On Kneser solutions of higher order nonlinear ordinary differential equations

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.