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Asymptotic stability for impulsive functional differential equations

  • Zhi-guo Luo (罗治国)Email author
  • Yan Luo (罗艳)
Article

Abstract

In this paper, we investigate the stability of a class of impulsive functional differential equations by using Lyapunov functional and Jensen’s inequality. Some new stability theorems are obtained. Examples are given to demonstrate the advantage of the obtained results.

Key words

stability impulsive functional differential equation Lyapunov functional Jensen’s inequality 

Chinese Library Classification

O175 

2000 Mathematics Subject Classification

34D20 34K20 

References

  1. [1]
    Lakshmikantham, V., Bainov, D. D., and Simeonov, P. S. Theory of Impulsive Differential Equations, World Scientific Publishing Co., Singapore (1989)zbMATHGoogle Scholar
  2. [2]
    Bainov, D. D. and Simenov, P. S. Systems with Impulse Effect: Stability Theory and Applications, Ellis Horwood, Chichester (1989)zbMATHGoogle Scholar
  3. [3]
    Samoilenko, A. M. and Perestyuk, N. A. Impulive Differential Equations, World Scientific Publishing Co., Singapore (1995)Google Scholar
  4. [4]
    Shen, J. and Yan, J. Razumikhin type stability theorems for impulsive functional differential equations. Nonlinear Analysis 33(5), 519–531 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    Liu, X. and Shen, J. Razumikhin type theorems on boundedness for impulsive functional differential equations. Dynamic Systems and Applications 12, 265–281 (2000)MathSciNetGoogle Scholar
  6. [6]
    Zhang, Y. and Sun, J. Stability of impulsive functional differential equations. Nonlinear Analysis 68(12), 3665–3678 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    Luo, Z. and Shen, J. New Razumikhin type theorems for impulsive functional differential equations. Appl. Math. Comput. 125(2–3), 375–386 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    Liu, X. and Ballinger, G. Uniformly asymptotic stability of impulsive delay differential equations. Comput. Math. Appl. 41(7–8), 903–915 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    Luo, Z. and Shen, J. Impulsive stabilization of functional differential equations with infinite delays. Appl. Math. Lett. 16(5), 695–701 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    Luo, Z. and Shen, J. Stability of impulsive functional differential equations via the Liapunov functional. Appl. Math. Lett. 22(2), 163–169 (2009)CrossRefMathSciNetGoogle Scholar
  11. [11]
    Shen, J., Luo, Z., and Liu, X. Impulsive stabilization of functional differential equations via Liapunov functionals. J. Math. Anal. Appl. 240(1–5), 1–15 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    Luo, Z. and Shen, J. Stability and boundedness for impulsive functional differential equations with infinite delays. Nonlinear Analysis 46(4), 475–493 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    Stamova, I. M. and Stamov, G. T. Lyapunov-Razumikhin method for impulsive functional differential equations and applications to the population dynamics. J. Comput. Appl. Math. 130(1–2), 163–171 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    Ballinger, G. and Liu, X. Existence and uniqueness results for impulsive delay differential equations. DCDIS 5, 579–591 (1999)zbMATHMathSciNetGoogle Scholar
  15. [15]
    Becker, L. C., Burton, T. A., and Zhang, S. Functional differential equations and Jensen’s inequality. J. Math. Anal. Appl. 138(1), 137–156 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    Becker, L. C. and Burton, T. A. Jensen’s inequality and Liapunov’s direct method. CUBO A Mathematical Journal 6(3), 65–87 (2004)MathSciNetGoogle Scholar
  17. [17]
    Natansoa, I. P. Theory of Functions of a Real Variable, Vol. II, Ungar, New York (1960)Google Scholar

Copyright information

© Shanghai University and Springer Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Department of MathematicsHunan Normal UniversityChangshaP. R. China

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