Second-Order Delay Differential Equations

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


Equations of the second order are popular due to their numerous applications. For equations of the second order with several delays and not including explicitly the first derivative, Chap. 7 presents nonoscillation criteria in the terms of the fundamental function of the equation and the generalized Riccati inequality. In addition, the chapter includes comparison results, explicit nonoscillation and oscillation conditions, the result that the equation is oscillatory if and only if it has a slowly oscillating solution, and explicit nonoscillation conditions which are obtained by substituting specific solutions of the generalized Riccati inequality. Sufficient conditions for positivity of a solution of the initial value problem are also presented in Chap. 7.


Second-order Delay Differential Equations Nonoscillation Criteria Nonoscillation Conditions Non-oscillatory Property Discontinuous Parameters 
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  1. 20.
    Azbelev, N.V.: The zeros of the solutions of a second order linear differential equation with retarded argument. Differ. Uravn. 7, 1147–1157 (1971), 1339 (in Russian) MathSciNetzbMATHGoogle Scholar
  2. 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
  3. 80.
    Berezansky, L., Larionov, A.: Positivity of the Cauchy matrix of a linear functional-differential equation. Differ. Equ. 24, 1221–1230 (1988) MathSciNetGoogle Scholar
  4. 84.
    Brands, J.J.A.M., Oscillation theorems for second-order functional differential equations. J. Math. Anal. Appl. 63, 54–64 (1978) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 140.
    Domshlak, Y.: Sturmian Comparison Method in Investigation of Behavior of Solutions for Differential-Operator Equations. Elm, Baku (1986) (in Russian) Google Scholar
  6. 141.
    Domshlak, Y.: Comparison theorems of Sturm type for first and second order differential equations with sign variable deviations of the argument. Ukr. Mat. Zh. 34, 158–163 (1982) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 154.
    Erbe, L.H., Kong, Q., Zhang, B.G.: Oscillation Theory for Functional Differential Equations. Dekker, New York (1995) Google Scholar
  8. 159.
    Gil’, M.I.: On Aizerman-Myshkis problem for systems with delay. Automatica 36, 1669–1673 (2000) MathSciNetCrossRefGoogle Scholar
  9. 160.
    Gil’, M.I.: Boundedness of solutions of nonlinear differential delay equations with positive Green functions and the Aizerman-Myshkis problem. Nonlinear Anal. 49, 1065–1078 (2002) MathSciNetCrossRefGoogle Scholar
  10. 161.
    Gil’, M.I.: The Aizerman-Myshkis problem for functional-differential equations with causal nonlinearities. Funct. Differ. Equ. 11, 445–457 (2005) Google Scholar
  11. 163.
    Gil’, M.I.: Lower bounds and positivity conditions for Green’s functions to second order differential-delay equations. Electron. J. Qual. Theory Differ. Equ. 2009(65) (2009), 11 pp. Google Scholar
  12. 174.
    Grace, S.R., Lalli, B.S.: Oscillation theory for damped differential equations of even order with deviating argument. SIAM J. Math. Anal. 15, 308–316 (1984) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 192.
    Győri, I., Ladas, G.: Oscillation Theory of Delay Differential Equations with Applications. Clarendon Press, New York (1991) Google Scholar
  14. 193.
    Győri, I., Pituk, M.: Comparison theorems and asymptotic equilibrium for delay differential and difference equations. Dyn. Syst. Appl. 5, 277–303 (1996) Google Scholar
  15. 206.
    Hille, E.: Nonoscillation theorems. Trans. Am. Math. Soc. 64, 234–252 (1948) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 217.
    Kartsatos, A.G.: Recent results in oscillation of solutions of forced and perturbed nonlinear differential equations of even order. In: Stability of Dynamic Systems: Theory and Applications. Lecture Notes in Pure and Applied Mathematics. Springer, New York (1977) Google Scholar
  17. 221.
    Kiguradze, I.T., Partsvaniya, N.L., Stavroulakis, I.P.: On the oscillatory properties of higher-order advance functional-differential equations. Differ. Equ. 38, 1095–1107 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 227.
    Koplatadze, R., Kvinikadze, G., Stavroulakis, I.P.: Oscillation of second order linear delay differential equations. Funct. Differ. Equ. 7, 121–147 (2000) MathSciNetzbMATHGoogle Scholar
  19. 248.
    Ladde, G.S., Lakshmikantham, V., Zhang, B.G.: Oscillation Theory of Differential Equations with Deviating Arguments. Dekker, New York (1987) Google Scholar
  20. 263.
    Li, M., Wang, M., Yan, J.: On oscillation of nonlinear second order differential equation with damping term. J. Appl. Math. Comput. 13, 223–232 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 283.
    Manfoud, W.E.: Comparison theorems for delay differential equations. Pac. J. Math. 83, 187–197 (1979) Google Scholar
  22. 289.
    Myshkis, A.D.: Linear Differential Equations with Retarded Argument. Nauka, Moscow (1972) (in Russian) zbMATHGoogle Scholar
  23. 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
  24. 297.
    Parhi, N.: Sufficient conditions for oscillation and nonoscillation of solutions of a class of second order functional-differential equations. Analysis 13, 19–28 (1993) MathSciNetzbMATHGoogle Scholar
  25. 300.
    Philos, Ch.G.: Oscillatory and asymptotic behavior of the bounded solutions of differential equations with deviating arguments. Hiroshima Math. J. 8, 31–48 (1978) MathSciNetzbMATHGoogle Scholar
  26. 301.
    Philos, Ch.G.: A comparison result in oscillation theory. J. Pure Appl. Math. 11, 1–7 (1980) MathSciNetzbMATHGoogle Scholar
  27. 305.
    Philos, C.G., Purnaras, I.K., Sficas, Y.G.: Oscillations in higher-order neutral differential equations. Can. J. Math. 45, 132–158 (1993) MathSciNetzbMATHCrossRefGoogle Scholar
  28. 306.
    Philos, C.G., Sficas, Y.G.: Oscillatory and asymptotic behavior of second and third order retarded differential equations. Czechoslov. Math. J. 32, 169–182 (1982) MathSciNetGoogle Scholar
  29. 327.
    Tiryaki, A., Zafer, A.: Oscillation criteria for second order nonlinear differential equations with damping. Turk. J. Math. 24, 185–196 (2000) MathSciNetzbMATHGoogle Scholar
  30. 335.
    Yan, J.R.: Oscillation theory for second order linear differential equations with damping. Proc. Am. Math. Soc. 98, 276–284 (1986) zbMATHCrossRefGoogle Scholar

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© 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

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