On a general homogeneous three-dimensional system of difference equations

Nouressadat Touafek; Durhasan Turgut Tollu; Youssouf Akrour; Nouressadat Touafek; Durhasan Turgut Tollu; Youssouf Akrour

doi:10.3934/era.2021017

Electronic Research Archive

2021, Volume 29, Issue 5: 2841-2876. doi: 10.3934/era.2021017

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On a general homogeneous three-dimensional system of difference equations

1.
LMAM Laboratory, Department of Mathematics, Mohamed Seddik Ben Yahia University, Jijel, Algeria
2.
Department of Mathematics and Computer Sciences, Necmettin Erbakan University, Konya, Turkey
3.
ENS Assia Djebar Constantine and LMAM Laboratory, Mohamed Seddik Ben Yahia University, Jijel, Algeria

Received: 01 October 2020 Revised: 01 January 2021 Published: 15 March 2021
Primary: 39A05, 39A10, 39A21, 39A23, 39A30

In this work, we study the behavior of the solutions of following three-dimensional system of difference equations
$ \begin{equation*} x_{n+1} = f(y_{n}, y_{n-1}), \, y_{n+1} = g(z_{n}, z_{n-1}), \, z_{n+1} = h(x_{n}, x_{n-1}) \end{equation*} $
where $ n\in \mathbb{N}_{0} $, the initial values $ x_{-1} $, $ x_{0} $, $ y_{-1} $, $ y_{0} $ $ z_{-1} $, $ z_{0} $ are positive real numbers, the functions $ f, \, g, \, h:\, \left(0, +\infty\right)^{2}\rightarrow\left(0, +\infty\right) $ are continuous and homogeneous of degree zero. By proving some general convergence theorems, we have established conditions for the global stability of the corresponding unique equilibrium point. We give necessary and sufficient conditions on existence of prime period two solutions of the above mentioned system. Also, we prove a result on oscillatory solutions. As applications of the obtained results, some particular systems of difference equations defined by homogeneous functions of degree zero are investigated. Our results generalize some existing ones in the literature.
- Global and local stability,
- homogeneous functions,
- oscillation,
- periodicity,
- systems of difference equations
Citation: Nouressadat Touafek, Durhasan Turgut Tollu, Youssouf Akrour. On a general homogeneous three-dimensional system of difference equations[J]. Electronic Research Archive, 2021, 29(5): 2841-2876. doi: 10.3934/era.2021017

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Abstract

In this work, we study the behavior of the solutions of following three-dimensional system of difference equations

$ \begin{equation*} x_{n+1} = f(y_{n}, y_{n-1}), \, y_{n+1} = g(z_{n}, z_{n-1}), \, z_{n+1} = h(x_{n}, x_{n-1}) \end{equation*} $

where $ n\in \mathbb{N}_{0} $, the initial values $ x_{-1} $, $ x_{0} $, $ y_{-1} $, $ y_{0} $ $ z_{-1} $, $ z_{0} $ are positive real numbers, the functions $ f, \, g, \, h:\, \left(0, +\infty\right)^{2}\rightarrow\left(0, +\infty\right) $ are continuous and homogeneous of degree zero. By proving some general convergence theorems, we have established conditions for the global stability of the corresponding unique equilibrium point. We give necessary and sufficient conditions on existence of prime period two solutions of the above mentioned system. Also, we prove a result on oscillatory solutions. As applications of the obtained results, some particular systems of difference equations defined by homogeneous functions of degree zero are investigated. Our results generalize some existing ones in the literature.

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