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Singular eigenvalue problems for second-order linear ordinary differential equations. (English) Zbl 0914.34021
Given the second-order Sturm-Liouville equation $(p(t)x')'+\lambda q(t)=0,\quad p(t)>0,\;q(t)>0,\tag{$$*$$}$ with a real-valued parameter $$\lambda$$ and $$t\in [a,\infty)$$, the authors characterize the set of $$\lambda$$ for which the principal solution to $$(*)$$ satisfies $$x(a)=0$$. This problem can be considered as a “singular” extension of the classical Sturm-Liouville eigenvalue problem on a compact interval $$[a,b]$$ with Dirichlet boundary conditions.
Similar to this regular case, there exists a sequence $$0<\lambda _0<\lambda _1<\dots <\lambda _n\to \infty$$ and the corresponding eigenvalue functions $$u_n$$, $$n=0,1,\dots ,$$ have exactly $$n$$ zeros in $$(a,\infty)$$.
An open problem concerning a possible relaxation of the restriction on coefficients $$p,q$$ in main results of the paper is discussed.
Reviewer: O.Došlý (Brno)

##### MSC:
 34B24 Sturm-Liouville theory 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations 34B05 Linear boundary value problems for ordinary differential equations
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