Solutions of a System of Two Higher-Order Difference Equations in Terms of Lucas Sequence

Year 2019, , 202 - 211, 26.12.2019

Yacine Halim Amira Khelifa Massaoud Berkal

https://doi.org/10.32323/ujma.610399

Cited By: 8

Abstract

In this paper we give some theoretical explanations related to the representation for the general solution of the   system of the  higher-order rational difference equations $$ x_{n+1} = \frac{5 y_{n-k}-5}{y_{n-k}}, \qquad y_{n+1} = \frac{5 x_{n-k}-5}{x_{n-k}} ,\qquad n, k\in \mathbb{N}_0, $$ where  $\mathbb{N}_{0}=\mathbb{N}\cup \left\{0\right\}$,  and the initial conditions $x_{-k}$, $x_{-k+1},\ldots$, $x_{0}$, $y_{-k}$, $y_{-k+1},\ldots$, $y_{0}$ are non zero real numbers such that their solutions are associated to Lucas numbers. We also study the  stability character and asymptotic behavior of this system.

Keywords

General solution, Lucas numbers, stability, system of difference equations

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