On a difference equation which is a sum of rational and linear terms

Turki D. Alharbi; Nouressadat Touafek; Turki D. Alharbi; Nouressadat Touafek

doi:10.3934/math.20261221

AIMS Mathematics

2026, Volume 11, Issue 9: 30830-30857. doi: 10.3934/math.20261221

Previous Article Next Article

Research article

On a difference equation which is a sum of rational and linear terms

Turki D. Alharbi ¹,
Nouressadat Touafek ^{2
,
,}

1.
Department of Mathematics, Al-Leith University College, Umm Al-Qura University, Mecca 24382, Saudi Arabia
2.
LMAM Laboratory, Faculty of Exact Sciences and Informatics, University of Jijel, Jijel 18000, Algeria

Received: 04 April 2026 Revised: 24 August 2026 Accepted: 04 September 2026 Published: 21 September 2026
MSC : 39A05, 39A10, 39A20, 39A30

A third order-difference equation defined as sum of rational and linear terms is solved explicitly. The importance of periodic solutions for difference equations lies, for instance, in their ability to model seasonal cycles in biology, recurring fluctuations in economics, and oscillatory behaviors in engineering systems. For this reason, and by relying on the explicit formulas of the solutions of our equation, we propose a rigorous approach to studying the existence of prime periodic solutions. Furthermore, the obtained formulas are used to investigate the asymptotic behavior of the solutions. Finally, several numerical examples are presented to illustrate and support the theoretical results.
- difference equations,
- form of the solutions,
- periodic solutions,
- limiting behavior of solutions
Citation: Turki D. Alharbi, Nouressadat Touafek. On a difference equation which is a sum of rational and linear terms[J]. AIMS Mathematics, 2026, 11(9): 30830-30857. doi: 10.3934/math.20261221

Related Papers:

Abstract

A third order-difference equation defined as sum of rational and linear terms is solved explicitly. The importance of periodic solutions for difference equations lies, for instance, in their ability to model seasonal cycles in biology, recurring fluctuations in economics, and oscillatory behaviors in engineering systems. For this reason, and by relying on the explicit formulas of the solutions of our equation, we propose a rigorous approach to studying the existence of prime periodic solutions. Furthermore, the obtained formulas are used to investigate the asymptotic behavior of the solutions. Finally, several numerical examples are presented to illustrate and support the theoretical results.

References

[1]	R. Abo-Zeid, On the solutions of a higher order difference equation, Georgian Math. J., 27 (2020), 165–175. https://doi.org/10.1515/gmj-2018-0008 doi: 10.1515/gmj-2018-0008
[2]	F. Alsharif, R. Ahmed, I. Alraddadi, M. M. Alsubhi, M. Amer, Md. J. Uddin, A study of stability and bifurcation in a discretized predator-prey model with holling type Ⅲ response and prey refuge via piecewise constant argument method, Complexity, 2025 (2025), 4542190. https://doi.org/10.1155/cplx/4542190 doi: 10.1155/cplx/4542190
[3]	Q. Din, U. Saeed, Stability, discretization, and bifurcation analysis for a chemical reaction system, MATCH-Commun. Math. Co., 90 (2023), 151–174. https://doi.org/10.46793/match.90-1.151D doi: 10.46793/match.90-1.151D
[4]	A. Ghezal, H. J. Al Salman, A. A. Al Ghafli, Three-dimensional second-order rational difference equations: explicit formulas and simulations, Mathematics, 14 (2026), 876. https://doi.org/10.3390/math14050876 doi: 10.3390/math14050876
[5]	E. A. Grove, G. Ladas, Periodicities in nonlinear difference equations, New York: Chapman and Hall/CRC, 2005. https://doi.org/10.1201/9781420037722
[6]	M. Gumus, R. Abo-Zeid, An explicit formula and forbidden set for a higher order difference equation, J. Appl. Math. Comput., 63 (2020), 133–142. https://doi.org/10.1007/s12190-019-01311-9 doi: 10.1007/s12190-019-01311-9
[7]	N. Haddad, N. Touafek, J. F. T. Rabago, Well-defined solutions of a system of difference equations, J. Appl. Math. Comput., 56 (2018), 439–458. https://doi.org/10.1007/s12190-017-1081-8 doi: 10.1007/s12190-017-1081-8
[8]	Y. Halim, N. Touafek, E. M. Elsayed, Closed form solutions of some systems of rational difference equations in terms of Fibonacci numbers, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 21 (2014), 473–486.
[9]	M. Kara, Y. Yazlik, On the solutions of three-dimensional system of difference equations via recursive relations of order two and applications, J. Appl. Anal. Comput., 12 (2022), 736–753. https://doi.org/10.11948/20210305 doi: 10.11948/20210305
[10]	M. Kara, N. Touafek, Y. Yazlik, Well-defined solutions of a three-dimensional system of difference equations, Gazi U. J. Sci., 33 (2020), 767–778.
[11]	R. R. A. Khan, A. Q. Khan, T. D. Alharbi, J. G. AL-Juaid, Codimension-two bifurcation analysis and global dynamics of a discrete epidemic model, Axioms, 14 (2025), 463. https://doi.org/10.3390/axioms14060463 doi: 10.3390/axioms14060463
[12]	S. Li, Y. H. Long, On homoclinic solutions for a class of fourth-order $\phi_c$-Laplacian partial difference equations, J. Differ. Equ. Appl., 32 (2026), 482–508. https://doi.org/10.1080/10236198.2025.2606415 doi: 10.1080/10236198.2025.2606415
[13]	T. Y. Liu, H. X. Zhang, S. Wang, A new high-order compact CN-ADI scheme on graded meshes for three-dimensional nonlinear PIDEs with multiple weakly singular kernels, Appl. Math. Lett., 171 (2025), 109697. https://doi.org/10.1016/j.aml.2025.109697 doi: 10.1016/j.aml.2025.109697
[14]	Y. H. Long, Infinitely many homoclinic solutions for nonlinear p-Laplacian partial difference equations without Ambrosetti-Rabinowitz condition, Demonstr. Math., 59 (2026), 20250261. https://doi.org/10.1515/dema-2025-0261 doi: 10.1515/dema-2025-0261
[15]	Y. H. Long, X. F. Pang, Q. Q. Zhang, Codimension-one and codimension-two bifurcations of a discrete Leslie-Gower type predator-prey model, Discrete Cont. Dyn.-B, 30 (2025), 1357–1389. https://doi.org/10.3934/dcdsb.2024132 doi: 10.3934/dcdsb.2024132
[16]	J. Reunsumrit, N. Patanarapeelert, T. Sitthiwirattham, On a coupled system of fractional symmetric Hahn difference equations with nonlocal fractional symmetric Hahn integral boundary value conditions, J. Nonlinear Funct. Anal., 2025 (2025), 31. https://doi.org/10.23952/jnfa.2025.31 doi: 10.23952/jnfa.2025.31
[17]	S. Stevic, Solvability of a product-type system of difference equations with six parameters, Adv. Nonlinear Anal., 8 (2019), 29–51. https://doi.org/10.1515/anona-2016-0145 doi: 10.1515/anona-2016-0145
[18]	S. Stevic, Third-order product-type systems of difference equations solvable in closed form, Electronic Journal of Diﬀerential Equations, 2016 (2016), 285.
[19]	N. Touafek, J. G. AL-Juaid. On a second-order system of difference equations: expressions and behavior of the solutions, AIMS Math., 10 (2025), 28077–28099. https://doi.org/10.3934/math.20251234 doi: 10.3934/math.20251234
[20]	X. H. Yang, Y. Shi, H. X. Zhang, Z. M. Zhang, Analysis of a new time-space two-grid method for the two-dimensional nonlinear nonlocal mobile/immobile transport model, J. Sci. Comput., 107 (2026), 96. https://doi.org/10.1007/s10915-026-03314-8 doi: 10.1007/s10915-026-03314-8
[21]	H. Zabat, N. Touafek, I. Dekkar, Formulas of the solutions of a non-autonomous difference equation and two systems of difference equations, Journal of Innovative Applied Mathematics and Computational Sciences, 4 (2024), 188–200.

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)