Conjugacy criteria and principal solutions of self-adjoint differential equations

Ondřej Došlý; Jan Komenda

Archivum Mathematicum (1995)

  • Volume: 031, Issue: 3, page 217-238
  • ISSN: 0044-8753

Abstract

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Oscillation properties of the self-adjoint, two term, differential equation ( - 1 ) n ( p ( x ) y ( n ) ) ( n ) + q ( x ) y = 0 ( * ) are investigated. Using the variational method and the concept of the principal system of solutions it is proved that (*) is conjugate on R = ( - ∞ , ∞ ) if there exist an integer m ∈ { 0 , 1 , ⋯ , n - 1 } and c 0 , ⋯ , c m ∈ R such that ∫ ∞ 0 x 2 ( n - m - 1 ) p - 1 ( x ) d x = ∞ = ∫ 0 ∞ x 2 ( n - m - 1 ) p - 1 ( x ) d x and lim sup x 1 ↓ - ∞ , x 2 ↑ ∞ ∫ x 1 x 2 q ( x ) ( c 0 + c 1 x + ⋯ + c m x m ) 2 d x ≤ 0 , q ( x ) ¬ ≡ 0 . Some extensions of this criterion are suggested.

How to cite

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Došlý, Ondřej, and Komenda, Jan. "Conjugacy criteria and principal solutions of self-adjoint differential equations." Archivum Mathematicum 031.3 (1995): 217-238. <http://eudml.org/doc/247674>.

@article{Došlý1995,
abstract = {Oscillation properties of the self-adjoint, two term, differential equation \[(-1)^n(p(x)y^\{(n)\})^\{(n)\}+q(x)y=0\qquad \mathrm \{(*)\}\] are investigated. Using the variational method and the concept of the principal system of solutions it is proved that (*) is conjugate on $R=(-\infty ,\infty )$ if there exist an integer $m\in \lbrace 0,1,\dots ,n-1\rbrace $ and $c_0,\dots ,c_m\in R$ such that \[\int \_\infty ^0 x^\{2(n-m-1)\}p^\{-1\}(x)\,dx=\infty =\int \_0^\infty x^\{2(n-m-1)\}p^\{-1\}(x)\,dx\] and \[\limsup \_\{x\_1\downarrow -\infty ,x\_2\uparrow \infty \}\int \_\{x\_1\}^\{x\_2\}q(x)(c\_0+c\_1x+\dots + c\_mx^m)^2\,dx\le 0,\quad q(x)\lnot \equiv 0.\] Some extensions of this criterion are suggested.},
author = {Došlý, Ondřej, Komenda, Jan},
journal = {Archivum Mathematicum},
keywords = {conjugate points; principal system of solutions; variational method; conjugacy criteria; selfadjoint two-term differential equation; conjugate},
language = {eng},
number = {3},
pages = {217-238},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Conjugacy criteria and principal solutions of self-adjoint differential equations},
url = {http://eudml.org/doc/247674},
volume = {031},
year = {1995},
}

TY - JOUR
AU - Došlý, Ondřej
AU - Komenda, Jan
TI - Conjugacy criteria and principal solutions of self-adjoint differential equations
JO - Archivum Mathematicum
PY - 1995
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 031
IS - 3
SP - 217
EP - 238
AB - Oscillation properties of the self-adjoint, two term, differential equation \[(-1)^n(p(x)y^{(n)})^{(n)}+q(x)y=0\qquad \mathrm {(*)}\] are investigated. Using the variational method and the concept of the principal system of solutions it is proved that (*) is conjugate on $R=(-\infty ,\infty )$ if there exist an integer $m\in \lbrace 0,1,\dots ,n-1\rbrace $ and $c_0,\dots ,c_m\in R$ such that \[\int _\infty ^0 x^{2(n-m-1)}p^{-1}(x)\,dx=\infty =\int _0^\infty x^{2(n-m-1)}p^{-1}(x)\,dx\] and \[\limsup _{x_1\downarrow -\infty ,x_2\uparrow \infty }\int _{x_1}^{x_2}q(x)(c_0+c_1x+\dots + c_mx^m)^2\,dx\le 0,\quad q(x)\lnot \equiv 0.\] Some extensions of this criterion are suggested.
LA - eng
KW - conjugate points; principal system of solutions; variational method; conjugacy criteria; selfadjoint two-term differential equation; conjugate
UR - http://eudml.org/doc/247674
ER -

References

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  9. Existence of conjugate points for second and fourth order differential equations, Proc. Roy. Soc. Edinburgh 89A (1981), 281-91. (1981) MR0635764
  10. An oscillation theorem for self- adjoint differential equations, Math. Nachr. 108 (1982), 79-92. (1982) MR0695118
  11. Nodal Domains for One- and Two- Dimensional Elliptic Differential Equations, Z. Anal. Anwendungen 7 (1988), 135-139. (1988) MR0951346
  12. [unknown], Sturmian Theory for Ordinary Differential Equations, Acad. Press, New-York 1980. Zbl1168.83300MR0606199
  13. General relativity and conjugate ordinary differential equations, J. Diff. Equations 30 (1978), 165-174. (1978) Zbl0362.34023MR0513268

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