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

Authors : Yacine HALİM, Amira KHELİFA, Massaoud BERKAL

Pages : 202-211

Doi:10.32323/ujma.610399

View : 17 | Download : 7

Publication Date : 2019-12-26

Article Type : Research Paper

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