doi: 10.3934/era.2021029

Global behavior of P-dimensional difference equations system

1. 

Department of Mathematics and LMAM laboratory, Mohamed Seddik Ben Yahia University, BP 98 Ouled Aissa 18000, Jijel, Algeria

2. 

Department of Mathematics and Computer Sciences, Abdelhafid Boussouf University Center, RP 26 Mila 43000, Mila, Algeria

* Corresponding author: Yacine Halim

Received  January 2021 Revised  March 2021 Published  April 2021

The global asymptotic stability of the unique positive equilibrium point and the rate of convergence of positive solutions of the system of two recursive sequences has been studied recently. Here we generalize this study to the system of $ p $ recursive sequences $x_{n+1}^{(j)}=A+\left(x_{n-m}^{(j+1) mod (p)} \;\;/ x_{n}^{(j+1) mod (p)}\;\;\;\right) $, $ n = 0,1,\ldots, $ $ m,p\in \mathbb{N} $, where $ A\in(0,+\infty) $, $ x_{-i}^{(j)} $ are arbitrary positive numbers for $ i = 1,2,\ldots,m $ and $ j = 1,2,\ldots,p. $ We also give some numerical examples to demonstrate the effectiveness of the results obtained.

Citation: Amira Khelifa, Yacine Halim. Global behavior of P-dimensional difference equations system. Electronic Research Archive, doi: 10.3934/era.2021029
References:
[1]

I. DekkarN. Touafek and Y. Yazlik, Global stability of a third-order nonlinear system of difference equations with period-two coefficients, Rev. R. Acad. Cienc. Exactas Fis. Nat., Ser. A Mat., 111 (2017), 325-347.  doi: 10.1007/s13398-016-0297-z.  Google Scholar

[2]

R. DevaultC. Kent and W. Kosmala, On the recursive sequence $x_{n+1} = p+(x_{n-k}/x_{n})$, J. Difference Equ. Appl., 9 (2003), 721-730.   Google Scholar

[3]

S. N. Elaydi, An Introduction to Difference Equations, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4757-9168-6.  Google Scholar

[4]

E. M. Elsayed, On a system of two nonlinear difference equations of order two, Proc. Jangeon Math. Soc., 18 (2015), 353-368.   Google Scholar

[5]

E. M. Elsayed, Solutions of rational difference systems of order two, Math. Comput. Modelling, 55 (2012), 378-384.  doi: 10.1016/j.mcm.2011.08.012.  Google Scholar

[6]

E. M. Elsayed and T. F. Ibrahim, Periodicity and solutions for some systems of nonlinear rational difference equations, Hacet. J. Math. Stat., 44 (2015), 1361-1390.   Google Scholar

[7] E. A. Grove and G. Ladas, Periodicities in Nonlinear Difference Equations, Chapman and Hall/CRC Press, Boca Raton, FL, 2005.   Google Scholar
[8]

M. Gümüş, The global asymptotic stability of a system of difference equations, J. Difference Equ. Appl., 24 (2018), 976-991.  doi: 10.1080/10236198.2018.1443445.  Google Scholar

[9]

Y. Halim, Global character of systems of rational difference equations, Electron. J. Math. Anal. Appl., 3 (2015), 204-214.   Google Scholar

[10]

Y. Halim and M. Bayram, On the solutions of a higher-order difference equation in terms of generalized Fibonacci sequences, Math. Methods. Appl. Sci., 39 (2016), 2974-2982.  doi: 10.1002/mma.3745.  Google Scholar

[11]

Y. HalimN. Touafek and Y. Yazlik, Dynamic behavior of a second-order nonlinear rational difference equation, Turkish J. Math., 39 (2015), 1004-1018.  doi: 10.3906/mat-1503-80.  Google Scholar

[12]

M. Kara and Y. Yazlik, Solvability of a system of nonlinear difference equations of higher order, Turk. J. Math., 43 (2019), 1533-1565.  doi: 10.3906/mat-1902-24.  Google Scholar

[13]

V. L. Kocić and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers Group, Dordrecht, 1993. doi: 10.1007/978-94-017-1703-8.  Google Scholar

[14]

G. Papaschinopoulos and C. J. Schinas, On the behavior of the solutions of a system of two nonlinear difference equations, Commun. Appl. Nonlinear Anal., 5 (1998), 47-59.   Google Scholar

[15]

G. Papaschinopoulos and C. J. Schinas, Oscillation and asymptotic stability of two systems of difference equations of rational form, J. Differ. Equations Appl., 7 (2001), 601-617.  doi: 10.1080/10236190108808290.  Google Scholar

[16]

M. Pituk, More on Poincaré's and Perron's theorems for difference equations, J. Difference Equ. Appl., 8 (2002), 201-216.  doi: 10.1080/10236190211954.  Google Scholar

[17]

D. T. TolluY. Yazlik and N. Taşkara, Behavior of positive solutions of a difference equation, J. Appl. Math. Inform., 35 (2017), 217-230.  doi: 10.14317/jami.2017.217.  Google Scholar

[18]

D. T. TolluY. Yazlik and N. Taşkara, On a solvable nonlinear difference equation of higher order, Turkish J. Math., 42 (2018), 1765-1778.  doi: 10.3906/mat-1705-33.  Google Scholar

[19]

Y. YazlikD. T. Tollu and N. Taskara, On the behaviour of solutions for some systems of difference equations, J. Comput. Anal. Appl., 18 (2015), 166-178.   Google Scholar

[20]

D. ZhangW. JiL. Wang and X. Li, On the symmetrical system of rational difference equation $x_{n+1}=A+y_{n-k}/y_{n}, y_{n+1}=A+x_{n-k}/x_{n}$, Appl. Math., 4 (2013), 834-837.   Google Scholar

show all references

References:
[1]

I. DekkarN. Touafek and Y. Yazlik, Global stability of a third-order nonlinear system of difference equations with period-two coefficients, Rev. R. Acad. Cienc. Exactas Fis. Nat., Ser. A Mat., 111 (2017), 325-347.  doi: 10.1007/s13398-016-0297-z.  Google Scholar

[2]

R. DevaultC. Kent and W. Kosmala, On the recursive sequence $x_{n+1} = p+(x_{n-k}/x_{n})$, J. Difference Equ. Appl., 9 (2003), 721-730.   Google Scholar

[3]

S. N. Elaydi, An Introduction to Difference Equations, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4757-9168-6.  Google Scholar

[4]

E. M. Elsayed, On a system of two nonlinear difference equations of order two, Proc. Jangeon Math. Soc., 18 (2015), 353-368.   Google Scholar

[5]

E. M. Elsayed, Solutions of rational difference systems of order two, Math. Comput. Modelling, 55 (2012), 378-384.  doi: 10.1016/j.mcm.2011.08.012.  Google Scholar

[6]

E. M. Elsayed and T. F. Ibrahim, Periodicity and solutions for some systems of nonlinear rational difference equations, Hacet. J. Math. Stat., 44 (2015), 1361-1390.   Google Scholar

[7] E. A. Grove and G. Ladas, Periodicities in Nonlinear Difference Equations, Chapman and Hall/CRC Press, Boca Raton, FL, 2005.   Google Scholar
[8]

M. Gümüş, The global asymptotic stability of a system of difference equations, J. Difference Equ. Appl., 24 (2018), 976-991.  doi: 10.1080/10236198.2018.1443445.  Google Scholar

[9]

Y. Halim, Global character of systems of rational difference equations, Electron. J. Math. Anal. Appl., 3 (2015), 204-214.   Google Scholar

[10]

Y. Halim and M. Bayram, On the solutions of a higher-order difference equation in terms of generalized Fibonacci sequences, Math. Methods. Appl. Sci., 39 (2016), 2974-2982.  doi: 10.1002/mma.3745.  Google Scholar

[11]

Y. HalimN. Touafek and Y. Yazlik, Dynamic behavior of a second-order nonlinear rational difference equation, Turkish J. Math., 39 (2015), 1004-1018.  doi: 10.3906/mat-1503-80.  Google Scholar

[12]

M. Kara and Y. Yazlik, Solvability of a system of nonlinear difference equations of higher order, Turk. J. Math., 43 (2019), 1533-1565.  doi: 10.3906/mat-1902-24.  Google Scholar

[13]

V. L. Kocić and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers Group, Dordrecht, 1993. doi: 10.1007/978-94-017-1703-8.  Google Scholar

[14]

G. Papaschinopoulos and C. J. Schinas, On the behavior of the solutions of a system of two nonlinear difference equations, Commun. Appl. Nonlinear Anal., 5 (1998), 47-59.   Google Scholar

[15]

G. Papaschinopoulos and C. J. Schinas, Oscillation and asymptotic stability of two systems of difference equations of rational form, J. Differ. Equations Appl., 7 (2001), 601-617.  doi: 10.1080/10236190108808290.  Google Scholar

[16]

M. Pituk, More on Poincaré's and Perron's theorems for difference equations, J. Difference Equ. Appl., 8 (2002), 201-216.  doi: 10.1080/10236190211954.  Google Scholar

[17]

D. T. TolluY. Yazlik and N. Taşkara, Behavior of positive solutions of a difference equation, J. Appl. Math. Inform., 35 (2017), 217-230.  doi: 10.14317/jami.2017.217.  Google Scholar

[18]

D. T. TolluY. Yazlik and N. Taşkara, On a solvable nonlinear difference equation of higher order, Turkish J. Math., 42 (2018), 1765-1778.  doi: 10.3906/mat-1705-33.  Google Scholar

[19]

Y. YazlikD. T. Tollu and N. Taskara, On the behaviour of solutions for some systems of difference equations, J. Comput. Anal. Appl., 18 (2015), 166-178.   Google Scholar

[20]

D. ZhangW. JiL. Wang and X. Li, On the symmetrical system of rational difference equation $x_{n+1}=A+y_{n-k}/y_{n}, y_{n+1}=A+x_{n-k}/x_{n}$, Appl. Math., 4 (2013), 834-837.   Google Scholar

Figure 1.  The plot of system (23) with $ A = 1.2>1 $
Figure 2.  The plot of system (23) with $ A = 1 $
Figure 3.  The plot of system (23) with $ A = 0.9<1 $
Figure 4.  The plot of system (24) with $ A = 1.4>1 $
Figure 5.  The plot of system (24) with $ A = 1 $
Figure 6.  The plot of system (24) with $ A = 0.7<1 $
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