On solutions of a two-dimensional (m + 1)-order system of difference equations via Pell numbers

Ahmed Ghezal; Khalil Zerari; Imane Zemmouri

doi:10.5269/bspm.67451

Ahmed Ghezal
Khalil Zerari
Imane Zemmouri Department of Mathematics, University of Annaba, Elhadjar 23

Abstract

In this paper, we are interested in the closed-form solution of the following two-dimensional system of difference equations of $\left( m+1\right) -$order,% \[ x_{n+1}^{\left( i\right) }=\frac{1}{14+x_{n-m}^{\left( i+1\right) \operatorname{mod}2}},i\in\left\{ 1,2\right\} ,n,m\in\mathbb{N}_{0}, \] and the initial values $x_{-j}^{\left( i\right) },$ $i\in\left\{1,2\right\} ,$ $j\in\left\{ 0,1,...,m\right\} $ are real numbers do not equal $-1/14$. We show that the solutions of this system are associated with Pell numbers and some other numbers. It is shown that the global stability of positive solutions of this system holds. Our results are illustrated via numerical examples.

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