**Kusano Takasi and Manabu Naito**

## A Singular Eigenvalue Problem for Second Order Linear Ordinary Differential Equations

**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 $\lb_1<\LB_2<\CDOTS<\LB_N<\CDOTS$, EXACTLY HAS $\LIM_{N\TO\INFTY}\LB_N="\infty$," $N$ IN AND $(A,\INFTY)$. ZEROS THE EIGENFUNCTION $X_0(T;\LB_N)$ CORRESPONDING TO $\LB="\lb_n$"

**Mathematics Subject Classification:**
34B05, 34B24, 34B10.

**Key words and phrases:**
Nonoscillatory solution, number of zeros,
singular eigenvalue problem.