MMN-4340
Solvability of a third-order system of nonlinear difference equations via a generalized Fibonacci sequence
Hamida Hamioud; Imane Dekkar; Nouressadat Touafek;Abstract
In this paper, we solve in closed-form the following third-order system of nonlinear difference equations
\begin{align*}
x_{n+1}=\dfrac{y_ny_{n-1}x^p_{n-1}}{x_{n}(a_n y^q_{n-2}+b_n y_n y_{n-1})},\quad
y_{n+1}=\dfrac{x_nx_{n-1}y^q_{n-1}}{y_{n}(c_n x^p_{n-2}+d_n x_n x_{n-1})}, \quad
p, q \in \mathbb{N}, n \in \mathbb{N}_0
\end{align*}
where the initial values $x_{-i}, y_{-i}, i = 0, 1,2\,$ and the parameters $(a_n)_{n\in\mathbb{N}_0}, (b_n)_{n\in\mathbb{N}_0}, (c_n)_{n\in\mathbb{N}_0},$ $ (d_n)_{n\in\mathbb{N}_0}$ are non-zero real numbers. The form of the solutions of the one dimensional case of our system and a more general System defined by one to one functions are also presented.
Vol. 25 (2024), No. 1, pp. 271-285