A solvable system of difference equations
Commun. Korean Math. Soc. 2020 Vol. 35, No. 1, 301-319
https://doi.org/10.4134/CKMS.c180472
Published online January 31, 2020
Necati Taskara, Durhasan T. Tollu, Nouressadat Touafek, Yasin Yazlik
Selcuk University; Necmettin Erbakan University; Mohamed Seddik Ben Yahia University; Nevsehir Haci Bektas Veli University
Abstract : In this paper, we show that the system of difference equations \begin{equation*} x_{n}=\frac{ay_{n-1}^{p}+b\left( x_{n-2}y_{n-1}\right) ^{p-1}}{ cy_{n-1}+dx_{n-2}^{p-1}},\ y_{n}=\frac{\alpha x_{n-1}^{p}+\beta \left( y_{n-2}x_{n-1}\right) ^{p-1}}{\gamma x_{n-1}+\delta y_{n-2}^{p-1}}, \end{equation*} $n \in \mathbb{N}_{0}$ where the parameters $a,b,c,d,\alpha ,\beta ,\gamma ,\delta,p$ and the initial values $x_{-2}$, $x_{-1}$, $y_{-2}$, $y_{-1}$ are real numbers, can be solved. Also, by using obtained formulas, we study the asymptotic behaviour of well-defined solutions of aforementioned system and describe the forbidden set of the initial values. Our obtained results significantly extend and develop some recent results in the literature.
Keywords : Difference equations, solution in closed-form, forbidden set, asymptotic behaviour
MSC numbers : Primary 39A10, 39A20, 39A23
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