Here, the authors consider half-linear ordinary differential equations involving a parameter of the form \[ \left(|x'|^\alpha\text{sgn }x'\right)'+\lambda p(t) |x|^\alpha\text{sgn }x=0,\quad t\geq a,\tag{\(H_\lambda\)} \] where \(\alpha>0\) is a constant, \(\lambda>0\) is a parameter; \(p(t)\) is a continuous function on \([a, \infty), a>0\), and \(p(t)>0\) for \(t\in [a, \infty)\). The main purpose of this paper is to show that, in the case \(\alpha\geq 1\), precise information on the number of zeros can be drawn for some special type of solutions \(x_\lambda(t)\) to (\(H_\lambda\)) such that \[ \lim_{t\rightarrow \infty}\frac{x_\lambda(t)}{\sqrt{t}}=0.\tag{*} \] The existence and the essential uniqueness of a nonoscillatory solution \(x_\lambda(t)\) satisfying \((*)\) are guaranteed by the following theorem:
Let \(\alpha>0\) and suppose that \[ \int_a^\infty p(s) ds < +\infty\quad\text{and}\quad\lim_{t\rightarrow \infty}t^\alpha\int_t^\infty p(s) ds=0.\tag{**} \] Then, for each \(\lambda>0\), there is a nonoscillatory solution \(x_\lambda(t)\) to (\(H_\lambda\)) satisfying \((*)\). Furthermore, if \(\alpha\geq 1\), then such a solution \(x_\lambda(t)\) is uniquely determined up to a nonzero constant multiple.
For the solutions \(x_\lambda(t)\) by employing the generalized Pr\(\ddot{\text{u}}\)fer transformation, which consists of the generalized sine and cosine functions, the authors establish the next theorem concerning the number of zeros:
Let \(\alpha\geq 1\) and suppose that \((**)\) holds. Then there exists a sequence \(\{\lambda_n\}_{n=1}^{\infty}\) of positive parameters with the properties that (i) \(0=\lambda_0<\lambda_1<\cdots<\lambda_n<\cdots\), \(\lim_{n\rightarrow \infty}\lambda_n=+\infty\); (ii) if \(\lambda\in (\lambda_{n-1}, \lambda_n), n=1, 2, \cdots\), then \(x_\lambda(t)\) has exactly \(n-1\) zeros in the open interval \((a, \infty)\) and \(x_\lambda(a)\neq 0\); (iii)  if \(\lambda=\lambda_n, n=1, 2, \cdots\), then \(x_\lambda(t)\) has exactly \(n-1\) zeros in the open interval \((a, \infty)\) and \(x_\lambda(a) = 0\).
This last theorem, for the case \(\alpha=1\), is given by T. Kusano and the second author [Differ. Equations 34, 302-311 (1998; Zbl 0951.34015)]. This theorem also gives a partial extension of a previous result by A. Elbert and the authors [On the number of zeros of nonoscillatory solutions to second-order half-linear differential equations, Ann. Univ. Sci. Budapest. Eőtvős Sect. Math. 42 (1999), 101-131 (2000)], where it has been shown that if \(\alpha>0\) and \[ \int_a^\infty p(s) ds < +\infty\quad\text{and}\quad\int_a^\infty\left(\int_s^\infty p(r) dr\right)^{1/\alpha} ds< +\infty, \] then equation (\(H_\lambda\)) has a unique solution \(x_\lambda(t)\) satisfying \( \lim_{t\rightarrow \infty}x_\lambda(t)=1\) and there exists a sequence \(\{\lambda_n\}_{n=1}^{\infty}\) having the same properties as in this theorem. Note that this theorem is considered for the case \(\alpha\geq 1\). The case \(0<\alpha< 1\) is presently open.