On some rational systems of difference equations

Volume 11, Issue 1, pp 49--72 http://dx.doi.org/10.22436/jnsa.011.01.05

Publication Date: December 24, 2017 Submission Date: August 18, 2017 Revision Date: September 19, 2017 Accteptance Date: November 05, 2017

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 2245 Downloads
• 5237 Views

Authors

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

Abstract

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.

Share and Cite

Keywords

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

MSC

•  39A10

References

• [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.
  • MathSciNet
  • Google Scholar
  • MATH


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


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


• [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.
  • View Article
  • Google Scholar


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


• [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.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [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.
  • Google Scholar


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


• [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.
  • View Article
  • Google Scholar
  • MATH


• [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.
  • MathSciNet


• [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.
  • MathSciNet
  • Google Scholar
  • MATH


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


• [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.
  • MathSciNet
  • Google Scholar
  • MATH


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


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


• [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.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


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


• [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.
  • View Article
  • MathSciNet
  • Google Scholar


• [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.
  • Google Scholar


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


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


• [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.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [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.
  • MathSciNet
  • Google Scholar


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


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


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


• [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.
  • View Article
  • MathSciNet
  • Google Scholar