Author:
Touafek Nouressadat,Tollu Durhasan Turgut,Akrour Youssouf
Abstract
<p style='text-indent:20px;'>In this work, we study the behavior of the solutions of following three-dimensional system of difference equations</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \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*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ n\in \mathbb{N}_{0} $\end{document}</tex-math></inline-formula>, the initial values <inline-formula><tex-math id="M2">\begin{document}$ x_{-1} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ x_{0} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ y_{-1} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ y_{0} $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M6">\begin{document}$ z_{-1} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ z_{0} $\end{document}</tex-math></inline-formula> are positive real numbers, the functions <inline-formula><tex-math id="M8">\begin{document}$ f, \, g, \, h:\, \left(0, +\infty\right)^{2}\rightarrow\left(0, +\infty\right) $\end{document}</tex-math></inline-formula> 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.</p>
Publisher
American Institute of Mathematical Sciences (AIMS)
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