The Form of the Solutions of System of Rational Difference Equation

Marwa M. Alzubaidi; E. M. Elsayed

doi:10.33187/jmsm.427368

Research Article

The Form of the Solutions of System of Rational Difference Equation

Year 2018, Volume: 1 Issue: 3, 181 - 191, 30.12.2018

Marwa M. Alzubaidi , E. M. Elsayed

https://doi.org/10.33187/jmsm.427368

Cited By: 2

Abstract

In this article, we study the form of the solutions of the system of difference equations $x_{n+1}=((y_{n-8})/(1+y_{n-2}x_{n-5}y_{n-8}))$, $y_{n+1}=((x_{n-8})/(\pm1\pm x_{n-2}y_{n-5}x_{n-8}))$, with the initial conditions are real numbers. Also, we give the numerical examples of some of difference equations and got some related graphs and figures using by Matlab.

Keywords

Difference Equation, System difference equations, Periodicity

References

[1] A. M. Ahmed, E. M. Elsayed, The expressions of solutions and the periodicity of some rational difference equations system, J. Appl. Math. Inform. 34(1-2) (2016), 35–48.
[2] M. M. El-Dessoky, The form of solutions and periodicity for some systems of third-order rational difference equations, Math. Meth. Appl. Sci., 39 (2016), 1076–1092.
[3] M. M. El-Dessoky, E. M. Elsayed, On the solutions and periodic nature of some systems of rational difference equations, J. Comput. Anal. Appl., 18(2) (2015), 206–218.
[4] M. M. El-Dessoky, A. Khaliq, A. Asiri, On some rational system of difference equations, J. Nonlinear Sci. Appl., 11 (2018), 49–72.
[5] E. M. Elsayed, T. F. Ibrahim, Periodicity and solutions for some systems of nonlinear rational difference equations, Hacettepe Journal of Mathematics and Statistics, 44 (6) (2015), 1361–1390.
[6] E. M. Elsayed, A. Alghamdi, The form of the solutions of nonlinear difference equations systems, J. Nonlinear Sci. Appl., 9 (2016), 3179–3196.
[7] N. Haddad , N. Touafek, J. F. T. Rabago, Solution form of a higher-order system of difference equations and dynamical behavior of its special case, Math. Methods Appl. Sci., 40 (2017), 3599–3607.
[8] A. S. Kurbanli, On the behavior of solutions of the system of rational difference equations $x_{n+1}=x_{n-1}/(y_{n}x_{n-1}-1),$; $y_{n+1}=y_{n-1}/(x_{n}y_{n-1}-1)$, World Appl. Sci. J., 10(11) (2010), 1344-1350.
[9] A. S. Kurbanli, C. C¸ inar, D. S¸ims¸ek, On the periodicity of solutions of the system of rational difference equations $x_{n+1}=x_{n-1}+y_{n}/(y_{n}x_{n-1}-1),$; $y_{n+1}=y_{n-1}+x_{n}/(x_{n}y_{n-1}-1)$, Appl. Math., 2 (2011), 410-413.
[10] A. S. Kurbanli, C. C¸ inar, I. Yalc¸inkaya, On the behavior of positive solutions of the system of rational difference equations $x_{n+1}=x_{n-1}/(y_{n}x_{n-1}+1),$; $y_{n+1}=y_{n-1}/(x_{n}y_{n-1}+1)$, Math. Comput. Modelling, 53 (2011), 1261–1267.
[11] M. Mansour, M. M. El-Dessoky, E. M. Elsayed, The form of the solutions and periodicity of some systems of difference equations, Discrete Dyn. Nat. Soc., 2012 (2012), Article ID 406821, 1-17.
[12] N. Touafek and E. M. Elsayed, On the solutions of systems of rational difference equations, Math. Comput. Modelling, 55 (2012), 1987-1997.
[13] S. A. Abramov, D. E. Khmelnov, Denominators of rational solutions of linear difference systems of an arbitrary order, Program. Comput. Softw., 38 (2) (2012), 84–91.
[14] R. Abu-Saris, C. Cinar, I. Yalcinkaya, On the Asymptotic Stability of $x_{n+1}=a+x_{n}x_{n-k}/x_{n}+x_{n-k},$; Comput. Math. Appl., 56(5) (2008), 1172-1175.
[15] N. Battaloglu, C. Cinar, I. Yalcinkaya, The dynamics of the difference equation $x_{n+1}=(\alpha x_{n-m})/(\beta+\gamma x_{n-(k+1)}^{p}),\ $; , Ars Combin., 97 (2010), 281–288.
[16] O. H. Criner, W. E. Taylor, J. L. Williams, On the solutions of a system of nonlinear difference equations, Int. J. Difference Equ., 10(2) (2015), 161–166.
[17] I. Dekkar, N. Touafek, Y. Yazlik, Global stability of a third-order nonlinear system of difference equations with period-two coefficients, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat., 111(2) (2017), 325–347.
[18] Q. Din, Qualitative nature of a discrete predator-prey system, Contem. Methods Math. Phys. Grav., 1 (2015), 27-42.
[19] E. M. Elabbasy, H. El-Metwally, E. M. Elsayed, Some properties and expressions of solutions for a class of nonlinear difference equation, Util. Math., 87 (2012), 93–110.
[20] E. M. Elabbasy, S. M. Eleissawy, Periodicty and stability of solutions of rational difference systems, Appl. Comput. Math., 1(4) (2012), 1-6.
[21] E. M. Elabbasy, A. A. Elsadany, S. Ibrahim, Behavior and periodic solutions of a two-dimensional systems of rational difference equations, J. Interpolat. Approx. Sci. Comput., 2016(2) (2016), 87-104.
[22] M. M. El-Dessoky, Solution for rational systems of difference equations of order three, Mathematics, 4(3) (2016), 1-12.
[23] M. M. El-Dessoky, E. M. Elsayed, E. O. Alzahrani, The form of solutions and periodic nature for some rational difference equations systems, J. Nonlinear Sci. Appl., 9 (2016), 5629-5647.
[24] E. M. Elsayed, Dynamics of recursive sequence of order two, Kyungpook Math. J., 50(4) (2010), 483-497.
[25] E. M. Elsayed, Solution and attractivity for a rational recursive sequence, Discrete Dyn. Nat. Soc., 2011 (2011), Article ID 982309, 17 pages.
[26] E. M. Elsayed, On the dynamics of a higher-order rational recursive sequence, Commun. Math. Anal., 12(1) (2012), 117–133.
[27] E. M. Elsayed, M. M. El-Dessoky, A. Alotaibi, On the solutions of a general system of difference equations, Discrete Dyn. Nat. Soc., 2012 (2012), Article ID 892571, 12 pages.
[28] M. Gumus, O. Ocalan, The qualitative analysis of a rational system of difference equations, J. Fract. Calc. Appl., 9(2) (2018), 113-126.
[29] N. Haddad, N. Touafek, E. M. Elsayed, A note on a system of difference equations, Analele Stiintifice ale universitatii al i cuza din iasi serie noua matematica, LXIII(3) (2017), 599-606.
[30] N. Haddad, N. Touafek, J. T. Rabago, Well-defined solutions of a system of difference equations, J. Appl. Math. Comput., 56 (1-2) (2018), 439–458.
[31] Y. Halim, N. Touafek, Y. Yazlik, Dynamic behavior of a second-order nonlinear rational difference equation, Turkish J. Math., 39 (2015), 1004-1018.
[32] Y. Halim, A system of difference equations with solutions associated to Fibonacci numbers, Int. J. Difference Equ., 11(1) (2016), 65–77.
[33] G. Hu, Global behavior of a system of two nonlinear difference equation, World J. Res. Rev., 2(6) (2016), 36-38.
[34] W. Ji, D. Zhang, L. Wang, Dynamics and behaviors of a third-order system of difference equation, Math. Sci., 7(34) (2013), 1-6.
[35] A. Khaliq, E. M. Elsayed, The dynamics and solution of some difference equations, J. Nonlinear Sci. Appl., 9 (2016), 1052-1063.
[36] A. S. Kurbanli, On the behavior of solutions of the system of rational difference equations $x_{n+1}=x_{n-1}/(y_{n}x_{n-1}-1),$% \emph{\ }$y_{n+1}=y_{n-1}/(x_{n}y_{n-1}-1),$; $z_{n+1}=1/y_{n}z_{n},$; Adv. Difference Equ., 2011 (2011), 40.
[37] A. S. Kurbanli, C. C¸ inar, M. E. Erdogan, On the behavior of solutions of the system of rational difference equations $x_{n+1}=x_{n-1}/(y_{n}x_{n-1}-1),$ \, $y_{n+1}=y_{n-1}/(x_{n}y_{n-1}-1), $\, $z_{n+1}=x_{n}/(y_{n}z_{n-1}),$; Appl. Math., 2 (2011), 1031-1038.
[38] A. S. Kurbanli, I. Yalcinkaya, A. Gelisken, On the behavior of the Solutions of the system of rational difference equations $x_{n+1}=x_{n-1}/(y_{n}x_{n-1}-1),$ $y_{n+1}=y_{n-1}/(x_{n}y_{n-1}-1), $\emph{\ }$z_{n+1}=x_{n}z_{n-1}/y_{n},$; Int. J. Phys. Sci., 8(2) (2013), 51-56.
[39] D. Simsek, B. Demir,C. Cinar, On the solutions of the system of difference equations $x_{n+1}=\max\left\{ \frac{1}{x_{n}},\frac{y_{n}}{x_{n}}\right\} ,$ $y_{n+1}=\max\left\{ \frac{1}{y_{n}},\frac{x_{n}}{y_{n}}\right\} ,$; Discrete Dyn. Nat.Soc., 2009 (2009), Article ID 325296, 11 pages.
[40] J. L. Williams, On a class of nonlinear max-type difference equations, Cogent Math., 3 (2016), 1269597, 1-11.
[41] I. Yalcinkaya, C. Cinar, On the solutions of a systems of difference equations, Int. J. Math. Stat., 9 (S11) (2011), 62–67.
[42] Y. Yazlik, D. T. Tollu, N. Taskara, On the solutions of difference equation systems with Padovan numbers, Appl. Math., 4 (2013), 15-20.
[43] Y. Yazlik, D. T. Tollu, N. Taskara, On the solutions of a max-type difference equation system, Math. Meth. Appl. Sci., 38 (2015), 4388–4410.
[44] Q. Zhang, J. Liu, Z. Luo, Dynamical behavior of a system of third-order rational difference equation, Discrete Dyn. Nat. Soc., 2015 (2015), Article ID 530453, 6 pages.

Year 2018, Volume: 1 Issue: 3, 181 - 191, 30.12.2018

Marwa M. Alzubaidi , E. M. Elsayed

https://doi.org/10.33187/jmsm.427368

Cited By: 2

Abstract

References

[1] A. M. Ahmed, E. M. Elsayed, The expressions of solutions and the periodicity of some rational difference equations system, J. Appl. Math. Inform. 34(1-2) (2016), 35–48.
[2] M. M. El-Dessoky, The form of solutions and periodicity for some systems of third-order rational difference equations, Math. Meth. Appl. Sci., 39 (2016), 1076–1092.
[3] M. M. El-Dessoky, E. M. Elsayed, On the solutions and periodic nature of some systems of rational difference equations, J. Comput. Anal. Appl., 18(2) (2015), 206–218.
[4] M. M. El-Dessoky, A. Khaliq, A. Asiri, On some rational system of difference equations, J. Nonlinear Sci. Appl., 11 (2018), 49–72.
[5] E. M. Elsayed, T. F. Ibrahim, Periodicity and solutions for some systems of nonlinear rational difference equations, Hacettepe Journal of Mathematics and Statistics, 44 (6) (2015), 1361–1390.
[6] E. M. Elsayed, A. Alghamdi, The form of the solutions of nonlinear difference equations systems, J. Nonlinear Sci. Appl., 9 (2016), 3179–3196.
[7] N. Haddad , N. Touafek, J. F. T. Rabago, Solution form of a higher-order system of difference equations and dynamical behavior of its special case, Math. Methods Appl. Sci., 40 (2017), 3599–3607.
[8] A. S. Kurbanli, On the behavior of solutions of the system of rational difference equations $x_{n+1}=x_{n-1}/(y_{n}x_{n-1}-1),$; $y_{n+1}=y_{n-1}/(x_{n}y_{n-1}-1)$, World Appl. Sci. J., 10(11) (2010), 1344-1350.
[9] A. S. Kurbanli, C. C¸ inar, D. S¸ims¸ek, On the periodicity of solutions of the system of rational difference equations $x_{n+1}=x_{n-1}+y_{n}/(y_{n}x_{n-1}-1),$; $y_{n+1}=y_{n-1}+x_{n}/(x_{n}y_{n-1}-1)$, Appl. Math., 2 (2011), 410-413.
[10] A. S. Kurbanli, C. C¸ inar, I. Yalc¸inkaya, On the behavior of positive solutions of the system of rational difference equations $x_{n+1}=x_{n-1}/(y_{n}x_{n-1}+1),$; $y_{n+1}=y_{n-1}/(x_{n}y_{n-1}+1)$, Math. Comput. Modelling, 53 (2011), 1261–1267.
[11] M. Mansour, M. M. El-Dessoky, E. M. Elsayed, The form of the solutions and periodicity of some systems of difference equations, Discrete Dyn. Nat. Soc., 2012 (2012), Article ID 406821, 1-17.
[12] N. Touafek and E. M. Elsayed, On the solutions of systems of rational difference equations, Math. Comput. Modelling, 55 (2012), 1987-1997.
[13] S. A. Abramov, D. E. Khmelnov, Denominators of rational solutions of linear difference systems of an arbitrary order, Program. Comput. Softw., 38 (2) (2012), 84–91.
[14] R. Abu-Saris, C. Cinar, I. Yalcinkaya, On the Asymptotic Stability of $x_{n+1}=a+x_{n}x_{n-k}/x_{n}+x_{n-k},$; Comput. Math. Appl., 56(5) (2008), 1172-1175.
[15] N. Battaloglu, C. Cinar, I. Yalcinkaya, The dynamics of the difference equation $x_{n+1}=(\alpha x_{n-m})/(\beta+\gamma x_{n-(k+1)}^{p}),\ $; , Ars Combin., 97 (2010), 281–288.
[16] O. H. Criner, W. E. Taylor, J. L. Williams, On the solutions of a system of nonlinear difference equations, Int. J. Difference Equ., 10(2) (2015), 161–166.
[17] I. Dekkar, N. Touafek, Y. Yazlik, Global stability of a third-order nonlinear system of difference equations with period-two coefficients, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat., 111(2) (2017), 325–347.
[18] Q. Din, Qualitative nature of a discrete predator-prey system, Contem. Methods Math. Phys. Grav., 1 (2015), 27-42.
[19] E. M. Elabbasy, H. El-Metwally, E. M. Elsayed, Some properties and expressions of solutions for a class of nonlinear difference equation, Util. Math., 87 (2012), 93–110.
[20] E. M. Elabbasy, S. M. Eleissawy, Periodicty and stability of solutions of rational difference systems, Appl. Comput. Math., 1(4) (2012), 1-6.
[21] E. M. Elabbasy, A. A. Elsadany, S. Ibrahim, Behavior and periodic solutions of a two-dimensional systems of rational difference equations, J. Interpolat. Approx. Sci. Comput., 2016(2) (2016), 87-104.
[22] M. M. El-Dessoky, Solution for rational systems of difference equations of order three, Mathematics, 4(3) (2016), 1-12.
[23] M. M. El-Dessoky, E. M. Elsayed, E. O. Alzahrani, The form of solutions and periodic nature for some rational difference equations systems, J. Nonlinear Sci. Appl., 9 (2016), 5629-5647.
[24] E. M. Elsayed, Dynamics of recursive sequence of order two, Kyungpook Math. J., 50(4) (2010), 483-497.
[25] E. M. Elsayed, Solution and attractivity for a rational recursive sequence, Discrete Dyn. Nat. Soc., 2011 (2011), Article ID 982309, 17 pages.
[26] E. M. Elsayed, On the dynamics of a higher-order rational recursive sequence, Commun. Math. Anal., 12(1) (2012), 117–133.
[27] E. M. Elsayed, M. M. El-Dessoky, A. Alotaibi, On the solutions of a general system of difference equations, Discrete Dyn. Nat. Soc., 2012 (2012), Article ID 892571, 12 pages.
[28] M. Gumus, O. Ocalan, The qualitative analysis of a rational system of difference equations, J. Fract. Calc. Appl., 9(2) (2018), 113-126.
[29] N. Haddad, N. Touafek, E. M. Elsayed, A note on a system of difference equations, Analele Stiintifice ale universitatii al i cuza din iasi serie noua matematica, LXIII(3) (2017), 599-606.
[30] N. Haddad, N. Touafek, J. T. Rabago, Well-defined solutions of a system of difference equations, J. Appl. Math. Comput., 56 (1-2) (2018), 439–458.
[31] Y. Halim, N. Touafek, Y. Yazlik, Dynamic behavior of a second-order nonlinear rational difference equation, Turkish J. Math., 39 (2015), 1004-1018.
[32] Y. Halim, A system of difference equations with solutions associated to Fibonacci numbers, Int. J. Difference Equ., 11(1) (2016), 65–77.
[33] G. Hu, Global behavior of a system of two nonlinear difference equation, World J. Res. Rev., 2(6) (2016), 36-38.
[34] W. Ji, D. Zhang, L. Wang, Dynamics and behaviors of a third-order system of difference equation, Math. Sci., 7(34) (2013), 1-6.
[35] A. Khaliq, E. M. Elsayed, The dynamics and solution of some difference equations, J. Nonlinear Sci. Appl., 9 (2016), 1052-1063.
[36] A. S. Kurbanli, On the behavior of solutions of the system of rational difference equations $x_{n+1}=x_{n-1}/(y_{n}x_{n-1}-1),$% \emph{\ }$y_{n+1}=y_{n-1}/(x_{n}y_{n-1}-1),$; $z_{n+1}=1/y_{n}z_{n},$; Adv. Difference Equ., 2011 (2011), 40.
[37] A. S. Kurbanli, C. C¸ inar, M. E. Erdogan, On the behavior of solutions of the system of rational difference equations $x_{n+1}=x_{n-1}/(y_{n}x_{n-1}-1),$ \, $y_{n+1}=y_{n-1}/(x_{n}y_{n-1}-1), $\, $z_{n+1}=x_{n}/(y_{n}z_{n-1}),$; Appl. Math., 2 (2011), 1031-1038.
[38] A. S. Kurbanli, I. Yalcinkaya, A. Gelisken, On the behavior of the Solutions of the system of rational difference equations $x_{n+1}=x_{n-1}/(y_{n}x_{n-1}-1),$ $y_{n+1}=y_{n-1}/(x_{n}y_{n-1}-1), $\emph{\ }$z_{n+1}=x_{n}z_{n-1}/y_{n},$; Int. J. Phys. Sci., 8(2) (2013), 51-56.
[39] D. Simsek, B. Demir,C. Cinar, On the solutions of the system of difference equations $x_{n+1}=\max\left\{ \frac{1}{x_{n}},\frac{y_{n}}{x_{n}}\right\} ,$ $y_{n+1}=\max\left\{ \frac{1}{y_{n}},\frac{x_{n}}{y_{n}}\right\} ,$; Discrete Dyn. Nat.Soc., 2009 (2009), Article ID 325296, 11 pages.
[40] J. L. Williams, On a class of nonlinear max-type difference equations, Cogent Math., 3 (2016), 1269597, 1-11.
[41] I. Yalcinkaya, C. Cinar, On the solutions of a systems of difference equations, Int. J. Math. Stat., 9 (S11) (2011), 62–67.
[42] Y. Yazlik, D. T. Tollu, N. Taskara, On the solutions of difference equation systems with Padovan numbers, Appl. Math., 4 (2013), 15-20.
[43] Y. Yazlik, D. T. Tollu, N. Taskara, On the solutions of a max-type difference equation system, Math. Meth. Appl. Sci., 38 (2015), 4388–4410.
[44] Q. Zhang, J. Liu, Z. Luo, Dynamical behavior of a system of third-order rational difference equation, Discrete Dyn. Nat. Soc., 2015 (2015), Article ID 530453, 6 pages.

There are 44 citations in total.

Details

Primary Language	English
Subjects	Mathematical Sciences
Journal Section	Articles
Authors	Marwa M. Alzubaidi E. M. Elsayed
Publication Date	December 30, 2018
Submission Date	May 26, 2018
Acceptance Date	November 3, 2018
Published in Issue	Year 2018 Volume: 1 Issue: 3

Cite

APA	Alzubaidi, M. M., & Elsayed, E. M. (2018). The Form of the Solutions of System of Rational Difference Equation. Journal of Mathematical Sciences and Modelling, 1(3), 181-191. https://doi.org/10.33187/jmsm.427368
AMA	Alzubaidi MM, Elsayed EM. The Form of the Solutions of System of Rational Difference Equation. Journal of Mathematical Sciences and Modelling. December 2018;1(3):181-191. doi:10.33187/jmsm.427368
Chicago	Alzubaidi, Marwa M., and E. M. Elsayed. “The Form of the Solutions of System of Rational Difference Equation”. Journal of Mathematical Sciences and Modelling 1, no. 3 (December 2018): 181-91. https://doi.org/10.33187/jmsm.427368.
EndNote	Alzubaidi MM, Elsayed EM (December 1, 2018) The Form of the Solutions of System of Rational Difference Equation. Journal of Mathematical Sciences and Modelling 1 3 181–191.
IEEE	M. M. Alzubaidi and E. M. Elsayed, “The Form of the Solutions of System of Rational Difference Equation”, Journal of Mathematical Sciences and Modelling, vol. 1, no. 3, pp. 181–191, 2018, doi: 10.33187/jmsm.427368.
ISNAD	Alzubaidi, Marwa M. - Elsayed, E. M. “The Form of the Solutions of System of Rational Difference Equation”. Journal of Mathematical Sciences and Modelling 1/3 (December 2018), 181-191. https://doi.org/10.33187/jmsm.427368.
JAMA	Alzubaidi MM, Elsayed EM. The Form of the Solutions of System of Rational Difference Equation. Journal of Mathematical Sciences and Modelling. 2018;1:181–191.
MLA	Alzubaidi, Marwa M. and E. M. Elsayed. “The Form of the Solutions of System of Rational Difference Equation”. Journal of Mathematical Sciences and Modelling, vol. 1, no. 3, 2018, pp. 181-9, doi:10.33187/jmsm.427368.
Vancouver	Alzubaidi MM, Elsayed EM. The Form of the Solutions of System of Rational Difference Equation. Journal of Mathematical Sciences and Modelling. 2018;1(3):181-9.