Solvability of a Second-Order Rational System of Difference Equations

Authors : Messaoud Berkal, R Abo-zeid

Pages : 232-242

Doi:10.33401/fujma.1383434

View : 70 | Download : 94

Publication Date : 2023-12-31

Article Type : Research

Abstract :In this paper, we represent the admissible solutions of the system of second-order rational difference equations given below in terms of Lucas and Fibonacci sequences: \\begin{eqnarray*} \\begin{split} x_{n+1}=\\dfrac{L_{m+2}+L_{m+1}y_{n-1}}{L_{m+3}+L_{m+2}y_{n-1}},\\quad y_{n+1}=\\dfrac{L_{m+2}+L_{m+1}z_{n-1}}{L_{m+3}+L_{m+2}z_{n-1}},\\\\ z_{n+1}=\\dfrac{L_{m+2}+L_{m+1}w_{n-1}}{L_{m+3}+L_{m+2}w_{n-1}},\\quad w_{n+1}=\\dfrac{L_{m+2}+L_{m+1}x_{n-1}}{L_{m+3}+L_{m+2}x_{n-1}}. \\end{split} \\end{eqnarray*} where $n\\in\\mathbb{N}_0$, $\\{L_m\\}_{m=-\\infty}^{+\\infty}$ is Lucas sequence and the initial conditions $x_{-1}$, $x_{0}$, $y_{-1}$, $y_{0}$, $z_{-1}$, $z_{0}$, $w_{-1}$, $w_{0}$ are arbitrary real numbers such that $\\displaystyle v_{-i}\\neq-\\frac{L_{m+3}}{L_{m+2}}$, where $v_{-i}=x_{-i},y_{-i},z_{-i},w_{-i}$, $i=0,1$ and $m\\in\\mathbb{Z}$.
Keywords : General solution, Lucas numbers, Fibonacci numbers, system of difference equations.