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On the global character of the difference equation \(x_{n+1}=\frac{\alpha+\gamma x_{n-(2k+1)}+\delta x_{n-2l}}{A+x_{n-2l}}\). (English) Zbl 1038.39004

The paper is devoted to investigate the behavior of the solutions to the higher order difference equation \[ x_{n+1}=\frac{\alpha+\gamma x_{n-(2k+1)}+\delta x_{n-2l}}{A+x_{n-2l}},\qquad n=0,1,\dots\tag{1} \] where \(k,l\) are nonnegative integers, \(\alpha,\gamma,\delta, A\geq 0\), \(\alpha+\gamma+\delta>0\), and the initial conditions are nonnegative real numbers such that \(A+x_{n-2l}>0\) for all \(n\geq 0\).
The main result states a trichotomy character in the behavior of the solutions, namely, every solution of (1) has a finite limit if \(\gamma<\delta +A\); if \(\gamma>\delta +A\), then there exist unbounded solutions, and, finally, every solution converges to a positive periodic solution when \(\gamma=\delta +A\).
The proof is made by the consideration of different particular cases of (1) depending on the involved parameters.
Reviewer: Eduardo Liz (Vigo)

MSC:

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