Abstract
In this paper, we investigate the local and global stability and the period two solutions of all nonnegative solutions of the difference equation,
where a, b, A, B are all positive real numbers, \(k \ge 1\) is a positive integer, and the initial conditions \(x_{-k},x_{-k+1},...,x_{0}\) are nonnegative real numbers. It is shown that the zero equilibrium point is globally asymptotically stable under the condition \(a+b \le A\), and the unique positive solution is also globally asymptotically stable under the condition \(a-b \le A \le a+b\). By the end, we study the global stability of such an equation through numerically solved examples.
Similar content being viewed by others
References
Abu-Saris, R., DeVault, R.: Global stability of \(y_{n+1}=A+\frac{y_n}{y_{n-k}}\). Appl. Math. Lett. 16, 173–178 (2003)
Amleh, A., Grove, E., Ladas, G., Georgiou, G.: On the recursive sequence \(x_{n+1}=\alpha +\frac{x_{n-1}}{x_{n}}\). J. Math. Anal. Appl. 533, 790–798 (1999)
Dannan, F.: The Asymptotic Stability of \(x_ { n + k }+ ax_ { n }+ bx_ { n -l}=0\). J. Differ. Equ. Appl. 10, 589–599 (2004)
DeVault, R., Schultz, S.W., Ladas, G.: On the recursive sequence \(x_{n+1}=\frac{A}{x_{n}}+\frac{1}{x_{n-2}}\). Proc. Am. Math. Soc. 126, 3257–3261 (1998)
Yan, Xing-Xue, Li, Wan-Tong, Zhao, Zhu: Global asymptotic stability for a higher order nonlinear rational differnce equations. Appl. Math. Comput. 182, 1819–1831 (2006)
DeVault, R., Kosmala, W., Ladas, G., Schultz, S.W.: On the Recursive Sequence Global behavior of \(y_{n+1}=\frac{p+y_{n-k}}{ qy_{n}+y_{n-k}}\). Nonlinear Anal. 47, 4743–4751 (2001)
Douraki, M., Dehghan, M., Razzaghi, M.: On the higher order rational difference equation \(x_{n}=\frac{A}{x_{n-k}}+\frac{B}{x_{n-3k}}\). Appl. Math. Comput. 173, 710–723 (2006)
Elaydi, S.: An Introduction to Difference Equations, 3rd edn. Springer, New York (2005)
El-Owaidy, H., Ahmed, A., Mousa, M.: On asymptotic behaviour of the difference equation \(x_{n+1}=\alpha +\frac{x_{n-k}}{x_{n}}\). Appl. Math. Comput. 147, 163–167 (2004)
Kocic, V.L., Ladas, G.: Global Asymptotic Behavior of Nonlinear Difference Equations of Higher Order with Applications. Kluwer Academic Publishers, Dordrect (1993)
Kosmala, W., Kulenovic, M.R.S., Ladas, G., Teixeira, C.T.: On the Recursive Sequence \(y_{n+1}=\frac{p+y_{n-1}}{qy_{n}+y_{n-1}}\). J. Math. Anal. Appl. 251, 571–586 (2000)
Kuruklis, S.A.: The asymptotic stability of \( x_{n+1}-ax_{n}+bx_{n-k}=0\). J. Math. Anal. Appl. 188, 719–731 (1994)
Papanicolaou, V.G.: On the asymptotic stability of a class of linear difference equations. Math. Mag. 69, 34–43 (1996)
Saleh, M., Aloqeili, M.: On the rational difference equation \(y_{n+1}=A+\frac{y_{n-k}}{ y_{n}}\), Appl. Math. Comput. 171(1), 862–869 (2005)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Saleh, M., Farhat, A. Global asymptotic stability of the higher order equation \(x_{n+1} = \frac{ ax_{n}+bx_{n-k}}{A+Bx_{n-k}}\) . J. Appl. Math. Comput. 55, 135–148 (2017). https://doi.org/10.1007/s12190-016-1029-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12190-016-1029-4