Singular eigenvalue problems for second order linear ordinary differential equations

Elbert, 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 -