On Kneser solutions of higher order nonlinear ordinary differential equations

The equation x((n)) (t)=(-1)(n)\x(t)\(k) with k>1 is considered. In the case n less than or equal to 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 Ic. 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.