On the Solutions of Systems of Rational Difference Equations

  • E. M. Elsayed Department of Mathematics, Faculty of Sciences, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, KSA https://orcid.org/0000-0003-0894-8472
  • J. G. Al-Juaid Department of Mathematics, Faculty of Sciences, Mansoura University, Mansoura 35516, Egypt https://orcid.org/0000-0001-6062-5916
  • H. Malaikah Department of Mathematics, Faculty of Sciences, Taif University, P. O. Box 11099, Taif 21944, KSA
Keywords: Solutions of difference equations, periodic solution, recursive sequences.

Abstract

In this paper we study the form of the solutions of the following systems of difference equations

w_{n+1}=\frac{s_{n}(w_{n-3}+s_{n-4})}{s_{n-4}+w_{n-3} -s_{n}} ,   s_{n+1}=\frac{w_{n-2}(w_{n-2}+s_{n-3})}{2*w_{n-2}+s_{n-3}}.

w_{n+1}=\frac{(s_{n-4} - w_{n-3})s_{n}}{s_{n-4} -w_{n-3} +s_{n}} ,   s_{n+1}=\frac{(s_{n-3} - w_{n-2})w_{n-2}}{s_{n-3}}.

With nonzero real numbers initial conditions.

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References

[1] R. P. Agarwal, Difference Equations and Inequalities, 1st edition, Marcel Dekker, New York, 1992, 2nd
edition, 2000.
[2] R. P. Agarwal and E. M. Elsayed, On the solution of fourth-order rational recursive sequence, Adv.
Stud. Contemp. Math., 20 (4), (2010), 525-545.
[3] A. Asiri, M. M. El-Dessoky, and E. M. Elsayed, Solution of a third order fractional system of difference
equations, Journal of Computational Analysis and Applications, vol. 24, no. 3, pp. 444-453, 2018.
[4] N. Battaloglu, C. Cinar and I. Yalcnkaya, The dynamics of the difference equation, ARS Combinatoria,
97 (2010), 281-288.
[5] C. Cinar, I. Yal¸cinkaya and R. Karatas, On the positive solutions of the difference equation system xn+1 = m
yn, yn+1 = pyn xn−1yn−1, J. Inst. Math. Comp. Sci., 18(2005), 135-136.
[6] Q. Din, M. N. Qureshi, and A. Q. Khan, Dynamics of a fourth-order system of rational difference
equations, Advances in Di’erence Equations, vol. 2012, 15 pages, 2012.
[7] E. M. Elabbasy , H. El-Metwally and E. M. Elsayed, Global behavior of the solutions of difference
equation, Adv. Differ. Equ., 2011, 2011:28.
[8] M. M. El-Dessoky, The form of solutions and periodicity for some systems of third-order rational
difference equations, Mathematical Methods in the Applied Sciences, vol. 39, no. 5, pp. 1076-1092, 2016.
[9] M. M. El-Dessoky and E. M. Elsayed, On the solutions and periodic nature of some systems of rational
difference equations, Journal of Computational Analysis and Applications, vol. 18, no. 2, pp. 206-218,
2015.
[10] M. M. El-Dessoky, E. M. Elsayed, and M. Alghamdi, Solutions and periodicity for some systems of
fourth order rational difference equations, Journal of Computational Analysis and Applications, vol. 18,
no. 1, pp. 179-194, 2015.
[11] M. M. El-Dessoky, A. Khaliq, and A. Asiri, On some rational systems of difference equations, Journal
of Nonlinear Sciences and Applications, vol. 11, no. 01, pp. 49-72, 2017.
[12] H. El-Metwally and E. M. Elsayed, Solution and behavior of a third rational difference equation,
Utilitas Mathematica, 88 (2012), 27-42.
[13] M. El-Moneam, On the dynamics of the solutions of the rational recursive sequences, British Journal of
Mathematics and Computer Science, vol. 5, no. 5, pp. 654-665, 2015.
[14] E. M. Elsayed, Solutions of rational di¤erence system of order two, Math. Comp. Mod., 55 (2012),
378-384.
[15] E. M. Elsayed, On the solutions of a rational system of difference equations, Fasciculi Mathematici, 45
(2010), 25-36.
[16] E. M. Elsayed, Solution and attractivity for a rational recursive sequence, Disc.Dyn. Nat. Soc., vol.
2011, article ID 982309, 17 pages.
[17] E. M. Elsayed and M. M. El-Dessoky, Dynamics and behavior of a higher order rational recursive
sequence, Adv. Differ. Equ., 2012, 2012:69.
[18] A.S. Kurbanli,C. Cinar and D. Simsek, On the periodicity of solutions of the system of rational
difference equations xn+1 = xn−1+yn
xn−1yn−1 , yn+1 = yn−1+xn
yn−1xn−1 . Applied Mathematics 2 (2011), 410-413.
[19] A. Kurbanli, C. Cinar and I. Yalcnkaya, On the behavior of positive solutions of the system of rational
difference equations, Math. Comp. Mod., 53 (2011), 1261-1267.
[20] M. Mansour, M. M. El-Dessoky and E. M. Elsayed, The form of the solutions and periodicity of some
systems of difference equations, Disc. Dyn. Nat. Soc., vol. 2012, article ID 406821, 17 pages.
[21] N. Touafek and E. M. Elsayed, On the solutions of systems of rational difference equations. Math.
Comput. Mod. 55(2012), 1987-1997.
[22] N. Touafek and E. Elsayed, On a second order rational systems of difference equations, Hokkaido
Mathematical Journal, vol. 44, no. 1, pp. 29-45, 2015.
[23] I. Yalcnkaya, On the global asymptotic stability of a second-order system of difference equations, Disc.
Dyn. Nat. Soc., vol. 2008, article ID 860152, 12 pages.
[24] I. Yalcnkaya, On the global asymptotic behavior of a system of two nonlinear difference equations,
ARS Combinatoria, 95 (2010), 151-159.
[25] I. Yalcnkaya, C. Cinar and M. Atalay, On the solutions of systems of difference equations, Adv. Di¤er.
Equ., vol. 2008, article ID 143943, 9 pages.
[26] X. Yang, Y. Liu and S. Bai, On the system of high order rational difference equations xn = a yn−p, yn = byn−p
xn−qyn−q. Appl. Math. Comp. 171(2) (2005), 853-856.
[27] Q. Zhang, L. Yang and J.Liu, Dynamics system of rational third order difference equation, Adv.
Differ. Equ., 2012: 2012: 136.
Published
2022-11-01
How to Cite
Elsayed, E., Al-Juaid, J., & Malaikah, H. (2022). On the Solutions of Systems of Rational Difference Equations. Journal of Progressive Research in Mathematics, 19(2), 49-59. Retrieved from http://scitecresearch.com/journals/index.php/jprm/article/view/2166
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