Kusano Takasi and Manabu Naito

A Singular Eigenvalue Problem for Second Order Linear Ordinary Differential Equations

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.