- Universal Journal of Mathematics and Applications
- Vol: 2 Issue: 4
- Solutions of a System of Two Higher-Order Difference Equations in Terms of Lucas Sequence
Solutions of a System of Two Higher-Order Difference Equations in Terms of Lucas Sequence
Authors : Yacine Halim, Amira Khelifa, Massaoud Berkal
Pages : 202-211
Doi:10.32323/ujma.610399
View : 9 | Download : 5
Publication Date : 2019-12-26
Article Type : Research
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