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Keywords:
Singular eigenvalue problem; Sturm-Liouville equation; zeros of nonoscillatory solutions
Singular eigenvalue problem; Sturm-Liouville equation; zeros of nonoscillatory solutions
Summary:
We consider linear differential equations of the form \[ (p(t)x^{\prime })^{\prime }+\lambda q(t)x=0~~~(p(t)>0,~q(t)>0) \qquad \mathrm {(A)}\] on an infinite interval $[a,\infty )$ and study the problem of finding those values of $\lambda $ for which () has principal solutions $x_{0}(t;\lambda )$ vanishing at $t = a$. This problem may well be called a singular eigenvalue problem, since requiring $x_{0}(t;\lambda )$ 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 $\lbrace \lambda _{n}\rbrace $ of eigenvalues such that $\displaystyle 0<\lambda _{0}<\lambda _{1}<\cdots <\lambda _{n}<\cdots $, $\displaystyle \lim _{n\rightarrow \infty }\lambda _{n}=\infty $, and the eigenfunction $x_{0}(t;\lambda _{n})$ corresponding to $\lambda = \lambda _{n}$ has exactly $n$ zeros in $(a,\infty ),~n=0,1,2,\dots $. We also show that a similar situation holds for nonprincipal solutions of () under stronger assumptions on $p(t)$ and $q(t)$.
References:
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