On the global behavior of the rational difference equation \(y_{n+1}=\frac{\alpha_n+y_{n-r}}{\alpha_n+y_{n-k}}\)
On the global behavior of the rational difference equation \(y_{n+1}=\frac{\alpha_n+y_{n-r}}{\alpha_n+y_{n-k}}\)
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
___
- [1] R. Abo-Zeid. Global behavior and oscillation of a third order difference equation. Quaest. Math., 2020: 1–20.
- [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] 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] E. Camouzis. Global convergence in periodically forced rational difference equations. J. Difference Equ. Appl., vol(14), Nos. 10-11, 1011-1033, 2008.
- [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] 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] 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] S. Elaydi. An Introduction to Difference Equations, Undergraduate Texts in Mathematics. Springer, New York, 2005.
- [9] M. Gümüs. The periodic character in a higher order difference equation with delays. Math. Meth. Appl. Sci., 43(3): 1112-1123, 2020.
- [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] 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] 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] 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] V. L. Kocic, G. Ladas, and L. W. Rodrigues. On rational recursive sequences. J. Math. Anal. Appl., 173:127–157, 1993.
- [15] V. Lakshmikantham and D. Trigiante. Theory of Difference Equations, Numerical Methods and Applications. Marcel Dekker, Inc., New York, 2002.
- [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] 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] S. Stevic. Solving a class of nonautonomous difference equations by generalized invariants Math. Methods Appl. Sci. 42, No. 18, 6315–6338 (2019).
- [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.
- Başlangıç: 2018
- Yayıncı: Erdal KARAPINAR