On the global behavior and the periodicity of the solutions of a $ k $-dimensional system of difference equations

Mouataz Billah Mesmouli; Nouressadat Touafek; Ioan-Lucian Popa; Abdelkader Moumen; Taher S. Hassan; Mouataz Billah Mesmouli; Nouressadat Touafek; Ioan-Lucian Popa; Abdelkader Moumen; Taher S. Hassan

doi:10.3934/math.2025799

AIMS Mathematics

2025, Volume 10, Issue 8: 17940-17953. doi: 10.3934/math.2025799

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On the global behavior and the periodicity of the solutions of a $ k $-dimensional system of difference equations

1.
Department of Mathematics, College of Science, University of Ha'il, Ha'il 2440, Saudi Arabia; mesmoulimouataz@hotmail.com, mo.abdelkader@uoh.edu.sa, tshassan@mans.edu.eg
2.
LMAM Laboratory, Faculty of Exact Sciences and Informatics, University of Jijel, Jijel 18000, Algeria
3.
Department of Computing, Mathematics and Electronics, 1 Decembrie 1918 University of Alba Iulia, 510009 Alba Iulia, Romania; lucian.popa@uab.ro
4.
Faculty of Mathematics and Computer Science, Transilvania University of Brasov, Iuliu Maniu Street 50, 500091 Brasov, Romania

Received: 03 May 2025 Revised: 03 June 2025 Accepted: 17 June 2025 Published: 07 August 2025
MSC : 39A05, 39A23, 39A30

In this work, we provide a detailed study on the behavior of solutions of a $ k $-dimensional system of rational difference equations, extending some existing results in the literature. We use the linearized stability theory to establish conditions for the local stability of the corresponding unique equilibrium point, and to show its global stability, we prove that the equilibrium point is a global attractor. Also, conditions for the existence of periodic solutions are provided. Our obtained results are confirmed by some examples.
- system of difference equations,
- equilibrium points,
- stability,
- boundedness,
- periodicity
Citation: Mouataz Billah Mesmouli, Nouressadat Touafek, Ioan-Lucian Popa, Abdelkader Moumen, Taher S. Hassan. On the global behavior and the periodicity of the solutions of a $ k $-dimensional system of difference equations[J]. AIMS Mathematics, 2025, 10(8): 17940-17953. doi: 10.3934/math.2025799

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Abstract

In this work, we provide a detailed study on the behavior of solutions of a $ k $-dimensional system of rational difference equations, extending some existing results in the literature. We use the linearized stability theory to establish conditions for the local stability of the corresponding unique equilibrium point, and to show its global stability, we prove that the equilibrium point is a global attractor. Also, conditions for the existence of periodic solutions are provided. Our obtained results are confirmed by some examples.

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