# Singular eigenvalue problems for second order linear ordinary differential equations

Archivum Mathematicum (1998)

• Volume: 034, Issue: 1, page 59-72
• ISSN: 0044-8753

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## Abstract

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We consider linear differential equations of the form ${\left(p\left(t\right){x}^{\text{'}}\right)}^{\text{'}}+\lambda q\left(t\right)x=0\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\left(p\left(t\right)>0,\phantom{\rule{3.33333pt}{0ex}}q\left(t\right)>0\right)\phantom{\rule{2.0em}{0ex}}\left(\mathrm{A}\right)$ on an infinite interval $\left[a,\infty \right)$ and study the problem of finding those values of $\lambda$ for which () has principal solutions ${x}_{0}\left(t;\lambda \right)$ vanishing at $t=a$. This problem may well be called a singular eigenvalue problem, since requiring ${x}_{0}\left(t;\lambda \right)$ to be a principal solution can be considered as a boundary condition at $t=\infty$. Similarly to the regular eigenvalue problems for () on compact intervals, we can prove a theorem asserting that there exists a sequence $\left\{{\lambda }_{n}\right\}$ of eigenvalues such that $0<{\lambda }_{0}<{\lambda }_{1}<\cdots <{\lambda }_{n}<\cdots$, $\underset{n\to \infty }{lim}{\lambda }_{n}=\infty$, and the eigenfunction ${x}_{0}\left(t;{\lambda }_{n}\right)$ corresponding to $\lambda ={\lambda }_{n}$ has exactly $n$ zeros in $\left(a,\infty \right),\phantom{\rule{3.33333pt}{0ex}}n=0,1,2,\cdots$. We also show that a similar situation holds for nonprincipal solutions of () under stronger assumptions on $p\left(t\right)$ and $q\left(t\right)$.

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