**Volume 11, Issue 1, pp 49--72**

**Publication Date**: 2017-12-24

http://dx.doi.org/10.22436/jnsa.011.01.05

M. M. El-Dessoky - Mathematics Department, Faculty of Science, King AbdulAziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia

A. Khaliq - Department of Mathematics, Riphah International University, Lahore, Pakistan

A. Asiri - Mathematics Department, Faculty of Science, King AbdulAziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia

Our goal in this paper is to find the form of solutions for the following systems of rational difference equations: \[ x_{n+1}=\frac{x_{n-3}y_{n-4}}{y_{n}(\pm 1\pm x_{n-3}y_{n-4})},\quad y_{n+1}=\frac{y_{n-3}x_{n-4}}{x_{n}(\pm 1\pm y_{n-3}x_{n-4})},\quad n=0,1,\ldots, \] where the initial conditions have non-zero real numbers.

Form of solution, stability, rational difference equations, rational systems

[1] E. O. Alzahrani, M. M. El-Dessoky, E. M. Elsayed, Y. Kuang, Solutions and Properties of Some Degenerate Systems of Difference Equations, J. Comput. Anal. Appl., 18 (2015), 321–333.

[2] H. Bao, On a System of Second-Order Nonlinear Difference Equations, J. Appl. Math. Phys., 3 (2015), 903–910.

[3] E. Camouzis, Open problems and Conjectures, J. Difference Equ. Appl., 15 (2009), 203–323.

[4] C. A. Clark, M. R. S. Kulenović, J. F. Selgrade, On a system of rational difference equations, J. Difference Equ. Appl., 11 (2005), 565–580.

[5] Q. Din, E. M. Elsayed, Stability analysis of a discrete ecological model, Comput. Ecol. Softw., 4 (2014), 89–103.

[6] Q. Din, M. N. Qureshi, A. Q. Khan, Dynamics of a fourth-order system of rational difference equations, Adv. Difference Equ., 2012 (2012), 15 pages.

[7] Q. Din, M. N. Qureshi, A. Q. Khan, Qualitative behavior of an anti-competitive system of third-order rational difference equations, Comput. Ecol. Softw., 4 (2014), 104–115.

[8] M. M. El-Dessoky, On a systems of rational difference equations of Order Two, Proc. Jangjeon Math. Soc., 19 (2016), 271–284.

[9] M. M. El-Dessoky, Solution of a rational systems of difference equations of order three, Mathematics, 2016 (2016), 12 pages.

[10] M. M. El-Dessoky, The form of solutions and periodicity for some systems of third -order rational difference equations, Math. Method Appl. Sci., 39 (2016), 1076–1092.

[11] M. M. El-Dessoky, E. M. Elsayed, On a solution of system of three fractional difference equations, J. Comput. Anal. Appl., 19 (2015), 760–769.

[12] M. M. El-Dessoky, E. M. Elsayed, M. Alghamdi, Solutions and periodicity for some systems of fourth order rational difference equations, J. Comput. Anal. Appl., 18 (2015), 179–194.

[13] M. M. El-Dessoky, M. Mansour, E. M. Elsayed, Solutions of some rational systems of difference equations, Util. Math., 92 (2013), 329–336.

[14] E. M. Elsayed, M. M. El-Dessoky, E. O. Alzahrani, The Form of The Solution and Dynamics of a Rational Recursive Sequence, J. Comput. Anal. Appl., 17 (2014), 172–186.

[15] E. M. Elsayed, M. Mansour, M. M. El-Dessoky, Solutions of fractional systems of difference equations, Ars Combin., 110 (2013), 469–479.

[16] A. Q. Khan, M. N. Qureshi, Global dynamics of some systems of rational difference equations, J. Egyptian Math. Soc., 24 (2016), 30–36.

[17] A. Q. Khan, M. N. Qureshi, Q. Din, Global dynamics of some systems of higher-order rational difference equations, Adv. Difference Equ., 2013 (2013), 23 pages.

[18] V. L. Kocić, G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, (1993).

[19] M. Kulenovic, G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman & Hall CRC Press, U.S.A., (2001).

[20] A. S. Kurbanlı, On the Behavior of Solutions of the System of Rational Difference Equations, World Appl. Sci. J., 10 (2010), 1344–1350.

[21] A. S. Kurbanlı, C. Çinar, I. Yalçinkaya, On the behavior of positive solutions of the system of rational difference equations \(x_{n+1} = \frac{x_{n-1}}{ y_nx_{n-1}+1} , y_{n+1} = \frac{y_{n-1}}{ x_ny_{n-1}+1}\) , Math. Comput. Model., 53 (2011), 1261–1267.

[22] M. Mansour, M. M. El-Dessoky, E. M. Elsayed, On the solution of rational systems of difference equations, J. Comput. Anal. Appl., 15 (2013), 967–976.

[23] A. Neyrameh, H. Neyrameh, M. Ebrahimi, A. Roozi, Analytic solution diffusivity equation in rational form, World Appl. Sci. J., 10 (2010), 764–768.

[24] A. Y. Özban, On the positive solutions of the system of rational difference equations, J. Math. Anal. Appl., 323 (2006), 26–32.

[25] G. Papaschinopoulos, C. J. Schinas, G. Stefanidou, On a k-order system of Lyness-type difference equations, Adv. Difference Equ., 2007 (2007), 13 pages.

[26] G. Papaschinopoulos, G. Stefanidou, Asymptotic behavior of the solutions of a class of rational difference equations, Int. J. Difference Equ., 5 (2010), 233–249.

[27] S. Stević, On a system of difference equations, Appl. Math. Comput., 218 (2011), 3372–3378.

[28] S. Stević, B. Iričanin, Z. Šmarda, Boundedness character of a fourth-order system of difference equations, Adv. Difference Equ., 2015 (2015), 11 pages.

[29] I. Yalçinkaya, On the global asymptotic stability of a second-order system of difference equations, Discrete Dyn. Nat. Soc., 2008 (2008), 12 pages.

[30] X. Yang, Y. Liu, S. Bai, On the system of high order rational difference equations \(x_{n+1} = \frac{\alpha}{ y_{n-p}}, y_{n+1} = \frac{by_{n-p}}{ x_{n-q}y_{n-p}}\), Appl. Math. Comput., 171 (2005), 853–856.