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On the Periodic Solutions of Some Systems of Difference Equations

Year 2018, Volume: 1 Issue: 2, 126 - 136, 24.12.2018
https://doi.org/10.33434/cams.442662

Abstract

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.

References

  • [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.
Year 2018, Volume: 1 Issue: 2, 126 - 136, 24.12.2018
https://doi.org/10.33434/cams.442662

Abstract

References

  • [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.
There are 25 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

E. M. Elsayed 0000-0003-0894-8472

H. S. Gafel This is me

Publication Date December 24, 2018
Submission Date July 11, 2018
Acceptance Date August 1, 2018
Published in Issue Year 2018 Volume: 1 Issue: 2

Cite

APA Elsayed, E. M., & Gafel, H. S. (2018). On the Periodic Solutions of Some Systems of Difference Equations. Communications in Advanced Mathematical Sciences, 1(2), 126-136. https://doi.org/10.33434/cams.442662
AMA Elsayed EM, Gafel HS. On the Periodic Solutions of Some Systems of Difference Equations. Communications in Advanced Mathematical Sciences. December 2018;1(2):126-136. doi:10.33434/cams.442662
Chicago Elsayed, E. M., and H. S. Gafel. “On the Periodic Solutions of Some Systems of Difference Equations”. Communications in Advanced Mathematical Sciences 1, no. 2 (December 2018): 126-36. https://doi.org/10.33434/cams.442662.
EndNote Elsayed EM, Gafel HS (December 1, 2018) On the Periodic Solutions of Some Systems of Difference Equations. Communications in Advanced Mathematical Sciences 1 2 126–136.
IEEE E. M. Elsayed and H. S. Gafel, “On the Periodic Solutions of Some Systems of Difference Equations”, Communications in Advanced Mathematical Sciences, vol. 1, no. 2, pp. 126–136, 2018, doi: 10.33434/cams.442662.
ISNAD Elsayed, E. M. - Gafel, H. S. “On the Periodic Solutions of Some Systems of Difference Equations”. Communications in Advanced Mathematical Sciences 1/2 (December 2018), 126-136. https://doi.org/10.33434/cams.442662.
JAMA Elsayed EM, Gafel HS. On the Periodic Solutions of Some Systems of Difference Equations. Communications in Advanced Mathematical Sciences. 2018;1:126–136.
MLA Elsayed, E. M. and H. S. Gafel. “On the Periodic Solutions of Some Systems of Difference Equations”. Communications in Advanced Mathematical Sciences, vol. 1, no. 2, 2018, pp. 126-3, doi:10.33434/cams.442662.
Vancouver Elsayed EM, Gafel HS. On the Periodic Solutions of Some Systems of Difference Equations. Communications in Advanced Mathematical Sciences. 2018;1(2):126-3.

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