arXiv subjects

Nouressadat Touafek

Publications and source records attributed to Nouressadat Touafek.

3 recordsLinked to original sources

arXiv2019

On a system of difference equations of third order solved in closed form

In this note we show the the system of difference equations $$ x_{n+1}=\dfrac{ay_{n-2}x_{n-1}y_n+bx_{n-1}y_{n-2}+cy_{n-2}+d}{y_{n-2}x_{n-1}y_n},$$ $$y_{n+1}=\dfrac{ax_{n-2}y_{n-1}x_n+by_{n-1}x_{n-2}+cx_{n-2}+d}{x_{n-2}y_{n-1}x_n},$$ where $n\in \mathbb{N}_{0}$, the initial values $x_{-2}$, $x_{-1}$, $x_0$, $y_{-2}$, $y_{-1}$ and $y_0$ are arbitrary nonzero real numbers and the parameters $a$, $b$, $c$ and $d$ are arbitrary real numbers with $d\ne 0$, can be solved in a closed form. We will see that when $a=b=c=d=1$ the solutions are expressed using the famous Teteranacci numbers. In particular, the results obtained here extend those in our work \cite{arxiv}.

math.DS↗

arXiv2019

On a system of difference equations of second order solved in a closed from

In this work we solve in closed form the system of difference equations \begin{equation*} x_{n+1}=\dfrac{ay_nx_{n-1}+bx_{n-1}+c}{y_nx_{n-1}},\; y_{n+1}=\dfrac{ax_ny_{n-1}+by_{n-1}+c}{x_ny_{n-1}},\;n=0,1,..., \end{equation*} where the initial values $x_{-1}$, $x_0$, $y_{-1}$ and $y_0$ are arbitrary nonzero real numbers and the parameters $a$, $b$ and $c$ are arbitrary real numbers with $c\ne 0$. In particular we represent the solutions of some particular cases of this system in terms of Tribonacci and Padovan numbers and we prove the global stability of the corresponding positive equilibrium points. The result obtained here extend those obtained in some recent papers.

math.DS↗

arXiv2016

Solution Form of a Higher Order System of Difference Equation and Dynamical Behavior of Its Special Case

The solution form of the system of nonlinear difference equations \begin{equation*} x_{n+1} = \frac{x_{n-k+1}^{p}y_{n}}{a y_{n-k}^{p}+b y_{n}},\ y_{n+1} = \frac{y_{n-k+1}^{p}x_{n}}{αx_{n-k}^{p}+βx_{n}}, \quad n, p \in \mathbb{N}_{0},\ k\in \mathbb{N}, \end{equation*} where the coefficients $a, b, α, β$ and the initial values $x_{-i},y_{-i},i\in\{0,1,\ldots,k\}$ are real numbers, is obtained. Furthermore, the behavior of solutions of the above system when $p=1$ is examined. Numerical examples are presented to illustrate the results exhibited in the paper.

math.DS↗