On the global behavior of the rational difference equation \(y_{n+1}=\frac{\alpha_n+y_{n-r}}{\alpha_n+y_{n-k}}\)

Abstract

In this article, we study the global behavior of the following higher-order nonautonomous rational difference equation \[ y_{n+1}=\frac{\alpha_n+y_{n-r}}{\alpha_n+y_{n-k}},\quad n=0,1,..., \] where \(\left\{\alpha_n\right\}_{n\geq0}\) is a bounded sequence of positive numbers, \(k,r\) are nonnegative integers such that \(r

Keywords

• Rational difference equation
• oscillation
• local stability
• global attractivity

Abstract

References

1. [1] R. Abo-Zeid. Global behavior and oscillation of a third order difference equation. Quaest. Math., 2020: 1–20.
2. [2] A. Alshareef, F. Alzahrani, and A. Q. Khan. Dynamics and Solutions’ Expressions of a Higher-Order Nonlinear Fractional Recursive Sequence. Math. Probl. Eng., 2021.
3. [3] M. Berkal and J. F. Navarro. Qualitative behavior of a two-dimensional discrete-time prey–predator model. Comp and Math Methods. 3( 6):e1193, 2021.
4. [4] E. Camouzis. Global convergence in periodically forced rational difference equations. J. Difference Equ. Appl., vol(14), Nos. 10-11, 1011-1033, 2008.
5. [5] E. Camouzis and S. Kotsios. May’s Host–Parasitoid geometric series model with a variable coefficient. Results Appl. Math. 11(2021), Article ID 100160, 5 p.
6. [6] I. Dekkar, N. Touafek, and Q. Din. On the global dynamics of a rational difference equation with periodic coefficients. J. Appl. Math. Comput., 60(1):567–588, 2019.
7. [7] M. J. Douraki and J. Mashreghi. On the population model of the non-autonomous logistic equation of second order with period-two parameters. J. Difference Equ. Appl., Vol. 14, No. 3, March 2008, 231-257.
8. [8] S. Elaydi. An Introduction to Difference Equations, Undergraduate Texts in Mathematics. Springer, New York, 2005.

9. [9] M. Gümüs. The periodic character in a higher order difference equation with delays. Math. Meth. Appl. Sci., 43(3): 1112-1123, 2020.
10. [10] M. Gümüs and R. Abo-Zeid. Global behavior of a rational second order difference equation. J. Appl. Math. Comput., 62(1):119–133, 2020.
11. [11] M.A. Kerker and A. Bouaziz. On the global behavior of a higher-order nonautonomous rational difference equation. Electron. J. Math. Anal. Appl., 9(1), 302-309.
12. [12] M.A. Kerker, E. Hadidi, and A. Salmi. On the dynamics of a nonautonomous rational difference equation. Int. J. Nonlinear Anal. Appl., 12:15-26, 2021.
13. [13] A. Q. Khan and K. Sharif. Global dynamics, forbidden set, and transcritical bifurcation of a one-dimensional discrete-time laser model. Math. Meth. Appl. Sci., 2020;1-13.
14. [14] V. L. Kocic, G. Ladas, and L. W. Rodrigues. On rational recursive sequences. J. Math. Anal. Appl., 173:127–157, 1993.
15. [15] V. Lakshmikantham and D. Trigiante. Theory of Difference Equations, Numerical Methods and Applications. Marcel Dekker, Inc., New York, 2002.
16. [16] E. Liz. Stability of non-autonomous difference equations: simple ideas leading to useful results. J. Difference Equ. Appl., Vol. 17, No. 2, February 2011, 203-220.
17. [17] O. Öcalan, H. Ogünmez, M. Gümüs.Global behavior test for a nonlinear difference equation with a period-two coefficient. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 21 (2014) 307-316.
18. [18] S. Stevic. Solving a class of nonautonomous difference equations by generalized invariants Math. Methods Appl. Sci. 42, No. 18, 6315–6338 (2019).
19. [19] A. Yildirim and D. T. Tollu. Global behavior of a second order difference equation with two-period coefficient. J. Math. Ext., Vol. 16, No. 4, (2022) (1)1-21.

Details

Primary Language

English

Subjects

Mathematical Sciences

Journal Section

Research Article

Authors

Sihem Oudina This is me
0000-0003-2937-3567
Algeria

Mohamed Amine Kerker ^*
Algeria

Abdelouahab Salmi
0000-0001-9607-970X
Algeria

Publication Date

September 30, 2022

Submission Date

July 24, 2021

Acceptance Date

July 28, 2022

Published in Issue

Year 2022 Volume: 5 Number: 3

DOI

https://doi.org/10.53006/rna.974156

IZ

https://izlik.org/JA98MD82XW

Cite

RIS / Bibtex

APA

Oudina, S., Kerker, M. A., & Salmi, A. (2022). On the global behavior of the rational difference equation \(y_{n+1}=\frac{\alpha_n+y_{n-r}}{\alpha_n+y_{n-k}}\). Results in Nonlinear Analysis, 5(3), 312-324. https://doi.org/10.53006/rna.974156

AMA

1.Oudina S, Kerker MA, Salmi A. On the global behavior of the rational difference equation \(y_{n+1}=\frac{\alpha_n+y_{n-r}}{\alpha_n+y_{n-k}}\). RNA. 2022;5(3):312-324. doi:10.53006/rna.974156

Chicago

Oudina, Sihem, Mohamed Amine Kerker, and Abdelouahab Salmi. 2022. “On the Global Behavior of the Rational Difference Equation \(y_{n+1}=\frac{\alpha_n+y_{n-R}}{\alpha_n+y_{n-K}}\)”. Results in Nonlinear Analysis 5 (3): 312-24. https://doi.org/10.53006/rna.974156.

EndNote

Oudina S, Kerker MA, Salmi A (September 1, 2022) On the global behavior of the rational difference equation \(y_{n+1}=\frac{\alpha_n+y_{n-r}}{\alpha_n+y_{n-k}}\). Results in Nonlinear Analysis 5 3 312–324.

IEEE

[1]S. Oudina, M. A. Kerker, and A. Salmi, “On the global behavior of the rational difference equation \(y_{n+1}=\frac{\alpha_n+y_{n-r}}{\alpha_n+y_{n-k}}\)”, RNA, vol. 5, no. 3, pp. 312–324, Sept. 2022, doi: 10.53006/rna.974156.

ISNAD

Oudina, Sihem - Kerker, Mohamed Amine - Salmi, Abdelouahab. “On the Global Behavior of the Rational Difference Equation \(y_{n+1}=\frac{\alpha_n+y_{n-R}}{\alpha_n+y_{n-K}}\)”. Results in Nonlinear Analysis 5/3 (September 1, 2022): 312-324. https://doi.org/10.53006/rna.974156.

JAMA

1.Oudina S, Kerker MA, Salmi A. On the global behavior of the rational difference equation \(y_{n+1}=\frac{\alpha_n+y_{n-r}}{\alpha_n+y_{n-k}}\). RNA. 2022;5:312–324.

MLA

Oudina, Sihem, et al. “On the Global Behavior of the Rational Difference Equation \(y_{n+1}=\frac{\alpha_n+y_{n-R}}{\alpha_n+y_{n-K}}\)”. Results in Nonlinear Analysis, vol. 5, no. 3, Sept. 2022, pp. 312-24, doi:10.53006/rna.974156.

Vancouver

1.Sihem Oudina, Mohamed Amine Kerker, Abdelouahab Salmi. On the global behavior of the rational difference equation \(y_{n+1}=\frac{\alpha_n+y_{n-r}}{\alpha_n+y_{n-k}}\). RNA. 2022 Sep. 1;5(3):312-24. doi:10.53006/rna.974156

References

1. [1] R. Abo-Zeid. Global behavior and oscillation of a third order difference equation. Quaest. Math., 2020: 1–20.
2. [2] A. Alshareef, F. Alzahrani, and A. Q. Khan. Dynamics and Solutions’ Expressions of a Higher-Order Nonlinear Fractional Recursive Sequence. Math. Probl. Eng., 2021.
3. [3] M. Berkal and J. F. Navarro. Qualitative behavior of a two-dimensional discrete-time prey–predator model. Comp and Math Methods. 3( 6):e1193, 2021.
4. [4] E. Camouzis. Global convergence in periodically forced rational difference equations. J. Difference Equ. Appl., vol(14), Nos. 10-11, 1011-1033, 2008.
5. [5] E. Camouzis and S. Kotsios. May’s Host–Parasitoid geometric series model with a variable coefficient. Results Appl. Math. 11(2021), Article ID 100160, 5 p.
6. [6] I. Dekkar, N. Touafek, and Q. Din. On the global dynamics of a rational difference equation with periodic coefficients. J. Appl. Math. Comput., 60(1):567–588, 2019.
7. [7] M. J. Douraki and J. Mashreghi. On the population model of the non-autonomous logistic equation of second order with period-two parameters. J. Difference Equ. Appl., Vol. 14, No. 3, March 2008, 231-257.
8. [8] S. Elaydi. An Introduction to Difference Equations, Undergraduate Texts in Mathematics. Springer, New York, 2005.

9. [9] M. Gümüs. The periodic character in a higher order difference equation with delays. Math. Meth. Appl. Sci., 43(3): 1112-1123, 2020.
10. [10] M. Gümüs and R. Abo-Zeid. Global behavior of a rational second order difference equation. J. Appl. Math. Comput., 62(1):119–133, 2020.
11. [11] M.A. Kerker and A. Bouaziz. On the global behavior of a higher-order nonautonomous rational difference equation. Electron. J. Math. Anal. Appl., 9(1), 302-309.
12. [12] M.A. Kerker, E. Hadidi, and A. Salmi. On the dynamics of a nonautonomous rational difference equation. Int. J. Nonlinear Anal. Appl., 12:15-26, 2021.
13. [13] A. Q. Khan and K. Sharif. Global dynamics, forbidden set, and transcritical bifurcation of a one-dimensional discrete-time laser model. Math. Meth. Appl. Sci., 2020;1-13.
14. [14] V. L. Kocic, G. Ladas, and L. W. Rodrigues. On rational recursive sequences. J. Math. Anal. Appl., 173:127–157, 1993.
15. [15] V. Lakshmikantham and D. Trigiante. Theory of Difference Equations, Numerical Methods and Applications. Marcel Dekker, Inc., New York, 2002.
16. [16] E. Liz. Stability of non-autonomous difference equations: simple ideas leading to useful results. J. Difference Equ. Appl., Vol. 17, No. 2, February 2011, 203-220.
17. [17] O. Öcalan, H. Ogünmez, M. Gümüs.Global behavior test for a nonlinear difference equation with a period-two coefficient. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 21 (2014) 307-316.
18. [18] S. Stevic. Solving a class of nonautonomous difference equations by generalized invariants Math. Methods Appl. Sci. 42, No. 18, 6315–6338 (2019).
19. [19] A. Yildirim and D. T. Tollu. Global behavior of a second order difference equation with two-period coefficient. J. Math. Ext., Vol. 16, No. 4, (2022) (1)1-21.