Abstract
It is shown that the following $\left( k+l\right) $-order nonlinear difference equation $$x_{n}=\frac{x_{n-k}x_{n-k-l}}{x_{n-l}\left( a_{n}+b_{n}x_{n-k}x_{n-k-l}\right)}, \ n\in \mathbb{N}_{0},$$ where $k,l\in \mathbb{N}$, $\left(a_{n} \right)_{n\in \mathbb{N}_{0}}$, $\left(b_{n} \right)_{n\in \mathbb{N}_{0}}$ and the initial values $x_{-i}$, $i=\overline {1,k+l}$, are real numbers, can be solved and extended some results in literature. Also, by using obtained formulas, we give the forbidden set of the initial values for aforementioned equation and study the asymptotic behavior of well-defined solutions of above difference equation for the case $k=3$, $l=k$.
Acknowledgment
This study is a part of the first authors Ph.D. Thesis.
Citation
Merve Kara. Yasin Yazlik. "Solvability of a $\left( k+l\right)$-order nonlinear difference equation." Tbilisi Math. J. 14 (2) 271 - 297, June 2021. https://doi.org/10.32513/tmj/19322008138
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