On the Well-Defined Solutions of a Nonlinear Difference Equation with Variable Coefficients

Raafat Abo-Zeid; Mehmet Gümüş; Ana Catarina Carapito; Abdul Qadeer Khan

doi:10.62298/advmath.34

Authors

Raafat Abo-Zeid Department of Basic Science, The Higher Institute for Engineering & Technology, Al-Obour, Cairo, Egypt https://orcid.org/0000-0002-1858-5583
Mehmet Gümüş Department of Mathematics, Faculty of Science, Zonguldak Bülent Ecevit University, 67100, Zonguldak, Türkiye https://orcid.org/0000-0002-7447-479X
Ana Catarina Carapito Center of Mathematics and Applications of University of Beira Interior (CMA-UBI), Department of Mathematics, University of Beira Interior, Portugal https://orcid.org/0000-0002-5868-6266
Abdul Qadeer Khan Department of Mathematics, University of Azad Jammu and Kashmir, Muzaffarabad 13100, Pakistan https://orcid.org/0000-0002-0278-1352

DOI:

https://doi.org/10.62298/advmath.34

Keywords:

Difference equation, Unbounded solution, Periodic solution, Forbidden set

Abstract

Obtaining analytical solutions of nonlinear difference equations is of great importance for a full understanding of the long-term behavior of dynamical systems. Such equations are widely used in mathematical modeling of complex systems, especially population dynamics, the spread of infectious diseases, economic models, and biological processes. Therefore, obtaining analytical solutions of nonlinear difference equations is indispensable not only for theoretical purposes but also for making reliable predictions in applied sciences.

Based on this undeniable fact, in this paper, we study the following non-linear difference equation:

ξₙ₊₁ = (ξₙ ξₙ₋₁) / (aₙ ξₙ − bₙ ξₙ₋₂), n ∈ ℕ₀.

Here, (aₙ)ₙ∈ℕ₀ and (bₙ)ₙ∈ℕ₀ are periodic sequences of positive real numbers with prime period two, and ξ₋₂, ξ₋₁, ξ₀ are real initial values. We introduce the well-defined solutions and study their global behavior.

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On the Well-Defined Solutions of a Nonlinear Difference Equation with Variable Coefficients

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