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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 \[ {x_{n+1}=\frac{a+bx_{n}}{Ax_{n}+Bx_{n-k}} ,\quad n=0,1,\dots,}\tag{1} \] 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) [M. R. S. Kulenovic, G. Ladas, L. F. Martins, and I. W. Rodrigues, Comput. Math. Appl. 45, No. 6–9, 1087–1099 (2003; Zbl 1077.39004)].

MSC:
39A11 Stability of difference equations (MSC2000)
39A20 Multiplicative and other generalized difference equations
39A12 Discrete version of topics in analysis
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