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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: nanjinghuang@hotmail.com)
*
✉For correspondence; e-mail: nanjinghuang@hotmail.com
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Abstract

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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

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[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
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