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On Kneser solutions of higher order nonlinear ordinary differential equations. (English) Zbl 1118.34317
Summary: The equation $$x^{(n)}(t)=(-1)^n|x(t)|^k$$, with $$k>1$$ is considered. In the case $$n\leq 4$$ 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.

##### MSC:
 34C11 Growth and boundedness of solutions to ordinary differential equations 34D05 Asymptotic properties of solutions to ordinary differential equations
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##### References:
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