A detailed study on a solvable system related to the linear fractional difference equation

Durhasan Turgut Tollu; İbrahim Yalçınkaya; Hijaz Ahmad; Shao-Wen Yao; Durhasan Turgut Tollu; İbrahim Yalçınkaya; Hijaz Ahmad; Shao-Wen Yao

doi:10.3934/mbe.2021273

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 5: 5392-5408. doi: 10.3934/mbe.2021273

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A detailed study on a solvable system related to the linear fractional difference equation

1.
Department of Mathematics and Computer Sciences, Necmettin Erbakan University, Konya, Turkey
2.
Department of Basic Sciences, University of Engineering and Technology, Peshawar, Pakistan
3.
School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, China

Academic Editor: Juan Luis García Guirao

Received: 15 March 2021 Accepted: 18 May 2021 Published: 17 June 2021

In this paper, we present a detailed study of the following system of difference equations

$ \begin{equation*} x_{n+1} = \frac{a}{1+y_{n}x_{n-1}}, \ y_{n+1} = \frac{b}{1+x_{n}y_{n-1}}, \ n\in\mathbb{N}_{0}, \end{equation*} $

where the parameters $ a $, $ b $, and the initial values $ x_{-1}, \; x_{0}, \ y_{-1}, \; y_{0} $ are arbitrary real numbers such that $ x_{n} $ and $ y_{n} $ are defined. We mainly show by using a practical method that the general solution of the above system can be represented by characteristic zeros of the associated third-order linear equation. Also, we characterized the well-defined solutions of the system. Finally, we study long-term behavior of the well-defined solutions by using the obtained representation forms.
- behavior of solutions,
- characteristic equation,
- general solution,
- system of difference equations,
- periodic solution
Citation: Durhasan Turgut Tollu, İbrahim Yalçınkaya, Hijaz Ahmad, Shao-Wen Yao. A detailed study on a solvable system related to the linear fractional difference equation[J]. Mathematical Biosciences and Engineering, 2021, 18(5): 5392-5408. doi: 10.3934/mbe.2021273

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Abstract

In this paper, we present a detailed study of the following system of difference equations

$ \begin{equation*} x_{n+1} = \frac{a}{1+y_{n}x_{n-1}}, \ y_{n+1} = \frac{b}{1+x_{n}y_{n-1}}, \ n\in\mathbb{N}_{0}, \end{equation*} $

where the parameters $ a $, $ b $, and the initial values $ x_{-1}, \; x_{0}, \ y_{-1}, \; y_{0} $ are arbitrary real numbers such that $ x_{n} $ and $ y_{n} $ are defined. We mainly show by using a practical method that the general solution of the above system can be represented by characteristic zeros of the associated third-order linear equation. Also, we characterized the well-defined solutions of the system. Finally, we study long-term behavior of the well-defined solutions by using the obtained representation forms.

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