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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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