Stability of a fractional difference equation of high order
Volume 12, Issue 2, pp 65--74
http://dx.doi.org/10.22436/jnsa.012.02.01
Publication Date: October 12, 2018
Submission Date: July 16, 2018
Revision Date: September 24, 2018
Accteptance Date: September 26, 2018
-
2142
Downloads
-
4729
Views
Authors
M. A. El-Moneam
- Mathematics Department, Faculty of Science, Jazan University, Saudi Arabia.
Tarek F. Ibrahim
- Mathematics Department, College of Sciences and Arts for Girls in sarat Abida, King Khalid University, Saudi Arabia.
- Mathematics Department, Faculty of Science, Mansoura University, Mansoura, Egypt.
S. Elamody
- Mathematics Department, Faculty of Science, Jazan University, Saudi Arabia.
Abstract
In this paper we investigate the local stability, global stability, and
boundedness of solutions of the recursive sequence%
\[
x_{n+1}=x_{n-p}\ \left( \frac{2\ x_{n-q}\ +a\ x_{n-r}}{x_{n-q}\ +a\ x_{n-r}}%
\right),
\]
where \(x_{-q+k}\ \neq -a\ x_{-r+k} \) for \( k=0,1,\dots,\min (q,r) , a\in \mathbb{R},\ p ,q, r \geq 0\) with the initial condition \(x_{-p},x_{-p+1} ,\dots, x_{-q},\) \(x_{-q+1} ,\dots, x_{-r},x_{-r+1} ,\dots, x_{-1}\)
and \(x_{0}\in (0,\infty )\). Some numerical examples will be given to
illustrate our results.
Share and Cite
ISRP Style
M. A. El-Moneam, Tarek F. Ibrahim, S. Elamody, Stability of a fractional difference equation of high order, Journal of Nonlinear Sciences and Applications, 12 (2019), no. 2, 65--74
AMA Style
El-Moneam M. A., Ibrahim Tarek F., Elamody S., Stability of a fractional difference equation of high order. J. Nonlinear Sci. Appl. (2019); 12(2):65--74
Chicago/Turabian Style
El-Moneam, M. A., Ibrahim, Tarek F., Elamody, S.. "Stability of a fractional difference equation of high order." Journal of Nonlinear Sciences and Applications, 12, no. 2 (2019): 65--74
Keywords
- Difference equations
- prime period two solution
- boundedness character
- locally asymptotically stable
- global attractor
- global stability
- high orders
MSC
References
-
[1]
R. P. Agarwal, E. M. Elsayed, On the solution of fourth-order rational recursive sequence , Adv. Stud. Contemp. Math. (Kyungshang), 20 (2010), 525–545.
-
[2]
R. Devault, W. Kosmala, G. Ladas, S. W. Schaultz, Global Behavior of \(y_{n+1} = \frac{p+y_{n-k}}{ qy_n+y_{n-k}}\), Nonlinear Anal., 47 (2001), 4743–4751.
-
[3]
E. M. Elabbasy, H. El-Metwally, E. M. Elsayed , On the difference equation \(x_{n+1} = ax_n - \frac{ bx_n}{ cx_{n }- dx_{n-1}}\), Adv. Difference Equ., 2006 (2006), 10 pages.
-
[4]
H. El-Metwally, E. A. Grove, G. Ladas, H. D. Voulov, On the global attractivity and the periodic character of some difference equations, J. Differ. Equations Appl., 7 (2001), 837–850.
-
[5]
M. A. El-Moneam , On the dynamics of the higher order nonlinear rational difference equation, Math. Sci. Lett., 3 (2014), 121–129.
-
[6]
M. A. El-Moneam, On the dynamics of the solutions of the rational recursive sequences, British J. Math. Computer Sci., 5 (2015), 654–665.
-
[7]
M. A. El-Moneam, S. O. Alamoudy, On study of the asymptotic behavior of some rational difference equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 21 (2014), 89–109.
-
[8]
M. A. El-Moneam, E. M. E. Zayed, Dynamics of the rational difference equation, Inf. Sci. Lett., 3 (2014), 1–9.
-
[9]
E. M. Elsayed, Solution and Attractivity for a Rational Recursive Sequence, Discrete Dyn. Nat. Soc., 2011 (2011), 17 pages.
-
[10]
E. M. Elsayed, T. F. Ibrahim, Solutions and Periodicity of a Rational Recursive Sequences of Order Five, Bull. Malays. Math. Sci. Soc., 38 (2015), 95–112.
-
[11]
E. A. Grove, G. Ladas, Periodicities in Nonlinear Difference Equations, Chapman & Hall/CRC , Boca Raton (2005)
-
[12]
T. F. Ibrahim , Boundedness and stability of a rational difference equation with delay, Rev. Roum. Math. Pures Appl., 57 (2012), 215–224.
-
[13]
T. F. Ibrahim, Periodicity and global attractivity of difference equation of higher order, J. Comput. Anal. Appl., 16 (2014), 552–564.
-
[14]
T. F. Ibrahim, Three-dimensional max-type cyclic system of difference equations , Int. J. Phys. Sci., 8 (2013), 629–634.
-
[15]
T. F. Ibrahim, N. Touafek, On a third-order rational difference equation with variable coefficients , Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 20 (2013), 251–264.
-
[16]
V. L. Kocic, G. Ladas , Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht (1993)
-
[17]
E. M. E. Zayed, M. A. El-Moneam, On the rational recursive two sequences \(x_{n+1} = ax_{n-k} +bx_{n-k}/ (cx_n + \delta dx_{n-k})\), Acta Math. Vietnam., 35 (2010), 355–369.
-
[18]
E. M. E. Zayed, M. A. El-Moneam, On the global attractivity of two nonlinear difference equations, J. Math. Sci., 177 (2011), 487–499.