Singular solutions of nonlinear ordinary differential equations (Q1387377)

The author investigates the existence of singular solutions to the \(m\)th-order differential equation \[ y^{(m)}=Q(t,y,\dots,y^{(m-1)}).\tag{*} \] Conditions on the function \(Q\) are given which guarantee that every Kneser solution to (*) (i.e. a solution satisfying \((-1)^iy^{(i)}(t)\geq 0\), \(i=0,\dots m-1\), for large \(t\)) is a singular solution of first kind. Conditions when every solution of another certain class is a singular solution of second kind are also given. Recall that a solution \(y\) to (*) is called singular of first (second) kind if \(y(t)\equiv 0\) eventually (there exists a finite number \(t^{*}\) such that \(\lim_{t\to t^*}| y^{(m-1)}(t)|=\infty\)). The results of the paper extend statements concerning the behavior of higher-order Emden-Fowler equations given e.g. in the monograph [\textit{I. T. Kiguradze} and \textit{T. A. Chanturia}, Asymptotic properties of solutions on nonautonomous ordinary differential equations. Transl. from the Russian. Mathematics and Its Applications. Sov. Ser. 89. Dordrecht: Kluwer (1993; Zbl 0782.34002)].