Arkiv för Matematik

  • Ark. Mat.
  • Volume 37, Number 2 (1999), 305-322.

On Kneser solutions of higher order nonlinear ordinary differential equations

Vladimir A. Kozlov

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Abstract

The equation x(n)(t)=(−1)n│x(t)│k with k>1 is considered. In the case n≦4 it is proved that solutions defined in a neighbourhood of infinity coincide with C(t−t0)−n/(k−1), where C is a constant depending only on n and k. In the general case such solutions are Kneser solutions and can be estimated from above and below by a constant times (t−t0)−n/(k−1). It is shown that they do not necessarily coincide with C(t−t0)−n/(k−1). This gives a negative answer to two conjectures posed by Kiguradze that Kneser solutions are determined by their value in a point and that blow-up solutions have prescribed asymptotics.

Dedication

Dedicated to Professor Vladimir Maz'ya on the occasion of his 60th birthday.

Note

The author was supported by the Swedish Natural Science Research Council (NFR) grant M-AA/MA 10879-304.

Article information

Source
Ark. Mat., Volume 37, Number 2 (1999), 305-322.

Dates
Received: 9 October 1997
Revised: 20 October 1998
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485898637

Digital Object Identifier
doi:10.1007/BF02412217

Mathematical Reviews number (MathSciNet)
MR1714766

Zentralblatt MATH identifier
1118.34317

Rights
1999 © Institut Mittag-Leffler

Citation

Kozlov, Vladimir A. On Kneser solutions of higher order nonlinear ordinary differential equations. Ark. Mat. 37 (1999), no. 2, 305--322. doi:10.1007/BF02412217. https://projecteuclid.org/euclid.afm/1485898637


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References

  • Astashova, I. V., Asymptotic behavior of solutions of certain nonlinear differential equations, in Reports of the extended sessions of a seminar of the I. N. Vekua Institute of Applied Mathematics 1∶3 (Kiguradze, I. T., ed.), pp. 9–11, Tbilis. Gos. Univ., Tbilisi, 1985 (Russian).
  • Guckenheimer, J. and Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Appl. Math. Sci. 42, Springer-Verlag, New York, 1983.
  • Kiguradze, I. T., On Kneser solutions of ordinary differential equations, Uspekhi Mat. Nauk 41:4 (1986), 211 (Russian).
  • Kiguradze, I. T. and Chanturia, T. A., Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations, Kluwer, Dordrecht, 1993.
  • Kneser, A., Untersuchung und asymptotische Darstellung der Integrale gewisser Differentialgleichungen bei grossen reelen Werten des Arguments, J. Reine Angew. Math. 116 (1896), 178–212.
    Zentralblatt MATH: 27.0253
  • Kozlov, V. and Maz'ya, V., Theory of a Higher-Order Sturm-Liouville Equations, Lecture Notes in Math. 1659, Springer-Verlag, Berlin-Heidelberg, 1997.
  • Kozlov, V. and Maz'ya, V., Differential equations with Operator Coefficients with Applications to Boundary Value Problems for Partial Differential Equations), Monogr. Math., Springer-Verlag, Berlin-Heidelberg, 1999.

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