Second-Order Delay Differential Equations with Damping Terms

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


Chapter 8 deals with nonoscillation problems for a scalar linear delay differential equation of the second order including explicitly a term with the first derivative which is usually called “a damping term”. Most of publications deal with equations not containing the term with the first derivative; for these equations, positivity of the coefficients and a solution on the semiaxis implies that its derivative is nonnegative. This fact is very important, and it is employed in most investigations on second-order delay differential equations. If the first derivative is included in the equation explicitly, i.e. the equation contains the damping term, then a sign of a solution does not uniquely define the sign of its derivative, which makes the study of oscillation properties of the equations with the damping term more complicated. This is the reason why such equations are much less studied than equations without the damping term.

The main result of the chapter is the following: if a generalized Riccati inequality has a nonnegative solution, then the differential equation has a positive solution with a nonnegative derivative, and the fundamental function of this equation is positive. If the damping term is not delayed, this immediately yields that the following four properties are equivalent: nonoscillation of solutions of this equation and the corresponding differential inequality, positivity of the fundamental function and existence of a nonnegative solution of the generalized Riccati inequality.

The generalized Riccati inequality is applied to compare oscillation properties of two equations without comparing their solutions. These results can be considered as a natural generalization of the well-known Sturm comparison theorem for a second-order ordinary differential equation. The chapter also contains explicit nonoscillation conditions which are obtained by substituting specific solutions of the generalized Riccati inequality.


Second-order Delay Differential Equations Damping Term Nonnegative Derivative Nonoscillation Conditions Oscillatory Properties 
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. 45.
    Berezansky, L., Braverman, E.: Nonoscillation of a second order linear delay differential equation with a middle term. Funct. Differ. Equ. 6, 233–247 (1999) MathSciNetzbMATHGoogle Scholar
  2. 46.
    Berezansky, L., Braverman, E.: Oscillation of a second-order delay differential equations with middle term. Appl. Math. Lett. 13, 21–25 (2000) MathSciNetCrossRefzbMATHGoogle Scholar
  3. 72.
    Berezansky, L., Braverman, E., Domoshnitsky, A.: Nonoscillation and stability of the second order ordinary differential equations with a damping term. Funct. Differ. Equ. 16, 169–197 (2009) MathSciNetzbMATHGoogle Scholar
  4. 73.
    Berezansky, L., Braverman, E., Domoshnitsky, A.: Stability of the second order delay differential equations with a damping term. Differ. Equ. Dyn. Syst. 16, 185–205 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  5. 77.
    Berezansky, L., Diblík, J., Šmarda, Z.: Positive solutions of second-order delay differential equations with a damping term. Comput. Math. Appl. 60, 1332–1342 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  6. 154.
    Erbe, L.H., Kong, Q., Zhang, B.G.: Oscillation Theory for Functional Differential Equations. Dekker, New York (1995) Google Scholar
  7. 164.
    Gil’, M.I.: Positivity of Green’s functions to Volterra integral and higher order integrodifferential equations. Anal. Appl. (Singap.) 7, 405–418 (2009) MathSciNetCrossRefGoogle Scholar
  8. 172.
    Grace, S.R.: On the oscillation of certain functional differential equations. J. Math. Anal. Appl. 194, 304–318 (1995) MathSciNetCrossRefzbMATHGoogle Scholar
  9. 173.
    Grace, S.R.: On the oscillation of certain forced functional differential equations. J. Math. Anal. Appl. 202, 555–577 (1996) MathSciNetCrossRefzbMATHGoogle Scholar
  10. 192.
    Győri, I., Ladas, G.: Oscillation Theory of Delay Differential Equations with Applications. Clarendon Press, New York (1991) Google Scholar
  11. 218.
    Kartsatos, A.G., Toro, J.: Comparison and oscillation theorems for equations with middle terms of order n−1. J. Math. Anal. Appl. 66, 297–312 (1978) MathSciNetCrossRefzbMATHGoogle Scholar
  12. 230.
    Kosmala, W.A.: Comparison results for functional differential equations with two middle terms. J. Math. Anal. Appl. 111, 243–252 (1985) MathSciNetCrossRefzbMATHGoogle Scholar
  13. 248.
    Ladde, G.S., Lakshmikantham, V., Zhang, B.G.: Oscillation Theory of Differential Equations with Deviating Arguments. Dekker, New York (1987) Google Scholar
  14. 289.
    Myshkis, A.D.: Linear Differential Equations with Retarded Argument. Nauka, Moscow (1972) (in Russian) zbMATHGoogle Scholar
  15. 292.
    Norkin, S.B.: Differential Equations of the Second Order with Retarded Argument. Translations of Mathematical Monographs, vol. 31. Am. Math. Soc., Providence (1972) zbMATHGoogle 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