Second-Order Linear Delay Impulsive Differential Equations

  • Ravi P. Agarwal
  • Leonid Berezansky
  • Elena Braverman
  • Alexander Domoshnitsky


In Chap. 13, second-order linear delay impulsive differential equations are considered. First, the equivalence of the four properties for a second-order impulsive delay equations is established: nonoscillation of the differential equation and of the corresponding differential inequality, positivity of the fundamental function and the existence of a solution of the generalized Riccati inequality. Further, comparison theorems, as well as explicit oscillation and nonoscillation conditions are justified. In the particular case when the values of impulses for the solution and its derivative match, a special nonimpulsive delay differential equation is constructed such that oscillation of an impulsive equation is equivalent to oscillation of the constructed nonimpulsive equation. As a consequence of this theorem, sharper nonoscillation results for this case of impulsive conditions are obtained, and oscillation properties of a second-order impulsive equation and some specially constructed nonimpulsive equation can be compared.


Linear Delay Impulsive Differential Equations Second-order Momentum Equation Nonoscillation Conditions Explicit Oscillation Riccati Inequality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 32.
    Bainov, D., Domshlak, Y., Simeonov, P.: Sturmian comparison theory for impulsive differential inequalities and equations. Arch. Math. 67, 35–49 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 33.
    Bainov, D., Domshlak, Y., Simeonov, P.: On the oscillation properties of first-order impulsive differential equations with deviating arguments. Isr. J. Math. 98, 167–187 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 40.
    Berezansky, L., Braverman, E.: Oscillation of a linear delay impulsive differential equation. Commun. Appl. Nonlinear Anal. 3, 61–77 (1996) MathSciNetzbMATHGoogle Scholar
  4. 43.
    Berezansky, L., Braverman, E.: Some oscillation problems for a second order linear delay differential equation. J. Math. Anal. Appl. 220, 719–740 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 44.
    Berezansky, L., Braverman, E.: On oscillation of a second order impulsive linear delay differential equation. J. Math. Anal. Appl. 233, 276–300 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 54.
    Berezansky, L., Braverman, E.: Oscillation and other properties of linear impulsive and nonimpulsive delay equations. Appl. Math. Lett. 16, 1025–1030 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 84.
    Brands, J.J.A.M., Oscillation theorems for second-order functional differential equations. J. Math. Anal. Appl. 63, 54–64 (1978) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 126.
    Domoshnitsky, A., Drakhlin, M.: Nonoscillation of first order impulse differential equations with delay. J. Math. Anal. Appl. 206, 254–269 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 167.
    Gopalsamy, K.: Stability and Oscillation in Delay Differential Equations of Population Dynamics. Kluwer Academic, Dordrecht, Boston, London (1992) Google Scholar
  10. 169.
    Gopalsamy, K., Zhang, B.G.: On delay differential equations with impulses. J. Math. Anal. Appl. 139, 110–122 (1989) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 249.
    Lakshmikantham, V., Bainov, D.D., Simeonov, P.S.: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989) zbMATHGoogle Scholar
  12. 326.
    Tian, Y.L., Weng, P.X., Yang, J.J.: Nonoscillation for a second order linear delay differential equation with impulses. Acta Math. Appl. Sin. (Engl. Ser.) 20, 101–114 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 341.
    Yong-shao, C., Wei-zhen, F.: Oscillations of second order nonlinear ODE with impulses. J. Math. Anal. Appl. 210, 150–169 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 347.
    Yu, J., Yan, J.R.: Positive solutions and asymptotic behavior of delay differential equations with nonlinear impulses. J. Math. Anal. Appl. 207, 388–396 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 353.
    Zhang, Y.: Oscillation criteria for impulsive delay differential equations. J. Math. Anal. Appl. 205, 461–470 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 359.
    Zhao, A., Yan, J.R.: Existence of positive solutions for delay differential equations with impulses. J. Math. Anal. Appl. 210, 667–678 (1997) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Ravi P. Agarwal
    • 1
  • Leonid Berezansky
    • 2
  • Elena Braverman
    • 3
  • Alexander Domoshnitsky
    • 4
  1. 1.Department of MathematicsTexas A&M University—KingsvilleKingsvilleUSA
  2. 2.Department of MathematicsBen-Gurion University of the NegevBeer-ShevaIsrael
  3. 3.Department of MathematicsUniversity of CalgaryCalgaryCanada
  4. 4.Department of Computer Sciences and MathematicsAriel University Center of SamariaArielIsrael

Personalised recommendations