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Global behavior of a higher order nonlinear difference equation. (English) Zbl 1066.39008
The authors consider the difference equation $$ x_{n+1} = f(x_n,x_{n-k}) $$ for $k>1$ fixed. The function $f(u,v)$ is continuous with respect to both arguments on $(0,\infty)$, decreasing in $u$ and increasing in $v$, with $f(\bar{x},v)/v$ non-increasing in $v$, where $\bar{x}$ is the unique positive equilibrium of the equation. Two kinds of global solutions are considered: oscillatory solutions and semi-cycles. Several results on existence of various global solutions are given, incorporating previous results concerning global behavior of equations that are special cases of the above.

39A11Stability of difference equations (MSC2000)
Full Text: DOI
[1] Ladas, G.: Open problems and conjectures. J. differential equations appl. 1, 317-321 (1995) · Zbl 0860.39018
[2] Ladas, G.: Progress report on xn+1=($\alpha +\beta xn+\gamma $xn - 1)/(A+Bxn+Cxn - 1). J. differential equations appl. 5, 211-215 (1995)
[3] Gibbons, C.; Kulenovic, M. R. S.; Ladas, G.: On the recursive sequence xn+1=($\alpha +\beta $xn - 1)/$(\gamma +xn)$. Math. sci. Res. hot-line 4, 1-11 (2000) · Zbl 1039.39004
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[8] Kocic, V. L.; Ladas, G.: Global behavior of nonlinear difference equations of higher order with applications. (1993) · Zbl 0787.39001
[9] Devault, R.; Kosmala, W.; Ladas, G.; Schultz, S. W.: Global behavior of yn+1=(p+yn - k)/(qyn+yn - k). Nonlinear anal. 47, 4743-4751 (2001) · Zbl 1042.39523