The form of solutions and periodic nature for some rational difference equations systems

Volume 9, Issue 10, pp 5629--5647 http://dx.doi.org/10.22436/jnsa.009.10.10

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Authors

M. M. El-Dessoky - Faculty of Science, Mathematics Department, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia. - Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt. E. M. Elsayed - Faculty of Science, Mathematics Department, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia. - Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt. E. O. Alzahrani - Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt.

Abstract

In this paper, we investigate the expressions of solutions and the periodic nature of the following systems of rational difference equations with order four \[x_{n+1} = \frac{y_{n-3} }{\pm 1\pm y_nz_{n-1}x_{n-2}y_{n-3}}, y_{n+1} = \frac{z_{n-3} }{\pm 1\pm z_nx_{n-1}y_{n-2}z_{n-3}}, z_{n+1} = \frac{x_{n-3} }{\pm 1\pm x_ny_{n-1}z_{n-2}x_{n-3}},\] with initial conditions \(x_{-3}; x_{-2}; x_{-1}; x_0; y_{-3}; y_{-2}; y_{-1}; y_0; z_{-3}; z_{-2}; z_{-1}\) and \(z_0\) which are arbitrary real numbers.

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Keywords

• Difference equations
• recursive sequences
• stability
• periodic solution
• system of difference equations.

MSC

•  39A20
•  39A23
•  39A30

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