Kusano Takasi and Manabu Naito
abstract:
The Sturm-Liouville equation of the form
% (A)
\begin{equation}
(p(t)x')'+\lb q(t)x=0\;\;\;(p(t)>0,\;\;q(t)>0), \tag{A}
\end{equation}
is considered on an infinite interval $[a,+\infty[$ and the problem of finding
the values of $\lb$ for which $(A)$ has a principal solution $x_0(t;\lb)$
satisfying $\al x_0(a;\lb)-\bt p(a)x_0'(a;\lb)=0$, $\al^2+\bt^2>0$, is studied:
Assuming that $(A)$ is strongly nonoscillatory in the sense of Nehari, a
general theorem is proved asserting that, similarly to the regular eigenvalue
problems on compact intervals, there exists a sequence $\{\lb_n\}$ of
eigenvalues such that $\lb_1<\lb_2<\cdots<\lb_n<\cdots$,
$\lim_{n\to\infty}\lb_n=\infty$, and the eigenfunction $x_0(t;\lb_n)$
corresponding to $\lb=\lb_n$ has exactly $n$ zeros in $(a,\infty)$.
Mathematics Subject Classification: 34B05, 34B24, 34B10.
Key words and phrases: Nonoscillatory solution, number of zeros, singular eigenvalue problem.