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ON A THREE-DIMENSIONAL SYSTEM OF DIFFERENCE EQUATIONS WITH VARIABLE COEFFICIENTS
KARA, MERVE,YAZLIK, YASIN,TOUAFEK, NOURESSADAT,AKROUR, YOUSSOUF The Korean Society for Computational and Applied M 2021 Journal of applied mathematics & informatics Vol.39 No.3
Consider the three-dimensional system of difference equations $x_{n+1}=\frac{{\prod_{j=0}^{k}}z_n-3j}{{\prod_{j=1}^{k}}x_n-(3j-1)\;\(a_n+b_n{\prod_{j=0}^{k}}z_n-3j\)}$, $y_{n+1}=\frac{{\prod_{j=0}^{k}}x_n-3j}{{\prod_{j=1}^{k}}y_n-(3j-1)\;\(c_n+d_n{\prod_{j=0}^{k}}x_n-3j\)}$, $z_{n+1}=\frac{{\prod_{j=0}^{k}}y_n-3j}{{\prod_{j=1}^{k}}z_n-(3j-1)\;\(e_n+f_n{\prod_{j=0}^{k}}y_n-3j\)}$, n ∈ ℕ<sub>0</sub>, where k ∈ ℕ<sub>0</sub>, the sequences $(a_n)_{n{\in}{\mathbb{N}}_0$, $(b_n)_{n{\in}{\mathbb{N}}_0$, $(c_n)_{n{\in}{\mathbb{N}}_0$, $(d_n)_{n{\in}{\mathbb{N}}_0$, $(e_n)_{n{\in}{\mathbb{N}}_0$, $(f_n)_{n{\in}{\mathbb{N}}_0$ and the initial values x<sub>-3k</sub>, x<sub>-3k+1</sub>, …, x<sub>0</sub>, y<sub>-3k</sub>, y<sub>-3k+1</sub>, …, y<sub>0</sub>, z<sub>-3k</sub>, z<sub>-3k+1</sub>, …, z<sub>0</sub> are real numbers. In this work, we give explicit formulas for the well defined solutions of the above system. Also, the forbidden set of solution of the system is found. For the constant case, a result on the existence of periodic solutions is provided and the asymptotic behavior of the solutions is investigated in detail.
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