# American Institute of Mathematical Sciences

• Previous Article
A multigrid based finite difference method for solving parabolic interface problem
• ERA Home
• This Issue
• Next Article
Three types of weak pullback attractors for lattice pseudo-parabolic equations driven by locally Lipschitz noise
November  2021, 29(5): 3121-3139. 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  November 2021 Early access  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, 2021, 29 (5) : 3121-3139. doi: 10.3934/era.2021029
##### References:
 [1] I. Dekkar, N. 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. Devault, C. 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. Halim, N. 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. Tollu, Y. 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. Tollu, Y. 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. Yazlik, D. 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. Zhang, W. Ji, L. 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. Dekkar, N. 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. Devault, C. 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. Halim, N. 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. Tollu, Y. 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. Tollu, Y. 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. Yazlik, D. 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. Zhang, W. Ji, L. 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
The plot of system (23) with $A = 1.2>1$
The plot of system (23) with $A = 1$
The plot of system (23) with $A = 0.9<1$
The plot of system (24) with $A = 1.4>1$
The plot of system (24) with $A = 1$
The plot of system (24) with $A = 0.7<1$
 [1] Zigen Ouyang, Chunhua Ou. Global stability and convergence rate of traveling waves for a nonlocal model in periodic media. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 993-1007. doi: 10.3934/dcdsb.2012.17.993 [2] Fuchen Zhang, Xiaofeng Liao, Chunlai Mu, Guangyun Zhang, Yi-An Chen. On global boundedness of the Chen system. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1673-1681. doi: 10.3934/dcdsb.2017080 [3] Alexandra Rodkina, Henri Schurz. On positivity and boundedness of solutions of nonlinear stochastic difference equations. Conference Publications, 2009, 2009 (Special) : 640-649. doi: 10.3934/proc.2009.2009.640 [4] Christian Lax, Sebastian Walcher. A note on global asymptotic stability of nonautonomous master equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2143-2149. doi: 10.3934/dcdsb.2013.18.2143 [5] Xu Pan, Liangchen Wang. Boundedness and asymptotic stability in a quasilinear two-species chemotaxis system with nonlinear signal production. Communications on Pure & Applied Analysis, 2021, 20 (6) : 2211-2236. doi: 10.3934/cpaa.2021064 [6] Tran Hong Thai, Nguyen Anh Dai, Pham Tuan Anh. Global dynamics of some system of second-order difference equations. Electronic Research Archive, 2021, 29 (6) : 4159-4175. doi: 10.3934/era.2021077 [7] Wei Mao, Liangjian Hu, Xuerong Mao. Asymptotic boundedness and stability of solutions to hybrid stochastic differential equations with jumps and the Euler-Maruyama approximation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 587-613. doi: 10.3934/dcdsb.2018198 [8] Michela Eleuteri, Pavel Krejčí. An asymptotic convergence result for a system of partial differential equations with hysteresis. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1131-1143. doi: 10.3934/cpaa.2007.6.1131 [9] Yulan Lu, Minghui Song, Mingzhu Liu. Convergence rate and stability of the split-step theta method for stochastic differential equations with piecewise continuous arguments. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 695-717. doi: 10.3934/dcdsb.2018203 [10] Gregory Berkolaiko, Cónall Kelly, Alexandra Rodkina. Sharp pathwise asymptotic stability criteria for planar systems of linear stochastic difference equations. Conference Publications, 2011, 2011 (Special) : 163-173. doi: 10.3934/proc.2011.2011.163 [11] Anna Cima, Armengol Gasull, Francesc Mañosas. Global linearization of periodic difference equations. Discrete & Continuous Dynamical Systems, 2012, 32 (5) : 1575-1595. doi: 10.3934/dcds.2012.32.1575 [12] Marcel Freitag. Global existence and boundedness in a chemorepulsion system with superlinear diffusion. Discrete & Continuous Dynamical Systems, 2018, 38 (11) : 5943-5961. doi: 10.3934/dcds.2018258 [13] Shiwang Ma, Xiao-Qiang Zhao. Global asymptotic stability of minimal fronts in monostable lattice equations. Discrete & Continuous Dynamical Systems, 2008, 21 (1) : 259-275. doi: 10.3934/dcds.2008.21.259 [14] Anatoli F. Ivanov, Musa A. Mammadov. Global asymptotic stability in a class of nonlinear differential delay equations. Conference Publications, 2011, 2011 (Special) : 727-736. doi: 10.3934/proc.2011.2011.727 [15] Cemil Tunç. Stability, boundedness and uniform boundedness of solutions of nonlinear delay differential equations. Conference Publications, 2011, 2011 (Special) : 1395-1403. doi: 10.3934/proc.2011.2011.1395 [16] Haibo Cui, Zhensheng Gao, Haiyan Yin, Peixing Zhang. Stationary waves to the two-fluid non-isentropic Navier-Stokes-Poisson system in a half line: Existence, stability and convergence rate. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 4839-4870. doi: 10.3934/dcds.2016009 [17] Yueh-Cheng Kuo, Huey-Er Lin, Shih-Feng Shieh. Asymptotic dynamics of hermitian Riccati difference equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 2037-2053. doi: 10.3934/dcdsb.2020365 [18] Andrejs Reinfelds, Klara Janglajew. Reduction principle in the theory of stability of difference equations. Conference Publications, 2007, 2007 (Special) : 864-874. doi: 10.3934/proc.2007.2007.864 [19] Tobias Black. Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1253-1272. doi: 10.3934/dcdsb.2017061 [20] Mengyao Ding, Wei Wang. Global boundedness in a quasilinear fully parabolic chemotaxis system with indirect signal production. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4665-4684. doi: 10.3934/dcdsb.2018328

2020 Impact Factor: 1.833