On the Periodic Solutions of Some Systems of Difference Equations

Yıl 2018, Cilt: 1 Sayı: 2, 126 - 136, 24.12.2018

E. M. Elsayed , H. S. Gafel

https://doi.org/10.33434/cams.442662

Öz

In this paper, we study the solution of the systems of difference equations \begin{equation*} x_{n+1}=\frac{1\pm (y_{n}+x_{n-1})}{y_{n-2}},\ \ \ y_{n+1}=\frac{1\pm (x_{n}+y_{n-1})}{x_{n-2}},\;\;n=0,1,..., \end{equation*}% {\Large \noindent }where the initial conditions $x_{-2},\ x_{-1},\ x_{0},$ $% y_{-2},\ y_{-1},\ y_{0}$ are arbitrary non zero real numbers.

Anahtar Kelimeler

Difference equation, Periodicity, System of difference equations

Kaynakça

• [1] F. Alzahrani, A. Khaliq, E. M. Elsayed, Dynamics and behaviour of some rational systems of difference equations, J. Comput. Theoret. Nanosci., 13(11) (2016), 8583-8599.
• [2] A. Asiri, M. M. El-Dessoky, E. M. Elsayed, Solution of a third order fractional system of difference equations, J. Comput. Anal. Appl., 24(3) (2018), 444-453.
• [3] H. Bao, On a system of second-order nonlinear difference equations, J. Appl. Math. Phys., 3 (2015), 903-910.
• [4] C. Cinar, On the positive solutions of the difference equation system, $x_{n+1}=\frac{1}{y_{n}},$ $y_{n+1}=\frac{y_{n}}{x_{n-1}y_{n-1}},$; Appl. Math. Comput.,158 (2004), 303-305.
• [5] C. A. Clark, M. R. S. Kulenovic, J. F. Selgrade, On a system of rational difference equations, J. Differ. Equ. Appl., 11 (2005), 565-580.
• [6] Q. Din, E. M. Elsayed, Stability analysis of a discrete ecological model, Comput. Ecol. Softw., 4 (2014), 89-103.
• [7] Q. Din, M. N. Qureshi, A. Q. Khan, Qualitative behaviour 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, 12 pages.
• [10] M. M. El-Dessoky, The form of solutions and periodicity for some systems of third - order rational difference equations, Math. Methods Appl., Sci., 39 (2016), 1076-1092.
• [11] M. M. El-Dessoky, E. M. Elsayed and M. Alghamdi, Solutions and periodicity for some systems of fourth order rational difference equations, J. Comput. Anal. Appl., 18 (2015), 179-194.
• [12] M. M. El-Dessoky, M. Mansour, E. M. Elsayed, Solutions of some rational systems of difference equations, Util. Math., 92 (2013), 329-336.
• [13] E. M. Elsayed, On the solutions of a rational system of difference equations, Fasc. Math., 45 (2010), 25-36.
• [14] E. M. Elsayed, A. Alotaibi, H. A. Almaylabi, On a solutions of fourth order rational systems of difference equations, J. Comput. Anal. Appl., 22(7) (2017), 1298-1308.
• [15] E. M. Elsayed, M. Mansour and 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] V. L. Kocic, G. Ladas, Global behavior of nonlinear difference equations of higher order with applications, Kluwer Academic Publishers, Dordrecht, 1993.
• [18] M. Kulenovic, G. Ladas, Dynamics of second order rational difference equations with open problems and conjectures, Chapman & Hall / CRC Press, U.S.A., 2001.
• [19] A. S. Kurbanlı, C. Cinar, I. Yalcinkaya, On the behavior of positive solutions of the system of rational difference equations $x_{n+1}= \frac{x_{n-1}}{1+x_{n-1}y_{n}},$ $y_{n+1}=\frac{y_{n-1}}{1+x_{n}y_{n-1}}$, Math. Comput. Model., 53(5-6) (2011), 1261-1267.
• [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] 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.
• [22] A. Neyrameh, H. Neyrameh, M. Ebrahimi, A. Roozi, Analytic solution diffusivity equation in rational form, World Appl. Sci. J., 10 (2010), 764-768.
• [23] G. Papaschinopoulos, C. J. Schinas, On a system of two nonlinear difference equations, J. Math. Anal. Appl., 219 (1998), 415-426.
• [24] S. Stevic, B. Iricanin, Z. Smarda, Boundedness character of a fourth-order system of difference equations, Adv. Differ. Equ., 2015 (2015), 11 pages.
• [25] I. Yalcinkaya, On the global asymptotic stability of a second-order system of difference equations, Discrete Dyn. Nat. Soc., 2008(2008), 12 pages.