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Formulas of the solutions of a non-autonomous difference equation and two systems of difference equations

Year: 2024
Type: article
Abstract: In this work, we explicitly solve the following:* A higher-order non-autonomous difference equation:\begin{equation*}x_{n+1} = \alpha_{n} x_{n-k} + \frac{\beta_{n}}{x_{n} x_{n-1} \cdots x_{n-k+1}},\end{equation*}where $n \in \mathbb{N}_{0}$, $k \in \mathbb{N}$, the sequences $\left(\alpha_{n}\right)_{n \in \mathbb{N}_{0}}$ and $\left(\beta_{n}\right)_{n \in \mathbb{N}_{0}}$ are real, and the initial values $x_{-k}, x_{-k+1}, \ldots, x_{0}$ are nonzero real numbers.* A three-dimensional system of second-order difference equations:\begin{equation*}x_{n+1} = \frac{a_{1} y_{n-1} z_{n-1}}{a x_{n-1} + b y_{n-1} + c z_{n-1}}, \quady_{n+1} = \frac{a_{2} x_{n-1} z_{n-1}}{a x_{n-1} + b y_{n-1} + c z_{n-1}},\end{equation*}\begin{equation*}z_{n+1} = \frac{a_{3} x_{n-1} y_{n-1}}{a x_{n-1} + b y_{n-1} + c z_{n-1}},\end{equation*}where $n \in \mathbb{N}_{0}$, the parameters $a, b, c, a_{1}, a_{2}, a_{3}$ are real numbers, and the initial values $x_{-1}, x_0, y_{-1}, y_0, z_{-1}, z_0$ are nonzero real numbers.* A three-dimensional system of first-order difference equations:\begin{equation*}x_{n+1} = \frac{a_{1} y_{n} z_{n}}{a x_{n} + b y_{n} + c z_{n}}, \quady_{n+1} = \frac{a_{2} x_{n} z_{n}}{a x_{n} + b y_{n} + c z_{n}}, \quadz_{n+1} = \frac{a_{3} x_{n} y_{n}}{a x_{n} + b y_{n} + c z_{n}},\end{equation*}where $n \in \mathbb{N}_{0}$, the parameters $a, b, c, a_{1}, a_{2}, a_{3}$ are real numbers, and the initial values $x_0, y_0, z_0$ are nonzero real numbers.}
DOI: 10.58205/jiamcs.v4i2.1884
Institution: University of Jijel
Language: English
Open Access status: diamond
OpenAlex ID: W6936573064
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