On the number of zeros of nonoscillatory solutions to half-linear ordinary differential equations involving a parameter

In this paper the following half-linear ordinary differential equation is considered:

where $\alpha > 0$ is a constant, $\lambda > 0$ is a parameter, and $p(t)$ is a continuous function on $[a, \infty)$, $a > 0$, and $p(t) > 0$ for $t \in [a, \infty)$. The main purpose is to show that precise information about the number of zeros can be drawn for some special type of solutions $x(t; \lambda)$ of (H $_{\lambda})$ such that

It is shown that, if $\alpha \geq 1$ and if (H $_{\lambda})$ is strongly nonoscillatory, then there exists a sequence $\{\lambda_{n}\}_{n=1}^{\infty}$ such that $0=\lambda_{0}<\lambda_{1}<\cdots< \lambda_{n}<\cdots$,   $\lambda_{n} \to +\infty$ as $n \to \infty$; and $x(t; \lambda)$ with $\lambda = \lambda_n$ has exactly $n-1$ zeros in the interval $(a,\infty)$ and $x(a; \lambda_n) = 0$; and $x(t; \lambda)$ with $\lambda \in (\lambda_{n-1}, \lambda_n)$ has exactly $n-1$ zeros in $(a,\infty)$ and $x(a; \lambda_n) \neq 0$. For the proof of the theorem, we make use of the generalized Prüfer transformation, which consists of the generalized sine and cosine functions.