Maximum Principles and Nonoscillation Intervals

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


In the previous chapters, as well as in the known monographs on nonoscillation of functional differential equations, nonoscillation was only interpreted as existence of eventually positive solutions. In this and the following two chapters, nonoscillation on an interval is considered. We start with a definition of homogeneous equations in such a form that they preserve one dimensional fundamental systems. This allows us to define nonoscillation interval for first-order functional differential equations as an interval where nontrivial solutions of the homogeneous equation do not have zeros. Nonoscillation results developed in this chapter consist of two parts: first, to discover a nonoscillation interval for functional differential equations, and, second, to find various corollaries and applications of nonoscillation results in maximum principles, boundary value problems and stability; in certain cases we can prove equivalence of nonoscillation and corresponding maximum principles.

One of the most important results obtained in this chapter for functional differential equations with a positive Volterra operator (in a particular case, for delay differential equations with positive coefficients) is the theorem on 8 equivalences which include nonoscillation, positivity of the Cauchy (fundamental) function, a differential inequality, an integral inequality, an estimate of the spectral radius of a corresponding auxiliary operator and, under the additional condition (for example, that this Volterra operator is nonzero in the case of the periodic problem), positivity of Green’s function of a wide class of boundary value problems. In particular, a generalized periodic problem and problems with integrals in boundary conditions can be considered; many explicit tests of nonoscillation and positivity of the Cauchy and Green’s functions are obtained. These tests require either a short of memory of the Volterra operator (a delay for equation with concentrated delay is small) or smallness of the interval where the functional differential equation is considered.

Chapter 15 also proposes new ideas on the study of nonoscillation and positivity of the Cauchy and Green’s functions in the case of a sum of positive and negative operators in first-order functional differential equations. Previous results deduced nonoscillation only in the case when the equation with the positive operator is nonoscillatory; our approach is based on the idea that positive and negative operators can compensate each other. As a result, we deduce nonoscillation in the cases when the equation with only the positive operator is oscillatory, and in the case when their solutions oscillate with amplitudes tending to infinity, we get exponential stability on the basis of nonoscillation.


Boundary Maximum Principle First-order Functional Differential Equations Volterra Operator Exponential Stability Non-oscillatory Results 
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. 3.
    Agarwal, R.P., Bohner, M., Li, W.-T.: Nonoscillation and Oscillation: Theory for Functional Differential Equations. Monographs and Textbooks in Pure and Applied Mathematics, vol. 267. Dekker, New York (2004) zbMATHCrossRefGoogle Scholar
  2. 4.
    Agarwal, R.P., Domoshnitsky, A.: Nonoscillation of the first order differential equations with unbounded memory for stabilization by control signal. Appl. Math. Comput. 173, 177–195 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 5.
    Agarwal, R.P., Domoshnitsky, A.: On positivity of several components of solution vector for systems of linear functional differential equations. Glasg. Math. J. 52, 115–136 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 7.
    Agarwal, R.P., Grace, S.R., O’Regan, D.: Oscillation Theory for Difference and Functional Differential Equations. Kluwer Academic, Dordrecht (2000) zbMATHGoogle Scholar
  5. 9.
    Agarwal, R.P., Grace, S.R., O’Regan, D.: Oscillation Theory for Dynamic Equations. Taylor & Francis, London (2003) zbMATHCrossRefGoogle Scholar
  6. 19.
    Azbelev, N.V.: About bounds of applicability of theorem of Tchaplygin about differential inequalities. Dokl. Acad. Nauk USSR 89, 589–591 (1953) (in Russian) MathSciNetzbMATHGoogle Scholar
  7. 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
  8. 22.
    Azbelev, N.V., Berezansky, L.M., Simonov, P.M., Chistykov, A.V.: Stability of linear systems with time-lag. Differ. Equ. 23, 493–500 (1987) zbMATHGoogle Scholar
  9. 23.
    Azbelev, N.V., Berezansky, L.M., Simonov, P.M., Chistykov, A.V.: Stability of linear systems with time-lag. Differ. Equ. 27, 383–388, 1165–1172 (1991) zbMATHGoogle Scholar
  10. 24.
    Azbelev, N.V., Berezansky, L.M., Simonov, P.M., Chistykov, A.V.: Stability of linear systems with time-lag. Differ. Equ. 29, 153–160 (1993) Google Scholar
  11. 25.
    Azbelev, N.V., Domoshnitsky, A.: A de la Vallée-Poussin differential inequality. Differ. Uravn. 22, 2042–2045, 2203 (1986) MathSciNetGoogle Scholar
  12. 26.
    Azbelev, N.V., Domoshnitsky, A.: On the question of linear differential inequalities. I. Differ. Equ. 27, 257–263 (1991) MathSciNetzbMATHGoogle Scholar
  13. 29.
    Azbelev, N.V., Maksimov, V.P., Rakhmatullina, L.F.: Introduction to the Theory of Linear Functional-Differential Equations. Advanced Series in Mathematical Science and Engineering, vol. 3. World Federation Publishers Company, Atlanta (1995) zbMATHGoogle Scholar
  14. 30.
    Azbelev, N.V., Simonov, P.M.: Stability of Differential Equations with Aftereffects. Stability Control Theory Methods and Applications, vol. 20. Taylor & Francis, London (2003) Google Scholar
  15. 31.
    Bainov, D., Domoshnitsky, A.: Nonnegativity of the Cauchy matrix and exponential stability of a neutral type system of functional differential equations. Extr. Math. 8, 75–82 (1992) MathSciNetGoogle Scholar
  16. 39.
    Berezansky, L.: Development of N.V. Azbelev’s W-method in problems of the stability of solutions of linear functional-differential equations. Differ. Uravn. 22, 739–750, 914 (1986) MathSciNetGoogle Scholar
  17. 61.
    Berezansky, L., Braverman, E.: On stability of some linear and nonlinear delay differential equations. J. Math. Anal. Appl. 314, 391–411 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 62.
    Berezansky, L., Braverman, E.: On exponential stability of linear differential equations with several delays. J. Math. Anal. Appl. 324, 1336–1355 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 63.
    Berezansky, L., Braverman, E.: Explicit exponential stability conditions for linear differential equations with several delays. J. Math. Anal. Appl. 332, 246–264 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 81.
    Berezansky, L., Domshlak, Y., Braverman, E.: On oscillation properties of delay differential equations with positive and negative coefficients. J. Math. Anal. Appl. 274, 81–101 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 89.
    Chaplygin, S.A.: Foundations of New Method of Approximate Integration of Differential Equations, Moscow, 1919 (Collected Works 1), pp. 348–368. GosTechIzdat, Moscow (1948) Google Scholar
  22. 96.
    Corduneanu, C.: Integral Equations and Applications. Cambridge University Press, Cambridge (1991) zbMATHCrossRefGoogle Scholar
  23. 97.
    Corduneanu, C.: Abstract Volterra equations: a survey. Nonlinear operator theory. Math. Comput. Model. 32, 1503–1528 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 98.
    Corduneanu, C.: Functional Equations with Causal Operators. Stability and Control: Theory, Methods and Applications, vol. 16. Taylor & Francis, London (2002) zbMATHCrossRefGoogle Scholar
  25. 102.
    de La Vallee Poussin, Ch.J.: Sur l’equation differentielle lineaire du second ordre. J. Math. Pures Appl. 8(9), 125–144 (1929) Google Scholar
  26. 108.
    Diblík, J., Svoboda, Z.: An existence criterion of positive solutions of p-type retarded functional differential equations. J. Comput. Appl. Math. 147, 315–331 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  27. 110.
    Domoshnitsky, A.: Extension of Sturm’s theorem to equations with time-lag. Differ. Uravn. 19, 1475–1482 (1983) MathSciNetGoogle Scholar
  28. 111.
    Domoshnitsky, A.: Conservation of sign of Cauchy function and stability of neutral equations. Bound. Value Probl., 44–48 (1986) (in Russian) Google Scholar
  29. 121.
    Domoshnitsky, A.: Maximum principles and nonoscillation intervals for first order Volterra functional differential equations. Dyn. Contin. Discrete Impuls. Syst., Ser. A Math. Anal. 15, 769–814 (2008) MathSciNetzbMATHGoogle Scholar
  30. 122.
    Domoshnitsky, A.: Maximum principles and boundary value problems. In: A. Cabada, E. Liz, J. Nieto (eds.) Mathematical Models in Engineering, Biology and Medicine, Proceedings of International Conference on Boundary Value Problems, pp. 89–100. Am. Inst. of Phys., Melville (2009) Google Scholar
  31. 123.
    Domoshnitsky, A.: Maximum principles, boundary value problems and stability for first order delay equations with oscillating coefficient. Int. J. Qualitative Theory Differ. Equ. Appl. 3, 33–42 (2009) Google Scholar
  32. 125.
    Domoshnitsky, A.: Differential inequalities for one component of solution vector for systems of functional differential equations. Adv. Differ. Equ. 2010 (2010). Art.ID 478020, 14 pp. Google Scholar
  33. 126.
    Domoshnitsky, A., Drakhlin, M.: Nonoscillation of first order impulse differential equations with delay. J. Math. Anal. Appl. 206, 254–269 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  34. 129.
    Domoshnitsky, A., Drakhlin, M., Litsyn, E.: Nonoscillation and positivity of solutions to first order state-dependent differential equations with impulses in variable moments. J. Differ. Equ. 228, 39–48 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  35. 130.
    Domoshnitsky, A., Drakhlin, M., Litsyn, E.: On equations with delay depending on solution. Nonlinear Anal. 49, 689–701 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  36. 132.
    Domoshnitsky, A., Goltser, Ya.: One approach to study stability of integro-differential equations. Nonlinear Anal. 47, 3885–3896 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  37. 138.
    Domoshnitsky, A., Maghakyan, A., Shklyar, R.: Maximum principles and boundary value problems for first order neutral functional differential equations. J. Inequal. Appl. 2009 (2009). Art.ID 141959, 26 pp. Google Scholar
  38. 139.
    Domoshnitsky, A., Sheina, M.V.: Nonnegativity of the Cauchy matrix and the stability of a system of linear differential equations with retarded argument. Differ. Equ. 25, 145–150 (1989) MathSciNetGoogle Scholar
  39. 146.
    Drakhlin, M.E.: Inner superposition operator in spaces of integrable functions. Izv. Vysš. Učebn. Zaved., Mat. 5, 18–24 (1986), 88 (in Russian) MathSciNetGoogle Scholar
  40. 147.
    Drakhlin, M.E., Plyshevskaya, T.K.: On the theory of functional-differential equations. Differ. Uravn. 14, 1347–1361 (1978) (in Russian) MathSciNetzbMATHGoogle Scholar
  41. 148.
    Drozdov, A.D., Kolmanovskii, V.B.: Stability in Viscoelasticity. North-Holland, Amsterdam (1994) zbMATHGoogle Scholar
  42. 154.
    Erbe, L.H., Kong, Q., Zhang, B.G.: Oscillation Theory for Functional Differential Equations. Dekker, New York (1995) Google Scholar
  43. 165.
    Golden, J.M., Graham, G.A.C.: Boundary Value Problems in Linear Viscoelasticity. Springer, Berlin (1988) zbMATHGoogle Scholar
  44. 178.
    Gusarenko, S.A., Domoshnitskii, A.I.: Asymptotic and oscillation properties of the first order linear scalar functional differential equations. Differ. Equ. 25, 1480–1491 (1989) MathSciNetzbMATHGoogle Scholar
  45. 192.
    Győri, I., Ladas, G.: Oscillation Theory of Delay Differential Equations with Applications. Clarendon Press, New York (1991) Google Scholar
  46. 194.
    Győri, I., Pituk, M.: Stability criteria for linear delay differential equations. Differ. Integral Equ. 10, 841–852 (1997) Google Scholar
  47. 197.
    Hakl, R., Lomtatidze, A., Ŝremr, J.: Some Boundary Value Problems for First Order Scalar Functional Differential Equations. FOLIA, Masaryk University, Brno, Czech Republic (2002) zbMATHGoogle Scholar
  48. 198.
    Hakl, R., Lomtatidze, A., Ŝremr, J.: On a boundary-value problem of periodic type for first-order linear functional differential equations. Nonlinear Oscil. (N.Y.) 5, 408–425 (2002) MathSciNetCrossRefGoogle Scholar
  49. 210.
    Islamov, G.G.: On an estimate of the spectral radius of the linear positive compact operator. In: Functional-Differential Equations and Boundary Value Problems in Mathematical Physics, pp. 119–122. Perm Politekhnical Institute, Perm (1978) (in Russian) Google Scholar
  50. 211.
    Islamov, G.G.: On an upper estimate of the spectral radius. Dokl. Acad. Nauk USSR 322, 836–838 (1992) MathSciNetGoogle Scholar
  51. 216.
    Kantorovich, L.V., Vulich, B.Z., Pinsker, A.G.: Functional Analysis in Semi-Ordered Spaces. GosTechIzdat, Moscow (1950) (in Russian) Google Scholar
  52. 223.
    Kiguradze, I., Pu̇z̆a, B.: Boundary Value Problems for Systems of Linear Functional Differential Equations. FOLIA, Masaryk University, Brno, Czech Republic (2003) zbMATHGoogle Scholar
  53. 248.
    Ladde, G.S., Lakshmikantham, V., Zhang, B.G.: Oscillation Theory of Differential Equations with Deviating Arguments. Dekker, New York (1987) Google Scholar
  54. 250.
    Lakshmikantham, V., Leela, S.: Differential and Integral Inequalities. Academic Press, New York (1969) zbMATHGoogle Scholar
  55. 251.
    Lakshmikantham, V., Wen, L., Zhang, B.G.: Theory of Differential Equations with Unbounded Delay. Kluwer Academic, Dordrecht (1994) zbMATHGoogle Scholar
  56. 256.
    Levin, A.J.: Nonoscillation of solution of the equation x (n)+p n−1(t)x (n−1)+⋯+p 0(t)x=0. Usp. Mat. Nauk 24, 43–96 (1969) Google Scholar
  57. 278.
    Luzin, N.N.: About method of approximate integration of Acad. S.A. Chaplygin. Usp. Mat. Nauk 6, 3–27 (1951) MathSciNetzbMATHGoogle Scholar
  58. 289.
    Myshkis, A.D.: Linear Differential Equations with Retarded Argument. Nauka, Moscow (1972) (in Russian) zbMATHGoogle Scholar
  59. 308.
    Polia, G.: On the mean-value theorem corresponding to a given linear homogeneous differential equations. Trans. Am. Math. Soc. 24, 312–324 (1924) CrossRefGoogle Scholar
  60. 309.
    Protter, M.H., Weinberger, H.F.: Maximum Principles in Differential Equations. Springer, New York (1984) zbMATHCrossRefGoogle Scholar
  61. 328.
    Tyshkevich, V.A.: Some Problems of the Stability Theory of Functional Differential Equations. Naukova Dumka, Kiev (1981) (in Russian) Google Scholar
  62. 333.
    Wilkins, J.E.: The converse of a theorem of Tchaplygin on differential inequalities. Bull. Am. Math. Soc. 53(4), 112–120 (1947) Google 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