On a solvable nonlinear difference equation of higher order

Authors

DURHASAN TURGUT TOLLU
YASİN YAZLIK
NECATİ TAŞKARA

DOI

10.3906/mat-1705-33

Abstract

In this paper we consider the following higher-order nonlinear difference equation $$ x_{n}=\alpha x_{n-k}+\frac{\delta x_{n-k}x_{n-\left( k+l\right) }}{\beta x_{n-\left( k+l\right) }+\gamma x_{n-l}},\ n\in \mathbb{N} _{0}, $$ where $k$ and $l$ are fixed natural numbers, and the parameters $\alpha $, $ \beta $, $\gamma $, $\delta $ and the initial values $x_{-i}$, $i=\overline{ 1,k+l}$, are real numbers such that $\beta ^{2}+\gamma ^{2}\neq 0$. We solve the above-mentioned equation in closed form and considerably extend some results in the literature. We also determine the asymptotic behavior of solutions and the forbidden set of the initial values using the obtained formulae for the case $l=1$.

Keywords

Difference equations, solution in closed form, forbidden set, asymptotic behavior

First Page

1765

Last Page

1778

Recommended Citation

TOLLU, DURHASAN TURGUT; YAZLIK, YASİN; and TAŞKARA, NECATİ (2018) "On a solvable nonlinear difference equation of higher order," Turkish Journal of Mathematics: Vol. 42: No. 4, Article 17. https://doi.org/10.3906/mat-1705-33
Available at: https://journals.tubitak.gov.tr/math/vol42/iss4/17