# Singular eigenvalue problems for second order linear ordinary differential equations

Árpád Elbert; Takaŝi Kusano; Manabu Naito

Archivum Mathematicum (1998)

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

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topElbert, Árpád, Kusano, Takaŝi, and Naito, Manabu. "Singular eigenvalue problems for second order linear ordinary differential equations." Archivum Mathematicum 034.1 (1998): 59-72. <http://eudml.org/doc/248186>.

@article{Elbert1998,

abstract = {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 $q(t)$.},

author = {Elbert, Árpád, Kusano, Takaŝi, Naito, Manabu},

journal = {Archivum Mathematicum},

keywords = {Singular eigenvalue problem; Sturm-Liouville equation; zeros of nonoscillatory solutions; singular eigenvalue problem; Sturm-Liouville equation; zeros of nonoscillatory solutions},

language = {eng},

number = {1},

pages = {59-72},

publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},

title = {Singular eigenvalue problems for second order linear ordinary differential equations},

url = {http://eudml.org/doc/248186},

volume = {034},

year = {1998},

}

TY - JOUR

AU - Elbert, Árpád

AU - Kusano, Takaŝi

AU - Naito, Manabu

TI - Singular eigenvalue problems for second order linear ordinary differential equations

JO - Archivum Mathematicum

PY - 1998

PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno

VL - 034

IS - 1

SP - 59

EP - 72

AB - 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 $q(t)$.

LA - eng

KW - Singular eigenvalue problem; Sturm-Liouville equation; zeros of nonoscillatory solutions; singular eigenvalue problem; Sturm-Liouville equation; zeros of nonoscillatory solutions

UR - http://eudml.org/doc/248186

ER -

## References

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- P. Hartman, Boundary value problems for second order ordinary differential equations involving a parameter, J. Differential Equations 12 (1972), 194–212. (1972) Zbl0255.34012MR0335927
- E. Hille, Lectures on Ordinary Differential Equations, Addison-Wesley, Reading, Massachusetts, 1969. (1969) Zbl0179.40301MR0249698
- Y. Kabeya, Uniqueness of nodal fast-decaying radial solutions to a linear elliptic equations on ${\mathbb{R}}^{n}$, preprint.
- M. Naito, Radial entire solutions of the linear equation $\Delta u+\lambda p\left(\right|x\left|\right)u=0$, Hiroshima Math. J. 19 (1989), 431–439. (1989) Zbl0716.35002MR1027944
- Z. Nehari, Oscillation criteria for second-order linear differential equations, Trans. Amer. Math. Soc. 85 (1957), 428–445. (1957) Zbl0078.07602MR0087816

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