Manabu Naito, Hiroyuki Usami

Singular Strongly Increasing Solutions of Higher-Order Quasilinear Ordinary Differential Equations

abstract:
In the present paper, we consider quasilinear ordinary differential equations of the form
\begin{equation*}
D(\alpha_n, \alpha_{n-1}, \dots, \alpha_1)x = p(t)|x|^{\beta}\sgn x, \;\; t \geq a, \eqno{(1.1)}
\end{equation*}
where $D(\alpha_n, \alpha_{n-1}, \dots, \alpha_1)$ is the $n$th-order iterated differential operator such that
\begin{equation*}
D(\alpha_n, \alpha_{n-1}, \dots, \alpha_1)x \equiv D(\alpha_n)D(\alpha_{n-1})\cdots D(\alpha_1)x
\end{equation*}
and, in general, $D(\alpha)$ is the first-order differential operator defined by $D(\alpha)x = (d/dt)(|x|^{\alpha}\sgn x)$ for $\alpha > 0$. For the case where $\alpha_1\alpha_2\cdots \alpha_n < \beta$, we present a new sufficient condition for all strongly increasing solutions of \eqref{1.1} to be singular. If $\alpha_1=\alpha_2=\cdots=\alpha_n=1$, then one of the main results, Corollary 3.2, gives an extension of the well-known theorem of Kiguradze and Chanturia [2].

Mathematics Subject Classification: 34C11

Key words and phrases: Singular solutions, strongly increasing solutions, quasilinear equations