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Dynamics of a higher order nonlinear rational difference equation. (English) Zbl 1071.39017
Authors’ abstract: We study the global attractivity, the invariant intervals, the periodic and oscillatory character of the difference equation $${x_{n+1}=\frac{a+bx_{n}}{Ax_{n}+Bx_{n-k}} ,\quad n=0,1,\dots,}\tag1$$ where $a,b,A,B$ are positive real numbers, $k\geq 1$ is a positive integer, and the initial conditions $x_{-k},\dots,x_{-1},x_{0}$ are nonnegative real numbers such that $x_{-k}$ or $x_{0}$ or both are positive real numbers. We show that the positive equilibrium of the difference equation is a global attractor. As a corollary, our main result confirms a conjecture proposed by Kulenovic et al. (2003) [{\it M. R. S. Kulenovic, G. Ladas, L. F. Martins}, and {\it I. W. Rodrigues}, Comput. Math. Appl. 45, No. 6--9, 1087--1099 (2003; Zbl 1077.39004)].

39A11Stability of difference equations (MSC2000)
39A20Generalized difference equations
39A12Discrete version of topics in analysis
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