Open AccessA detailed study on a solvable system related to the linear fractional difference equation
Author(s) -
Durhasan Turgut Tollu,
İbrahim Yalçınkaya,
Hijaz Ahmad,
Shao-Wen Yao
Publication year - 2021
Publication title -
mathematical biosciences and engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.451
H-Index - 45
eISSN - 1551-0018
pISSN - 1547-1063
DOI - 10.3934/mbe.2021273
Subject(s) - mathematics , order (exchange) , term (time) , class (philosophy) , combinatorics , linear equation , mathematical analysis , physics , quantum mechanics , finance , artificial intelligence , computer science , economics
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.