ,$

where $\alpha \ge 0$ and $p\ge 1$.

##### MSC:
 39A11 Stability of difference equations (MSC2000) 39A20 Generalized difference equations
##### References:
 [1] A.M. Amleh, E.A. Grove, D.A. Georgion and G. Ladas, On the recursive sequence ${x}_{n+1}=\alpha +\frac{{x}_{n-1}}{{x}_{n}}$ , J. Math. Anal. Appl. 233, (1999), 790–798. · Zbl 0962.39004 · doi:10.1006/jmaa.1999.6346 [2] C. Gibbons, M. Kulenović and G. Ladas,On the recursive sequence y n+1=($\alpha$+$\beta$yn)/($\gamma$+yn) Math. Sci. Res. Hot-Line 4,No 2. (2000), 1–11. [3] V. L. Kocic, and G. Ladas,Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, 1993. [4] V.L. Kocic, G. Ladas and I. Rodrigues,On the rational recursive sequences, J. Math. Anal. Appl. 173 (1993), 127–157. · Zbl 0777.39002 · doi:10.1006/jmaa.1993.1057 [5] W. Kosmala, M. Kulenović, G. Ladas and C. Teixeira,On the recursive sequence, y n+1= (p+yn)/(qyn+yn), J. Math. Anal. Appl. 251, (2000), 571–586. · Zbl 0967.39004 · doi:10.1006/jmaa.2000.7032 [6] Z. Zhang, B. Ping and W. Dong,Oscillatory of unstable type second-order neutral difference equations, Journal of Applied Mathematics and computing 9,No 1(2002), 87–100. [7] Z. Zhou, J. Yu and G. Lei,O scillations for even-order neutral difference equations, Journal of Applied Mathematics and Computing(old:KJCAM) 7, No 3(2000), 601–610.