Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-13T00:45:29.176Z Has data issue: false hasContentIssue false
  • English
  • Français

GLOBAL ASYMPTOTIC STABILITY FOR GENERAL SYMMETRIC RATIONAL DIFFERENCE EQUATIONS

Part of: Difference and functional equations Difference equations

Published online by Cambridge University Press:  02 October 2009

CONG ZHANG ,
HONG-XU LI  and
NAN-JING HUANG
CONG ZHANG
Affiliation:
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, PR China
HONG-XU LI
Affiliation:
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, PR China
NAN-JING HUANG*
Affiliation:
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, PR China (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We investigate the global asymptotic stability for positive solutions to a class of general symmetric rational difference equations and prove that the unique positive equilibrium 1 of the general symmetric rational difference equations is globally asymptotically stable. As a special case of our result, we solve the conjecture raised by Berenhaut, Foley and Stević [‘The global attractivity of the rational difference equation yn=(yn−k+yn−m)/(1+yn−kyn−m)’, Appl. Math. Lett.20 (2007), 54–58].

Keywords

rational difference equationglobal asymptotic stabilitypositive solutionequilibrium

MSC classification

Secondary: 39A20: Multiplicative and other generalized difference equations, e.g. of Lyness type
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 81 , Issue 2 , April 2010 , pp. 251 - 259
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

Footnotes

This work was supported by the Key Program of NSFC (Grant No. 70831005), the National Natural Science Foundation of China (10671135) and Specialized Research Fund for the Doctoral Program of Higher Education (20060610005).

References

[1]Agarwal, R. P., Difference Equations and Inequalities: Theory, Methods and Applications, 2nd edn (Marcel Dekker, New York, 2000).Google Scholar
[2]Berenhaut, K. S., Foley, J. D. and Stević, S., ‘The global attractivity of the rational difference equation y n=(y n−k+y n−m)/(1+y n−ky n−m)’, Appl. Math. Lett. 20 (2007), 54–58.Google Scholar
[3]Berenhaut, K. S. and Stević, S., ‘The global attractivity of a higher order rational difference equation’, J. Math. Anal. Appl. 326 (2007), 940–944.Google Scholar
[4]Grove, E. A. and Ladas, G., Periodicities in Nonlinear Difference Equations (Chapman & Hall/CRC Press, Boca Raton, FL, 2004).Google Scholar
[5]Kocić, V. and Ladas, G., Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Mathematics and its Applications, 256 (Kluwer Academic, Dordrecht, 1993).CrossRefGoogle Scholar
[6]Kulenović, M. R. S. and Ladas, G., Dynamics of Second Order Rational Difference Equations with Open Problems and Applications (Chapman & Hall/CRC Press, Boca Raton, FL, 2001).CrossRefGoogle Scholar
[7]Li, Z. and Zhu, D., ‘Global asymptotic stablility of a higher order nonlinear difference equation’, Appl. Math. Lett. 19 (2006), 926–930.CrossRefGoogle Scholar