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Volume 14
Issue 6
10.3390/axioms14060450
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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

Note on Iterations of Nonlinear Rational Functions

by
Michal Fečkan
1,2,*,
Amira Khelifa
3,
Yacine Halim
4,5 and
Ibraheem M. Alsulami
6
1
Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, 842 48 Bratislava, Slovakia
2
Mathematical Institute, Slovak Academy of Sciences, 814 73 Bratislava, Slovakia
3
Department of Mathematics, Mohamed Seddik Ben Yahia University, BP 78 Oueld Aissa, Jijel 18000, Algeria
4
Department of Mathematics, Abdelhafid Boussouf University Center, R.P 26, Mila 43000, Algeria
5
LMAM Laboratory, Mohamed Seddik Ben Yahia University, BP 78 Oueld Aissa, Jijel 18000, Algeria
6
Mathematics Department, Faculty of Science, Umm Al-Qura University, Makkah 21955, Saudi Arabia
*
Author to whom correspondence should be addressed.
Axioms 2025, 14(6), 450; https://doi.org/10.3390/axioms14060450 (registering DOI)
Submission received: 9 May 2025 / Revised: 4 June 2025 / Accepted: 5 June 2025 / Published: 7 June 2025
(This article belongs to the Section Mathematical Analysis)
Versions Notes

Abstract

This paper investigates a class of nonlinear rational difference equations with delayed terms, which often arise in various mathematical models. We analyze the iterative behavior of these rational functions and show how their iterations can be represented through second-order linear recurrence relations. By establishing a connection with generalized Balancing sequences, we derive explicit formulas that describe the system’s asymptotic behavior. Our main contribution is proving the existence of a unique globally asymptotically stable equilibrium point for all trajectories, regardless of initial conditions. We also provide analytical expressions for the solutions and support our findings with numerical examples. These results offer valuable insights into the dynamics of nonlinear rational systems and form a theoretical basis for further exploration in this area.
Keywords: iterations of nonlinear functions; system of difference equations; generalized Balancing sequences; stability iterations of nonlinear functions; system of difference equations; generalized Balancing sequences; stability

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MDPI and ACS Style

Fečkan, M.; Khelifa, A.; Halim, Y.; Alsulami, I.M. Note on Iterations of Nonlinear Rational Functions. Axioms 2025, 14, 450. https://doi.org/10.3390/axioms14060450

AMA Style

Fečkan M, Khelifa A, Halim Y, Alsulami IM. Note on Iterations of Nonlinear Rational Functions. Axioms. 2025; 14(6):450. https://doi.org/10.3390/axioms14060450

Chicago/Turabian Style

Fečkan, Michal, Amira Khelifa, Yacine Halim, and Ibraheem M. Alsulami. 2025. "Note on Iterations of Nonlinear Rational Functions" Axioms 14, no. 6: 450. https://doi.org/10.3390/axioms14060450

APA Style

Fečkan, M., Khelifa, A., Halim, Y., & Alsulami, I. M. (2025). Note on Iterations of Nonlinear Rational Functions. Axioms, 14(6), 450. https://doi.org/10.3390/axioms14060450

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