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Dynamics of a non-linear difference equation. (English) Zbl 1106.39011
The authors consider the dynamics of the difference equation \[ y_{n+1}=\frac{y_n + p y_{n-k}}{y_n+q},\quad n=0,1,2\dots \] where the initial conditions \(y_{-k},\dots,y_{-1}, y_0\) are non-negative, \(k\in \mathbb{N}\), and the parameters \(p\) and \(q\) are non-negative. They study several characteristics of the equation and obtain results about periodicity, oscillatory character, invariant interval and global stability. Firstly, they investigate the periodic solutions and obtain the invariant intervals. Next, they discuss the oscillatory characters and analysis of semicycles. Finally, they study the global asymptotic stability and show that, when \(p>q+1\), the equation has unbounded solutions.
The results obtained solve an open problem from the monograph by M. R. S. Kulenovic and G. Ladas [Dynamics of second-order rational difference equations with open problems and conjectures. Boca Raton, FL: Chapman & Hall/CRC. (2002; Zbl 0981.39011)].

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