On solutions of three-dimensional system of difference equations with constant coefficients

Year 2023, Volume: 72 Issue: 2, 462 - 481, 23.06.2023

Merve Kara , Ömer Aktaş

https://doi.org/10.31801/cfsuasmas.1163955

Cited By: 1

Abstract

In this study, we show that the system of difference equations
\begin{align}
x_{n}=\frac{x_{n-2}y_{n-3}}{y_{n-1}\left(a+bx_{n-2}y_{n-3} \right) }, \nonumber \\
y_{n}=\frac{y_{n-2}z_{n-3}}{z_{n-1}\left(c+dy_{n-2}z_{n-3} \right) },~n\in\mathbb{N}_{0}, ~ \nonumber \\
z_{n}=\frac{z_{n-2}x_{n-3}}{x_{n-1}\left(e+fz_{n-2}x_{n-3} \right) }, \nonumber \\
\end{align}
where the initial values $x_{-i}, y_{-i}, z_{-i}$, $i=\overline{1,3}$ and the parameters $a$, $b$, $c$, $d$, $e$, $f$ are non-zero real numbers, can be solved in closed form. Moreover, we obtain the solutions of above system in explicit form according to the parameters $a$, $c$ and $e$ are equal $1$ or not equal $1$. In addition, we get periodic solutions of aforementioned system. Finally, we define the forbidden set of the initial conditions by using the acquired formulas.

Keywords

Explicit form, forbidden set, periodicity

Supporting Institution

Karamanoglu Mehmetbey University

Project Number

13-YL-22

Thanks

This paper was presented in 4th International Conference on Pure and Applied Mathematics (ICPAM - VAN 2022), Van-Turkey, June 22-23, 2022. This work is supported by the Scientific Research Project Fund of Karamanoglu Mehmetbey University under the project number 13-YL-22.

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