z-logo
open-access-imgOpen Access
On a general homogeneous three-dimensional system of difference equations
Author(s) -
Nouressadat Touafek,
Durhasan Turgut Tollu,
Youssouf Akrour
Publication year - 2021
Publication title -
electronic research archive
Language(s) - English
Resource type - Journals
ISSN - 2688-1594
DOI - 10.3934/era.2021017
Subject(s) - mathematics , homogeneous , arithmetic , combinatorics
In this work, we study the behavior of the solutions of following three-dimensional system of difference equations\begin{document}$ \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*} $\end{document}where \begin{document}$ n\in \mathbb{N}_{0} $\end{document} , the initial values \begin{document}$ x_{-1} $\end{document} , \begin{document}$ x_{0} $\end{document} , \begin{document}$ y_{-1} $\end{document} , \begin{document}$ y_{0} $\end{document}\begin{document}$ z_{-1} $\end{document} , \begin{document}$ z_{0} $\end{document} are positive real numbers, the functions \begin{document}$ f, \, g, \, h:\, \left(0, +\infty\right)^{2}\rightarrow\left(0, +\infty\right) $\end{document} 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.

The content you want is available to Zendy users.

Already have an account? Click here to sign in.
Having issues? You can contact us here