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.
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
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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