Elbert, Árpád; Kusano, Takaŝi; Naito, Manabu Singular eigenvalue problems for second-order linear ordinary differential equations. (English) Zbl 0914.34021 Arch. Math., Brno 34, No. 1, 59-72 (1998). 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) Cited in 1 ReviewCited in 1 Document 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 Keywords:singular eigenvalue problem; Sturm-Liouville equation; zeros of nonoscillatory solutions PDFBibTeX XMLCite \textit{Á. Elbert} et al., Arch. Math., Brno 34, No. 1, 59--72 (1998; Zbl 0914.34021) Full Text: EuDML