Advertisement

Nonoscillation Intervals for n-th-Order Equations

  • Ravi P. Agarwal
  • Leonid Berezansky
  • Elena Braverman
  • Alexander Domoshnitsky
Chapter
  • 820 Downloads

Abstract

In Chap. 17, n-th-order functional differential equations are considered. In order to create a concept of nonoscillation for functional differential equations which will be an analogue of nonoscillation theory for ordinary differential equations, we define a nonoscillation interval for the n-th-order homogeneous functional differential equation as interval, where each nontrivial solution can have at most n−1 zeros, counting every zero with its multiplicity. The study of many classical questions of the qualitative theory of n-th-order equations such as existence and uniqueness of solutions of the interpolation boundary value problems, positivity or corresponding regular behavior of their Green’s functions, maximum principles and stability, was considered in the general framework based on the notion of nonoscillation intervals of corresponding linear functional differential equations. It is demonstrated that if the Wronskian of the fundamental system is not equal to zero at the initial point of the interval, then the nonoscillation interval exists for functional differential equations; this is true for a particular case of the n-th-order functional differential equations with Volterra operators (delay equations). For second-order functional differential equations, nonvanishing Wronskian is equivalent to the Sturm theorem on separation of zeros (between two adjacent zeros of any nontrivial solution there is a zero of every other solution). The analogue of the classical Mammana theorem, establishing necessary and sufficient nonoscillation conditions through nonvanishing normal chains of the Wronskians, is proven for n-th-order functional differential equations.

For an n-th-order linear ordinary differential equation the classical Levin-Chichkin theorems claims that Green’s functions of all interpolation (De la Vallee-Poussin) problems behave regularly (preserve their signs in corresponding zones) on the nonoscillation interval. Under corresponding sign conditions on the relevant operators, this theorem is extended to n-th-order functional differential equations. Several possibilities to avoid these sign conditions are proposed, and various explicit tests for positivity and negativity of Green’s functions of two and multipoint boundary value problems are obtained.

Keywords

Nonoscillation Functional Differential Equations Adjacent Zeros Volterra Operator Wronskian 
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.

References

  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. 26.
    Azbelev, N.V., Domoshnitsky, A.: On the question of linear differential inequalities. I. Differ. Equ. 27, 257–263 (1991) MathSciNetzbMATHGoogle Scholar
  3. 27.
    Azbelev, N.V., Domoshnitsky, A.: On the question of linear differential inequalities. II. Differ. Equ. 27, 641–647 (1991) MathSciNetzbMATHGoogle Scholar
  4. 28.
    Azbelev, N.V., Maksimov, V.P., Rakhmatullina, L.F.: Introduction to Theory of Functional-Differential Equations. Nauka, Moscow (1991) (in Russian) zbMATHGoogle Scholar
  5. 37.
    Beesack, P.R.: On the Green’s function of an n-point boundary value problem. Pac. J. Math. 12, 801–812 (1962) MathSciNetzbMATHGoogle Scholar
  6. 82.
    Berkowitz, K., Domoshnitsky, A., Maghakyan, A.: About functional differential generalization of Burger’s equation. Funct. Differ. Equ. 17, 53–60 (2010) MathSciNetGoogle Scholar
  7. 93.
    Chichkin, E.S.: Theorem about differential inequality for multipoint boundary value problems. Izv. Vysš. Učebn. Zaved., Mat. 2, 170–179 (1962) Google Scholar
  8. 100.
    Deift, V.A.: Conditions of nonoscillation for linear homogeneous differential equations with delayed argument. Differential Equations 10, 1957–1963 (1974) MathSciNetzbMATHGoogle Scholar
  9. 101.
    Deift, V.A.: Condition of nonoscillation for linear homogeneous differential equations. Thesis, Alma-Ata (1977) (in Russian) Google Scholar
  10. 110.
    Domoshnitsky, A.: Extension of Sturm’s theorem to equations with time-lag. Differ. Uravn. 19, 1475–1482 (1983) MathSciNetGoogle Scholar
  11. 112.
    Domoshnitsky, A.: Preserving the sign of the Green function of a two-point boundary value problem for an nth-order functional-differential equation. Differ. Equ. 25, 666–669 (1989) MathSciNetGoogle Scholar
  12. 114.
    Domoshnitsky, A.: Factorization of a linear boundary value problem and the monotonicity of the Green operator. Differ. Equ. 28, 323–327 (1992) MathSciNetGoogle Scholar
  13. 116.
    Domoshnitsky, A.: Sturm’s theorem for equation with delayed argument. Georgian Math. J. 1, 267–276 (1994) zbMATHCrossRefGoogle Scholar
  14. 117.
    Domoshnitsky, A.: Unboundedness of solutions and instability of second order equations with delayed argument. Differ. Integral Equ. 14, 559–576 (2001) MathSciNetzbMATHGoogle Scholar
  15. 118.
    Domoshnitsky, A.: Wronskian of fundamental system of delay differential equations. Funct. Differ. Equ. 9, 353–376 (2002) MathSciNetzbMATHGoogle Scholar
  16. 119.
    Domoshnitsky, A.: About asymptotic and oscillation properties of the Dirichlet problem for delay partial differential equations. Georgian Math. J. 10, 495–502 (2003) MathSciNetzbMATHGoogle Scholar
  17. 120.
    Domoshnitsky, A.: Maximum principle for functional equations in the space of discontinuous functions of three variables. J. Math. Anal. Appl. 329, 238–267 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 124.
    Domoshnitsky, A.: Nonoscillation interval for n-th order functional differential equations. Nonlinear Anal. 71, e2449–e2456 (2009) MathSciNetCrossRefGoogle Scholar
  19. 126.
    Domoshnitsky, A., Drakhlin, M.: Nonoscillation of first order impulse differential equations with delay. J. Math. Anal. Appl. 206, 254–269 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 127.
    Domoshnitsky, A., Drakhlin, M., Litsyn, E.: On Nth order functional-differential equations with impulses. Mem. Differ. Equ. Math. Phys. 12, 50–56 (1997) MathSciNetzbMATHGoogle Scholar
  21. 128.
    Domoshnitsky, A., Drakhlin, M., Litsyn, E.: On boundary value problems for Nth order functional differential equations with impulses. Adv. Math. Sci. Appl. 8, 987–996 (1998) MathSciNetzbMATHGoogle Scholar
  22. 158.
    Gantmakher, F.R., Kreĭn, M.R.: Oscillatory Matrices and Kernels and Small Oscillations of Mechanical Systems. GosTechIzdat, Moscow, Leningrad (1950) Google Scholar
  23. 192.
    Győri, I., Ladas, G.: Oscillation Theory of Delay Differential Equations with Applications. Clarendon Press, New York (1991) Google Scholar
  24. 203.
    Hartman, P.: Ordinary Differential Equations. Wiley, New York, London, Sydney (1964) zbMATHGoogle Scholar
  25. 204.
    Hartman, P., Winter, A.: On non-conservative linear oscillators of low frequency. Am. J. Math. 70, 529–539 (1948) zbMATHCrossRefGoogle Scholar
  26. 220.
    Kiguradze, I.T., Chanturia, T.A.: Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations. Kluwer Academic, Dordrecht, Boston, London (1993) zbMATHCrossRefGoogle Scholar
  27. 241.
    Labovskii, S.M.: Condition of nonvanishing of Wronskian of fundamental system of linear equation with delayed argument. Differ. Uravn. 10, 426–430 (1974) (in Russian) MathSciNetGoogle Scholar
  28. 248.
    Ladde, G.S., Lakshmikantham, V., Zhang, B.G.: Oscillation Theory of Differential Equations with Deviating Arguments. Dekker, New York (1987) Google Scholar
  29. 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
  30. 282.
    Mammana, G.: Decomposizione delle espressioni differenziali omogenee in prodotto di fattori simbolici e applicazione relativa allo studion delle equazioni differenzi ali lineari. Math. Z. 33, 186–231 (1931) MathSciNetCrossRefGoogle Scholar
  31. 289.
    Myshkis, A.D.: Linear Differential Equations with Retarded Argument. Nauka, Moscow (1972) (in Russian) zbMATHGoogle Scholar
  32. 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
  33. 314.
    Sansone, G.: Equazioni Differenziali nel Campo Reale, 2nd edn. Zanichelli, Bologna (1948) (in Italian) 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