 ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Dynamics of $K^{th}$ Order Rational Difference Equation

Journal of Applied Nonlinear Dynamics 10(1) (2021) 125--149 | DOI:10.5890/JAND.2021.03.008

Department of Mathematics, Birzeit University, West Bank

Abstract

In this paper we will investigate the dynamical behavior of the following rational difference equation \begin{equation} x_{n+1}= \frac{\alpha + \beta x_{n} + \gamma x_{n-k}} {A +B x_{n} + C x_{n-k}},\quad n=0,1,... \end{equation} where the parameters $\alpha, \beta, \gamma$ and A, B, C and the initial conditions $x_{-k},\dots,x_{-1},x_{0}$ are non-negative real numbers, and the denominator is nonzero. Our concentration here, is on the global stability, the periodic character, the analysis of semi-cycles and the invariant intervals of the positive solution of the above equation. It is worth mentioning that our difference equation is the general case of the rational equation which is studied by Kulenovic and Ladas in their monograph ( Dynamics of Second Order Rational Difference Equation with Open Problems and Conjectures, 2002 ).

References

1.   Kulenovic, M.R.S. and Ladas, G. (2002), Dynamics of Second Order Rational Difference Equations With Open Problems and Conjectures.
2.   Yan, X.X., Li, W.T., and Zhao, Z. (2006), Global asymptotic stability for a higher order nonlinear rational difference equations, Appl. Math. Comput., 182, 1819-1831.
3.   Devault, R., Kosmala, W., Ladas, G., and Schuults, S.W. (2001), Global Behavior, of $y_{n+1}=\frac{p+y_{n-k}}{q y_{n}+y_{n-k}}$, Nonlinear Analysis, 47, 4743-4751.
4.   Dehghan, M., Sebdani, R.M. (2006), Dynamics of a higher-order rational difference equation, Appl. Math. Comput., 178, 345-354.
5.   Dehghan, M. and Douraki, M.J. (2005), Dynamics of a rational difference equation using both theoretical and computational approaches, Appl. Math. Comput., 168, 756-775.
6.   Saleh, M. and Abu-Baha, S. (2006), Dynamics of higher order Rational Difference Equation, App. Math. Comp., 181, 84-102.
7.   Saleh, M., Alkoumi, N., and Farhat, A. (2017), ph{On the dynamics of a rational difference equation $x_{n+1}=\frac{ \alpha +\beta x_{n}+\gamma x_{n-k}}{Bx_{n}+Cx_{n-k}}$}, Chaos and Soliton, 76-84.
8.   Saleh, M. and Aloqeili, M. (2005), On the rational difference equation $y_{n+1}=A + \frac{y_{n-k}}{y_{n}}$, Appl. Math. Comput., 171, 862-869.
9.   Li, W.T. and Sun, H.R. (2005), Dynamics of a rational difference equation, Appl. Math. Comput., 163, 577-591.
10.   Saleh, M. and Aloqeili, M. (2006), On the rational difference equation $y_{n+1}=A + \frac{y_{n}}{y_{n-k}}$, Appl. Math. Comput., 177, 189-193.
11.   Salas Einarhille, S.L. (1982), Calculus One and Several Variables, fourth edition.
12.   Douraki, M.J., Dehghan, M., and Razzaghi, M. (2005), The qualitative behavior of solutions of nonlinear difference equation Appl. Math. Comput., 170, 485-502.
13.   Elaydi, S. (2000), Discrete Chaos, Springer-Newyork.
14.   Sebdani, R.M. and Dehghan, M. (2006), The study of a class of rational difference equation, Appl. Math. Comput., 179, 98-107.
15.   Camouzis, E. and Ladas, G. (2007), Dynamics of Third-Order Rational Difference Equations With Open Problems and Conjectures.
16.   Dehghan, M. and Sebdani, R.M. (2006), On the recursive sequence $x_{n+1}=\frac{a+b x_{n-k+1}}{A+Bx_{n-k+1}+Cx_{n-2k+1}}$, Appl. Math. Comput., 178, 273-286.
17.   Douraki, M.J., Dehghan, M., and Razzaghi, M. (2006), Global behavior of the difference equation $x_{n+1}=\frac{x_{n-l+1}}{1+a0x_{n}+a1x_{n-1}+\dots+alx_{n-l}+x_{n-l+1}}$, Article in press.
18.   Elaydi, S. (2005), An Introduction to Difference Equations , third edition.
19.   Farhat, A. (2007), Dynamics Of Some Rational Nonlinear Difference Equations, Master thesis, Birzeit University.
20.   Grove, E.A. and Ladas, G. (2005), Periodicities in Nonlinear Difference Equations.
21.   He, W.S., Li, W.T., and Yan, X.X. (2004), Global attractivity of the difference equation $x_{n+1}=\alpha+\frac{x_{n-k}}{x_{n}}$, Appl. Math. Comput., 151, 879-885.
22.   Saleh, M. and Aloqeili, M. (2005), On the rational difference equation $y_{n+1}= A + \frac{y_{n-k}}{y_{n}}$, Appl. Math. Comp., 171862-869.
23.   Saleh, M. and Farhat, A. (2017),{Global Asymptotic Stability of the Higher Order Equation $x_{n+1} = \frac{ ax_{n}+bx_{n-k}}{A+Bx_{n-k}}$}, ph {J. Appl. Math. Comput.}, 135-148, DOI 10.1007/s12190-016-1029-4.
24.   Jafar, J. and Saleh, M. (2018), {Dynamics of nonlinear difference equation $x_{n+1} = \frac{ \beta x_{n}+\gamma x_{n-k}}{A+Bx_{n}+Cx_{n-k}}$}, ph {J. Appl. Math. Comput.}, 493-524.
25.   Abu Alhalawa M. and Saleh, M. (2017), ph{Dynamics of higher order rational difference equation $x_{n+1} = \frac{\alpha +\beta x_{n}}{A+Bx_{n}+x_{n-k}}$}, Int. J. Nonlinear Anal. Appl., 363-379.
26.   Sebdani, R.M. and Dehghan, M. (2006), Global stability of $y_{n+1}=\frac{p + q y_{n}+ry_{n-k}}{1+y_{n}}$, Appl. Math. Comput., 182, 621-630.
27.   Sebdani, R.M. and Dehghan, M. (2006), Dynamics of a non-linear difference equation, Appl. Math. Comput., 178, 250-261.
28.   Zayed, E.M.E. and El-Moneam, M.A. (2006), On the Rational Recursive Sequence, Egypt.