Solvability of a system of higher order nonlinear difference equations

Year 2020, Volume: 49 Issue: 5, 1566 - 1593, 06.10.2020

Merve Kara Yasin Yazlik Durhasan Turgut Tollu

https://doi.org/10.15672/hujms.474649

Cited By: 15

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

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