On a two-dimensional nonlinear system of difference equations close to the bilinear system

Stevo Stević; Durhasan Turgut Tollu; Stevo Stević; Durhasan Turgut Tollu

doi:10.3934/math.20231048

AIMS Mathematics

2023, Volume 8, Issue 9: 20561-20575. doi: 10.3934/math.20231048

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On a two-dimensional nonlinear system of difference equations close to the bilinear system

Stevo Stević ^{1,2
,
,},
Durhasan Turgut Tollu ³

1.
Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/Ⅲ, Beograd 11000, Serbia
2.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
3.
Necmettin Erbakan University, Faculty of Science, Department of Mathematics and Computer Sciences, Konya, Turkey

Received: 27 April 2023 Revised: 06 June 2023 Accepted: 09 June 2023 Published: 26 June 2023
MSC : 39A20

We consider the two-dimensional nonlinear system of difference equations

$ x_n = x_{n-k}\frac{ay_{n-l}+by_{n-(k+l)}}{cy_{n-l}+dy_{n-(k+l)}},\quad y_n = y_{n-k}\frac{{\alpha} x_{n-l}+{\beta} x_{n-(k+l)}}{{\gamma} x_{n-l}+{\delta} x_{n-(k+l)}}, $

for $ n\in{\mathbb N}_0, $ where the delays $ k $ and $ l $ are two natural numbers, and the initial values $ x_{-j}, y_{-j} $, $ 1\le j\le k+l $, and the parameters $ a, b, c, d, {\alpha}, {\beta}, {\gamma}, {\delta} $ are real numbers. We show that the system of difference equations is solvable by presenting a method for finding its general solution in detail. Bearing in mind that the system of equations is a natural generalization of the corresponding one-dimensional difference equation, whose special cases appear in the literature from time to time, our main result presented here also generalizes many results therein.
- nonlinear system of difference equations,
- solvable system,
- closed-form formula
Citation: Stevo Stević, Durhasan Turgut Tollu. On a two-dimensional nonlinear system of difference equations close to the bilinear system[J]. AIMS Mathematics, 2023, 8(9): 20561-20575. doi: 10.3934/math.20231048

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Abstract

We consider the two-dimensional nonlinear system of difference equations

$ x_n = x_{n-k}\frac{ay_{n-l}+by_{n-(k+l)}}{cy_{n-l}+dy_{n-(k+l)}},\quad y_n = y_{n-k}\frac{{\alpha} x_{n-l}+{\beta} x_{n-(k+l)}}{{\gamma} x_{n-l}+{\delta} x_{n-(k+l)}}, $

for $ n\in{\mathbb N}_0, $ where the delays $ k $ and $ l $ are two natural numbers, and the initial values $ x_{-j}, y_{-j} $, $ 1\le j\le k+l $, and the parameters $ a, b, c, d, {\alpha}, {\beta}, {\gamma}, {\delta} $ are real numbers. We show that the system of difference equations is solvable by presenting a method for finding its general solution in detail. Bearing in mind that the system of equations is a natural generalization of the corresponding one-dimensional difference equation, whose special cases appear in the literature from time to time, our main result presented here also generalizes many results therein.

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