Georgian Mathematical Journal

, Volume 2, Issue 4, pp 395–418 | Cite as

On proper oscillatory and vanishing-at-infinity solutions of differential equations with a deviating argument

  • I. Kiguradze
  • D. Chichua
Article

Abstract

Sufficient conditions are found for the existence of multiparametric families of proper oscillatory and vanishing-at-infinity solutions of the differential equation
$$u^{(n)} (t) = g\left( {t, u(\tau _0 (t)), \ldots ,u^{(m - 1)} (\tau _{m - 1} (t))} \right)$$
, wheren≥4,m is the integer part of π/2,g:R+×R m R is a function satisfying the local Carathéodory conditions, and τ i :R+R(i=0,...,m−1) are measurable functions such that τ i (t) →+∞ fort→+∞(i=0,...,m−1).

1991 Mathematics Subject Classification

34K15 34K10 

Key words and phrases

Functional differential equation proper solution oscillatory solution vanishing-at-infinity solution 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    R. G. Koplatadze and T. A. Chanturia, On oscillatory properties of differential equations with a deviating argument. (Russian)Tbilisi Univ. Press, Tbilisi, 1977.Google Scholar
  2. 2.
    Christos G. Philos, An oscillatory and asymptotic classification of the solutions of differential equations with deviating arguments.Atti. Acad. Naz. Lincei. Rend. Cl. Sci. fis. mat. e natur. 63(1977), No. 3–4, 195–203.Google Scholar
  3. 3.
    V. N. Shevelo, Oscillation of solutions of differential equations with a deviating argument. (Russian)Naukova Dumka, Kiev, 1978.Google Scholar
  4. 4.
    Yu. I. Domshlak, A comparison method by Shturm for investigation of behavior of solutions of differential-operator equations. (Russian)Elm, Baku, 1986.Google Scholar
  5. 5.
    U. Kitamura, Oscillation of functional differential equations with general deviating arguments.Hiroshima Math. J. 15(1985), 445–491.Google Scholar
  6. 6.
    M. E. Drakhlin, On oscillation properties of some functional differential equations. (Russian)Differentisial'nyie Uravneniya 22(1986), No. 3, 396–402.Google Scholar
  7. 7.
    J. Jaroš and T. Kusano, Oscillation theory of higher order linear functional differential equations of neutral type.Hiroshima Math. J. 18(1988), 509–531.Google Scholar
  8. 8.
    R. G. Koplatadze, On differential equations with deviating arguments having propertiesA andB. (Russian)Differentsial'nye Uravneniya 25(1989), No. 11, 1897–1909.Google Scholar
  9. 9.
    R. G. Koplatadze, On monotone and oscillatory solutions ofnth order differential equations with deviating arguments. (Russian)Mathematica Bohemica 116(1991), No. 3, 296–308.Google Scholar
  10. 10.
    I. Kiguradze and D. Chichua, On some boundary value problems with integral conditions for functional differential equations.Georgian Math. J. 2(1995), No. 2, 165–188.CrossRefGoogle Scholar
  11. 11.
    I. T. Kiguradze and T. A. Chanturia, Asymptotic properties of solutions of nonautonomous ordinary differential equations.Kluwer Acadmic Publishers, Dodrecht, Boston, London, 1993.Google Scholar
  12. 12.
    M. Biernacki, Sur l'équation differentielley 4+A(x)y=0.Prace Ann. Univ. M. Curie-Skodowska 6(1952), 65–78.Google Scholar
  13. 13.
    M. Švec, Sur le comportement asymptotique des intégrales de l'équation differentielley (u) +Q(x)y=0.Czechosl. Math. J. 8(1958), No. 2, 450–462.Google Scholar
  14. 14.
    I. T. Kiguradze, On vanishing-at-infinity solutions of ordinary differential equations.Czechosl. Math. J. 33(1983), No. 4, 613–646.Google Scholar
  15. 15.
    M. Bartušek, Asymptotic properties of oscillatory solutions of differential equations of thenth order.Masaryk University, Brno, 1992.Google Scholar

Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • I. Kiguradze
    • 1
  • D. Chichua
    • 2
  1. 1.A. Razmadze Mathematical InstituteGeorgian Academy of SciencesTbilisiRepublic of Georgia
  2. 2.I. Vekua Institute of Applied MathematicsTbilisi State UniversityTbilisiRepublic of Georgia

Personalised recommendations