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

Volume 9, Issue 10, pp 5629--5647
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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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##### ISRP Style

M. M. El-Dessoky, E. M. Elsayed, E. O. Alzahrani, The form of solutions and periodic nature for some rational difference equations systems, Journal of Nonlinear Sciences and Applications, 9 (2016), no. 10, 5629--5647

##### AMA Style

El-Dessoky M. M., Elsayed E. M., Alzahrani E. O., The form of solutions and periodic nature for some rational difference equations systems. J. Nonlinear Sci. Appl. (2016); 9(10):5629--5647

##### Chicago/Turabian Style

El-Dessoky, M. M., Elsayed, E. M., Alzahrani, E. O.. "The form of solutions and periodic nature for some rational difference equations systems." Journal of Nonlinear Sciences and Applications, 9, no. 10 (2016): 5629--5647

### Keywords

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

•  39A20
•  39A23
•  39A30

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