On a second-order system of difference equations: expressions and behavior of the solutions

Nouressadat Touafek; Jawharah Ghuwayzi AL-Juaid; Nouressadat Touafek; Jawharah Ghuwayzi AL-Juaid

doi:10.3934/math.20251234

AIMS Mathematics

2025, Volume 10, Issue 11: 28077-28099. doi: 10.3934/math.20251234

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On a second-order system of difference equations: expressions and behavior of the solutions

Nouressadat Touafek ^{1
,
,},
Jawharah Ghuwayzi AL-Juaid ²

1.
LMAM Laboratory, Faculty of Exact Sciences and Informatics, University of Jijel, Jijel 18000, Algeria
2.
Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia

Received: 16 September 2025 Revised: 10 November 2025 Accepted: 24 November 2025 Published: 28 November 2025
MSC : 39A05, 39A10, 39A20, 39A30

We explicitly solve the following second-order system of difference equations:
$ \begin{equation*} x_{n+1} = \frac{ax_{n}y_{n-1}}{y_{n}-\beta y_{n-1}-\gamma}+b x_n+c, \, y_{n+1} = \frac{\alpha y_{n}x_{n-1}}{x_{n}-bx_{n-1}-c}+\beta y_n+\gamma, \end{equation*} $
where $ n\in\mathbb{N}_0 $, $ a $, $ \alpha $, and the initial conditions $ x_{-1} $, $ y_{-1} $, $ x_0 $, and $ y_0 $ are nonzero real numbers, while the remaining parameters $ b $, $ c $, $ \beta $, and $ \gamma $ are real numbers. A detailed analysis of the solutions of our system when $ \alpha = a $ with respect to the form, the periodicity and the limiting behavior is presented. To support and illustrate our theoretical results, numerical examples are provided. Our study, considerably generalizes some existing results in the literature.
- systems of difference equations,
- well-defined solutions,
- explicit formulas of solutions,
- periodic solutions,
- limiting behavior of solutions
Citation: Nouressadat Touafek, Jawharah Ghuwayzi AL-Juaid. On a second-order system of difference equations: expressions and behavior of the solutions[J]. AIMS Mathematics, 2025, 10(11): 28077-28099. doi: 10.3934/math.20251234

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Abstract

We explicitly solve the following second-order system of difference equations:

$ \begin{equation*} x_{n+1} = \frac{ax_{n}y_{n-1}}{y_{n}-\beta y_{n-1}-\gamma}+b x_n+c, \, y_{n+1} = \frac{\alpha y_{n}x_{n-1}}{x_{n}-bx_{n-1}-c}+\beta y_n+\gamma, \end{equation*} $

where $ n\in\mathbb{N}_0 $, $ a $, $ \alpha $, and the initial conditions $ x_{-1} $, $ y_{-1} $, $ x_0 $, and $ y_0 $ are nonzero real numbers, while the remaining parameters $ b $, $ c $, $ \beta $, and $ \gamma $ are real numbers. A detailed analysis of the solutions of our system when $ \alpha = a $ with respect to the form, the periodicity and the limiting behavior is presented. To support and illustrate our theoretical results, numerical examples are provided. Our study, considerably generalizes some existing results in the literature.

References

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