Formulas of the solutions of a non-autonomous difference equation and two systems of difference equations - Zabat, Hiba; Touafek, Nouressadat; Dekkar, Imane (Journal of Innovative Applied Mathematics and Computational Sciences, 2024)
In this work, we explicitly solve the following: * A higher-order non-autonomous difference equation: \begin{equation*} x_{n+1} = α_{n} x_{n-k} + \frac{β_{n}}{x_{n} x_{n-1} … x_{n-k+1}}, \end{equation*} where $n \in \mathbb{N}_{0}$, $k \in \mathbb{N}$, the sequences $\left(α_{n}\right)_{n \in \mathbb{N}_{0}}$ and $\left(β_{n}\right)_{n \in \mathbb{N}_{0}}$ are real, and the initial values $x_{-k}, x_{-k+1}, …, x_{0}$ 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}}, \quad y_{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}}, \quad y_{n+1} = \frac{a_{2} x_{n} z_{n}}{a x_{n} + b y_{n} + c z_{n}}, \quad z_{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.}