Well-Defined Solutions of a Three-Dimensional System of Difference Equations

Abstract

We show that the three-dimensional system of difference equations

x_{n+1}=\frac{ax_{n}z_{n-1}}{z_{n}-\beta}+\gamma,  y_{n+1}=\frac{by_{n}x_{n-1}}{x_{n}-\gamma}+\alpha, z_{n+1}=\frac{cz_{n}y_{n-1}}{y_{n}-\alpha}+\beta,

where the parameters a,b,x, \alpha, \beta, \gamma  and the initial conditions x_{-i}, y_{-i}, i\in\{0,1\}  are non-zero real numbers, can be solved. Using the obtained formulas, we determine the asymptotic behavior of solutions and give conditions for which periodic solutions exists. Some numerical examples are given to demonstrate the theoretical results.

Keywords

• Periodic solution,difference equation,three dimensional system of difference equations

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