Yacine HALİM, Amira KHELİFA, Massaoud BERKAL

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

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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