Solvability of a system of higher order nonlinear difference equations

Authors : Merve KARA, Yasin YAZLİK, Durhasan Turgut TOLLU

Pages : 1566-1593

Doi:10.15672/hujms.474649

View : 24 | Download : 10

Publication Date : 2020-10-06

Article Type : Research Paper

Abstract :In this paper we show that the system of difference equations \[ x_n= a y_{n-k}+\frac{dy_{n-k}x_{n-insert ignore into journalissuearticles values( k+l ); }}{b x_{n-insert ignore into journalissuearticles values(k+l);}+cy_{n-l}}=\alpha x_{n-k}+\frac{\delta x_{n-k}y_{n-insert ignore into journalissuearticles values(k+l);}}{\beta y_{n-insert ignore into journalissuearticles values(k+l);}}+\gamma x_{n-l}, \]   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