arXiv subjects

Yacine Halim

Publications and source records attributed to Yacine Halim.

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

On the solutions of a second-order difference equations in terms of generalized Padovan sequences

This paper deals with the solution, stability character and asymptotic behavior of the rational difference equation \begin{equation*} x_{n+1}=\frac{αx_{n-1}+β}{ γx_{n}x_{n-1}},\qquad n \in \mathbb{N}_{0}, \end{equation*} where $\mathbb{N}_{0}=\mathbb{N}\cup \left\{0\right\}$, $α,β,γ\in\mathbb{R}^{+}$, and the initial conditions $x_{-1}$ and $x_{0}$ are non zero real numbers such that their solutions are associated to generalized Padovan numbers. Also, we investigate the two-dimensional case of the this equation given by \begin{equation*} x_{n+1} = \frac{αx_{n-1} + β}{γy_n x_{n-1}}, \qquad y_{n+1} = \frac{αy_{n-1} +β}{γx_n y_{n-1}} ,\qquad n\in \mathbb{N}_0, \end{equation*} and this generalizes the results presented in \cite{yazlik}

math.DS↗