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Singular eigenvalue problem; Sturm-Liouville equation; zeros of nonoscillatory solutions
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 $q(t)$.
[1] P. Hartman: Ordinary Differential Equations. Wiley, New York, 1964. MR 0171038 | Zbl 0125.32102
[2] P. Hartman: Boundary value problems for second order ordinary differential equations involving a parameter. J. Differential Equations 12 (1972), 194–212. MR 0335927 | Zbl 0255.34012
[3] E. Hille: Lectures on Ordinary Differential Equations. Addison-Wesley, Reading, Massachusetts, 1969. MR 0249698 | Zbl 0179.40301
[4] Y. Kabeya: Uniqueness of nodal fast-decaying radial solutions to a linear elliptic equations on $\mathbb{R}^n$. preprint.
[5] M. Naito: Radial entire solutions of the linear equation $\Delta u + \lambda p(|x|)u = 0$. Hiroshima Math. J. 19 (1989), 431–439. MR 1027944 | Zbl 0716.35002
[6] Z. Nehari: Oscillation criteria for second-order linear differential equations. Trans. Amer. Math. Soc. 85 (1957), 428–445. MR 0087816 | Zbl 0078.07602
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