Theoretical and numerical analysis of solutions of some systems of nonlinear difference equations

E. M. Elsayed; Q. Din; N. A. Bukhary; E. M. Elsayed; Q. Din; N. A. Bukhary

doi:10.3934/math.2022851

AIMS Mathematics

2022, Volume 7, Issue 8: 15532-15549. doi: 10.3934/math.2022851

Previous Article Next Article

Research article Special Issues

Theoretical and numerical analysis of solutions of some systems of nonlinear difference equations

1.
King Abdulaziz University, Faculty of Science, Mathematics Department, P.O. Box 80203, Jeddah 21589, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
3.
Department of Mathematics, University of Poonch Rawalakot, Rawalakot 12350, Pakistan
4.
Mathematics Department, Majmaah University, Al Majmaah 15341, Saudi Arabia

Received: 19 April 2022 Revised: 05 June 2022 Accepted: 09 June 2022 Published: 22 June 2022
MSC : 39A10

In this paper, we obtain the form of the solutions of the following rational systems of difference equations

$ \begin{equation*} x_{n+1} = \dfrac{y_{n-1}z_{n}}{z_{n}\pm x_{n-2}}, \;y_{n+1} = \dfrac{z_{n-1}x_{n} }{x_{n}\pm y_{n-2}}, \ z_{n+1} = \dfrac{x_{n-1}y_{n}}{y_{n}\pm z_{n-2}}, \end{equation*} $

with initial values are non-zero real numbers.
- difference equations,
- periodic solutions,
- stability,
- recursive sequences,
- system of difference equations
Citation: E. M. Elsayed, Q. Din, N. A. Bukhary. Theoretical and numerical analysis of solutions of some systems of nonlinear difference equations[J]. AIMS Mathematics, 2022, 7(8): 15532-15549. doi: 10.3934/math.2022851

Related Papers:

Abstract

In this paper, we obtain the form of the solutions of the following rational systems of difference equations

$ \begin{equation*} x_{n+1} = \dfrac{y_{n-1}z_{n}}{z_{n}\pm x_{n-2}}, \;y_{n+1} = \dfrac{z_{n-1}x_{n} }{x_{n}\pm y_{n-2}}, \ z_{n+1} = \dfrac{x_{n-1}y_{n}}{y_{n}\pm z_{n-2}}, \end{equation*} $

with initial values are non-zero real numbers.

References

[1]	R. Abo-Zeid, On the solutions of a fourth order difference equation, Univ. J. Math. Appl., 4 (2021), 76–81. https://doi.org/10.1017/S0040298221000693 doi: 10.1017/S0040298221000693
[2]	Y. Akrour, N. Touafek, Y. Halim, On a system of difference equations of third order solved in closed form, J. Innov. Appl. Math. Comput. Sci., 1 (2021), 1–15. https://doi.org/10.48550/arXiv.1910.14365 doi: 10.48550/arXiv.1910.14365
[3]	M. B. Almatrafi, Analysis of solutions of some discrete systems of rational difference equations, J. Comput. Anal. Appl., 29 (2021), 355–368.
[4]	A. M. Alotaibi, M. A. El-Moneam, On the dynamics of the nonlinear rational difference equation ${ x_{n+1}} = \frac{{\alpha {x_{n-m}}}\ \ \ +\delta {{x_{n}}}}{{\beta +\gamma {x_{n-k}}\ \ \ { x_{n-l}}\ \ \ \ \left({{x_{n-k}}\ \ \ \ +{x_{n-l}}}\ \ \ \ \ \ \right) }}$, AIMS Math., 7 (2022), 7374–7384. https://doi.org/10.3934/math.2022411 doi: 10.3934/math.2022411
[5]	N. Battaloglu, C. Cinar, I. Yalçınkaya, The dynamics of the difference equation, ARS Combinatoria, 97 (2010), 281–288.
[6]	C. Cinar, I. Yalcinkaya, R. Karatas, On the positive solutions of the difference equation system $x_{n+1} = m/y_{n}, \; y_{n+1} = py_{n}/x_{n-1}y_{n-1}$, J. Inst. Math. Comp. Sci., 18 (2005), 135–136.
[7]	C. Cinar, I. Yalçinkaya, 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. Comp. Sci., 18 (2005), 91–93.
[8]	S. E. Das, M. Bayram, On a system of rational difference equations, World Appl. Sci. J., 10 (2010), 1306–1312.
[9]	E. M. Elabbasy, H. El-Metwally, E. M. Elsayed, On the solutions of a class of difference equations systems, Demonstr. Math., 41 (2008), 109–122. https://doi.org/10.1515/dema-2008-0111 doi: 10.1515/dema-2008-0111
[10]	E. M. Elsayed, Solution and attractivity for a rational recursive sequence, Discrete Dyn. Nat. Soc., 2011 (2011). https://doi.org/10.1155/2011/982309 doi: 10.1155/2011/982309
[11]	E. M. Elsayed, Solutions of rational difference system of order two, Math. Comput. Model., 55 (2012), 378–384. https://doi.org/10.1016/j.mcm.2011.08.012 doi: 10.1016/j.mcm.2011.08.012
[12]	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). https://doi.org/10.1155/2012/892571 doi: 10.1155/2012/892571
[13]	E. M. Elsayed, A. Alshareef, Qualitative behavior of a system of second order difference equations, Eur. J. Math. Appl., 1 (2021), 1–11.
[14]	E. M. Elsayed, B. S. Alofi, A. Q. Khan, Qualitative behavior of solutions of tenth-order recursive sequence equation, Math. Probl. Eng., 2022 (2022). https://doi.org/10.1155/2022/5242325 doi: 10.1155/2022/5242325
[15]	E. M. Elsayed, F. Alzahrani, Periodicity and solutions of some rational difference equations systems, J. Appl. Anal. Comput., 9 (2019), 2358–2380. https://doi.org/10.11948/20190100 doi: 10.11948/20190100
[16]	T. F. Ibrahim, A. Q. Khan, A. Ibrahim, Qualitative behavior of a nonlinear generalized recursive sequence with delay, Math. Probl. Eng., 2021 (2021). https://doi.org/10.1155/2021/6162320 doi: 10.1155/2021/6162320
[17]	T. F. Ibrahim, A. Q. Khan, Forms of solutions for some two-dimensional systems of rational partial recursion equations, Math. Probl. Eng., 2021 (2021). https://doi.org/10.1155/2021/9966197 doi: 10.1155/2021/9966197
[18]	M. Kara, Y. Yazlik, On a solvable three-dimensional system of difference equations, Filomat, 34 (2020), 1167–1186. https://doi.org/10.2298/FIL2004167K doi: 10.2298/FIL2004167K
[19]	K. Y. Liu, Z. J. Zhao, X. R. Li, P. Li, More on three-dimensional systems of rational difference equations, Discrete Dyn. Nat. Soc., 2011 (2011). https://doi.org/10.1155/2011/178483 doi: 10.1155/2011/178483
[20]	A. Khaliq, M. Shoaib, Dynamics of three-dimensional system of second order rational difference equations, Electron. J. Math. Anal. Appl., 9 (2021), 308–319.
[21]	A. Khelifa, Y. Halim, M. Berkal, Solutions of a system of two higher-order difference equations in terms of Lucas sequence, Univ. J. Math. Appl., 2 (2019), 202–211. https://doi.org/10.32323/ujma.610399 doi: 10.32323/ujma.610399
[22]	A. Khelifa, Y. Halim, Global behavior of P-dimensional difference equations system, Electron. Res. Arch., 29 (2021), 3121–3139. https://doi.org/10.3934/era.2021029 doi: 10.3934/era.2021029
[23]	A. S. Kurbanli, C. Cinar, I. Yalçınkaya, On the behavior of positive solutions of the system of rational difference equations, Math. Comput. Model., 53 (2011), 1261–1267. https://doi.org/10.1016/j.mcm.2010.12.009 doi: 10.1016/j.mcm.2010.12.009
[24]	A. S. Kurbanli, On the behavior of solutions of the system of rational difference equations, Adv. Differ. Equ., 2011 (2011), 40. https://doi.org/10.1186/1687-1847-2011-40 doi: 10.1186/1687-1847-2011-40
[25]	A. S. Kurbanli, On the behavior of solutions of the system of rational difference equations: $x_{n+1} = x_{n-1}/x_{n-1}y_{n}-1, \; y_{n+1} = y_{n-1}/y_{n-1}x_{n}-1, z_{n+1} = z_{n-1}/z_{n-1}y_{n}-1$, Discrete Dyn. Nat. Soc., 2011 (2011). https://doi.org/10.1186/1687-1847-2011-40 doi: 10.1186/1687-1847-2011-40
[26]	A. Kurbanli, C. Cinar, M. Erdoğan, On the behavior of solutions of the system of rational difference equations $x_{n+1} = \dfrac{ x_{n-1}}{x_{n-1}y_{n}-1}, \; y_{n+1} = \dfrac{y_{n-1}}{y_{n-1}x_{n}-1}, z_{n+1} = \dfrac{x_{n}}{z_{n-1}y_{n}}$, Appl. Math., 2 (2011), 1031–1038.
[27]	B. Oğul, D. Şimşek, On the recursive sequence $ x_{n+1} = x_{n-14}/1+x_{n-2}x_{n-5}x_{n-8}x_{n-11}$, MANAS J. Eng., 8 (2020), 155–163. https://hdl.handle.net/20.500.13091/1672
[28]	A. Y. Ozban, On the system of rational difference equations $ x_{n+1} = a/y_{n-3}, \; y_{n+1} = by_{n-3}/x_{n-q}y_{n-q}$, Appl. Math. Comput., 188 (2007), 833–837. https://doi.org/10.1016/j.amc.2006.10.034 doi: 10.1016/j.amc.2006.10.034
[29]	D. Tollu, İ. Yalçınkaya, H. Ahmad, S. Yao, A detailed study on a solvable system related to the linear fractional difference equation, Math. Biosci. Eng., 18 (2021), 5392–5408. https://doi.org/10.3934/mbe.2021273 doi: 10.3934/mbe.2021273
[30]	N. Touafek, E. M. Elsayed, On the solutions of systems of rational difference equations, Math. Comput. Model., 55 (2012), 1987–1997. https://doi.org/10.1016/j.mcm.2011.11.058 doi: 10.1016/j.mcm.2011.11.058
[31]	N. Touafek, D. Tollu, Y. Akrour, On a general homogeneous three-dimensional system of difference equations, Electron. Res. Arch., 29 (2021), 2841–2876. https://doi.org/10.3934/era.2021017 doi: 10.3934/era.2021017
[32]	I. Yalcinkaya, On the global asymptotic stability of a second-order system of difference equations, Discrete Dyn. Nat. Soc., 2008 (2008). https://doi.org/10.1155/2008/860152 doi: 10.1155/2008/860152
[33]	I. Yalçınkaya, On the global asymptotic behavior of a system of two nonlinear difference equations, ARS Combinatoria, 95 (2010), 151–159. https://doi.org/10.1016/j.ygeno.2009.12.003 doi: 10.1016/j.ygeno.2009.12.003
[34]	I. Yalçınkaya, C. Cinar, D. Simsek, Global asymptotic stability of a system of difference equations, Appl. Anal., 87 (2008), 689–699. https://doi.org/10.1097/PHM.0b013e31817e4b84 doi: 10.1097/PHM.0b013e31817e4b84
[35]	I. Yalcinkaya, C. Cinar, Global asymptotic stability of two nonlinear difference equations, Fasciculi Math., 43 (2010), 171–180.
[36]	I. Yalcinkaya, C. Cinar, M. Atalay, On the solutions of systems of difference equations, Adv. Differ. Equ., 2008 (2008). https://doi.org/10.1155/2008/143943 doi: 10.1155/2008/143943
[37]	X. Yang, Y. Liu, S. Bai, On the system of high order rational difference equations $x_{n} = a/y_{n-p}, \; y_{n} = by_{n-p}/x_{n-q}y_{n-q}$, Appl. Math. Comput., 171 (2005), 853–856. https://doi.org/10.1016/j.amc.2005.01.092 doi: 10.1016/j.amc.2005.01.092
[38]	Y. Yazlik, D. T. Tollu, N. Taskara, On the behavior of solutions for some systems of difference equations, J. Comput. Anal. Appl., 18 (2015), 166–178.
[39]	Y. Zhang, X. Yang, G. M. Megson, D. J. Evans, On the system of rational difference equations, Appl. Math. Comput., 176 (2006), 403–408. https://doi.org/10.1016/j.amc.2005.09.039 doi: 10.1016/j.amc.2005.09.039

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)