- Hacettepe Journal of Mathematics and Statistics
- Vol: 49 Issue: 5
- Solvability of a system of higher order nonlinear difference equations
Solvability of a system of higher order nonlinear difference equations
Authors : Merve Kara, Yasin Yazlik, Durhasan Turgut Tollu
Pages : 1566-1593
Doi:10.15672/hujms.474649
View : 14 | Download : 5
Publication Date : 2020-10-06
Article Type : Research
Abstract :In this paper we show that the system of difference equations \[ x_n= a y_{n-k}+\frac{dy_{n-k}x_{n-( k+l ) }}{b x_{n-(k+l)}+cy_{n-l}}=\alpha x_{n-k}+\frac{\delta x_{n-k}y_{n-(k+l)}}{\beta y_{n-(k+l)}}+\gamma x_{n-l}, \] where $n\in \mathbb{N}_{0},$ $k$ and $l$ are positive integers, the parameters $a$, $b$, $c$, $d$, $\alpha $, $\beta $, $\gamma $, $\delta $ are real numbers and the initial values $x_{-j}$, $y_{-j}$, $j=\overline{1,k+l}$, are real numbers, can be solved in the closed form. We also determine the asymptotic behavior of solutions for the case $l=1$ and describe the forbidden set of the initial values using the obtained formulas. Our obtained results significantly extend and develop some recent results in the literature.Keywords : System of difference equations, Asymptotic behavior, Fibonacci sequence, Forbidden set