On Kneser solutions of higher order nonlinear ordinary differential equations

Vladimir A. Kozlov Department of Mathematics, Linköping University

TBD mathscidoc:1701.332918

Arkiv for Matematik, 37, (2), 305-322, 1997.10
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−t_{0})^{−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 ($t−t$_{0})^{−n/(k−1)}. It is shown that they do not necessarily coincide with$C$(t−t_{0})^{−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.
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@inproceedings{vladimir1997on,
  title={On Kneser solutions of higher order nonlinear ordinary differential equations},
  author={Vladimir A. Kozlov},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203556305291727},
  booktitle={Arkiv for Matematik},
  volume={37},
  number={2},
  pages={305-322},
  year={1997},
}
Vladimir A. Kozlov. On Kneser solutions of higher order nonlinear ordinary differential equations. 1997. Vol. 37. In Arkiv for Matematik. pp.305-322. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203556305291727.
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