On a solvable for some systems of rational difference equations

Volume 9, Issue 6, pp 3744--3759 http://dx.doi.org/10.22436/jnsa.009.06.25

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)

• 1452 Downloads
• 2756 Views

Authors

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

Abstract

In this paper, we study the existence of solutions for a class of rational systems of difference equations of order four in four-dimensional case \[x_{n+1} = \frac {x_{n-3}}{\pm 1\pm t_nz_{n-1}y_{n-2}x_{n-3}}, \qquad y_{n+1} =\frac{ y_{n-3}} {\pm 1\pm x_nt_{n-1}z_{n-2}y_{n-3}},\] \[z_{n+1} =\frac{ z_{n-3}} {\pm 1\pm y_nx_{n-1}t_{n-2}z_{n-3}}, \qquad t_{n+1} =\frac{ t_{n-3}} {\pm 1\pm z_ny_{n-1}x_{n-2}t_{n-3}},\] with the initial conditions are real numbers. Also, we study some behavior such as the periodicity and boundedness of solutions for such systems. Finally, some numerical examples are given to confirm our theoretical results and graphed by Matlab.

Share and Cite

Keywords

• Recursive sequences
• difference equation system
• periodic solutions.

MSC

•  39A10
•  39A22
•  39A23

References

• [1] R. Abu-Saris, C. Çinar, I. Yalcinkaya, On the asymptotic stability of \(x_{n+1} = \frac{a+x_nx_{n-k}}{x_n+x_{n-k}}\), Comput. Math. Appl., 56 (2008), 1172-1175.


• [2] R. Abo-Zeid, Attractivity of two nonlinear third order difference equations, J. Egyptian Math. Soc., 21 (2013), 241-247.


• [3] Q. Din, On a system of rational difference equations, Demonstr. Math., 47 (2014), 324-335.


• [4] Q. Din, On a system of fourth-order rational difference equations, Acta Univ. Apulensis Math. Inform., 39 (2014), 137-150.


• [5] Q. Din, Global behavior of a plant-herbivore model , Adv. Difference Equ., 2015 (2015), 12 pages.


• [6] Q. Din, Asymptotic behavior of an anti-competitive system of second-order difference equations, J. Egyptian Math. Soc., 24 (2016), 37-43.


• [7] Q. Din, T. F. Ibrahim, K. A. Khan, Behavior of a competitive system of second-order difference equations, Sci. World J., 2014 (2014), 9 pages.


• [8] 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.


• [9] 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.


• [10] 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.


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


• [12] H. El-Metwally , Solutions form for some rational systems of difference equations, Discrete Dyn. Nat. Soc., 2013 (2013), 10 pages.


• [13] E. M. Elsayed, H. A. El-Metwally , On the solutions of some nonlinear systems of difference equations, Adv. Difference Equ., 2013 (2013), 14 pages.


• [14] Y. Halim, N. Touafek, Y. Yazlik, Dynamic behavior of a second-order nonlinear rational difference equation, Turkish J. Math., 39 (2015), 1004-1018.


• [15] L. X. Hu, X. M. Jia, Global asymptotic stability of a rational system , Abstr. Appl. Anal., 2014 (2014), 6 pages.


• [16] T. F. Ibrahim, N. Touafek, On a third order rational difference equation with variable coefficients, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 20 (2013), 251-264.


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


• [18] 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.


• [19] A. S. Kurbanli, On the behavior of solutions of the system of rational difference equations: \(x_{n+1} = x_{n-1}/x_{n-1}y_{n- 1}; y_{n+1} = y_{n-1}/y_{n-1}x_{n - 1}\); and \(z_{n+1} = x_n/z_{n-1}y_{n - 1}\), Discrete Dyn. Nat. Soc., 2011 (2011), 12 pages.


• [20] A. S. Kurbanli, C. Çinar, M. Erdoğan , On the behavior of solutions of the system of rational difference equations\( x_{n+1} =\frac{ x_{n-1}}{ x_{n-1}y_{n-1}} ; y_{n+1} =\frac{ y_{n-1}}{ y_{n-1}x_{n-1}} ; z_{n+1} = \frac{x_n}{ z_{n-1}y_n} \) , Appl. Math. (Irvine), 2 (2011), 1031-1038.


• [21] K. Liu, Z. Zhao, X. Li, P. Li , More on three-dimensional systems of rational difference equations, Discrete Dyn. Nat. Soc., 2011 (2011), 9 pages.


• [22] H. L. Ma, H. Feng, On Positive Solutions for the Rational Difference Equation Systems \(x_{n+1} = \frac{A x_n}{y^2_n}}\) and\( y_{n+1} = \frac{By_n }{x_{n-1}y_{n-1}}\), International Scholarly Research Notices, 2014 (2014), 4 pages.


• [23] O. Özkan, A. S. Kurbanli, On a system of difference equation, Discrete Dyn. Nat. Soc., 2013 (2013), 7 pages.


• [24] B. Sroysang, Dynamics of a system of rational higher-order difference equation, Discrete Dyn. Nat. Soc., 2013 (2013), 5 pages.


• [25] S. Stević, Boundedness and persistence of some cyclic-type systems of difference equations, Appl. Math. Lett., 56 (2016), 78-85.


• [26] S. Stević, J. Diblík, B. Iricanin, Z. Šmarda, On a solvable system of rational difference equations, J. Difference Equ. Appl., 20 (2014), 811-825.


• [27] N. Touafek, E. M. Elsayed , On a second order rational systems of difference equations, Hokkaido Math. J., 44 (2015), 29-45.


• [28] C. Y. Wang, S. Wang, W. Wang, Global asymptotic stability of equilibrium point for a family of rational difference equations , Appl. Math. Lett., 24 (2011), 714-718.


• [29] I. Yalcinkaya, C. Çinar, On the positive solutions of the difference equation system \(x_{n+1} = 1/z_n; y_{n+1} = y_n/x_{n-1}y_{n-1}; z_{n+1} = 1/x_{n-1}\), J. Inst. Math. Comput. Sci., 18 (2005), 91-93.


• [30] I. Yalcinkaya, C. Çinar , Global asymptotic stability of two nonlinear difference equations, Fasc. Math., 43 (2010), 171-180.


• [31] Y. Yazlik, D. T. Tollu, N. Taskara, On the behaviour of solutions for some systems of difference equations, J. Comput. Anal. Appl., 18 (2015), 166-178.


• [32] E. M. E. Zayed, M. A. El-Moneam, On the global attractivity of two nonlinear difference equations, J. Math. Sci., 177 (2011), 487-499.


• [33] Q. Zhang, J. Liu, Z. Luo, Dynamical behavior of system of third-order rational difference equation, Discrete Dyn. Nat. Soc., 2015 (2015), 6 pages.