Some properties of third order differential operators

Mariella Cecchi; Zuzana Došlá; Mauro Marini

Czechoslovak Mathematical Journal (1997)

  • Volume: 47, Issue: 4, page 729-748
  • ISSN: 0011-4642

Abstract

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Consider the third order differential operator L given by L ( · ) ≡ 1 a 3 ( t ) d d t 1 a 2 ( t ) d d t 1 a 1 ( t ) d d t ( · ) and the related linear differential equation L ( x ) ( t ) + x ( t ) = 0 . We study the relations between L , its adjoint operator, the canonical representation of L , the operator obtained by a cyclic permutation of coefficients a i , i = 1 , 2 , 3 , in L and the relations between the corresponding equations. We give the commutative diagrams for such equations and show some applications (oscillation, property A).

How to cite

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Cecchi, Mariella, Došlá, Zuzana, and Marini, Mauro. "Some properties of third order differential operators." Czechoslovak Mathematical Journal 47.4 (1997): 729-748. <http://eudml.org/doc/30395>.

@article{Cecchi1997,
abstract = {Consider the third order differential operator $L$ given by \[L(\cdot )\equiv \,\frac\{1\}\{a\_3(t)\}\frac\{\mbox\{d\}\}\{\mbox\{d\} t\}\frac\{1\}\{a\_2(t)\}\frac\{\mbox\{d\}\}\{\mbox\{d\} t\} \frac\{1\}\{a\_1(t)\}\frac\{\mbox\{d\}\}\{\mbox\{d\} t\}\,(\cdot ) \] and the related linear differential equation $L(x)(t)+x(t)=0$. We study the relations between $L$, its adjoint operator, the canonical representation of $L$, the operator obtained by a cyclic permutation of coefficients $a_i$, $ i=1,2,3$, in $L$ and the relations between the corresponding equations. We give the commutative diagrams for such equations and show some applications (oscillation, property A).},
author = {Cecchi, Mariella, Došlá, Zuzana, Marini, Mauro},
journal = {Czechoslovak Mathematical Journal},
keywords = {Differential operators; linear differential equation of third order; canonical forms; adjoint equation; cyclic permutation; oscillatory solution; Kneser solution; property $\mathrm \{A\}$; differential operators; linear differential equations of third order; canonical forms; adjoint equation; cyclic permutation; oscillatory solution; Kneser solution; property A},
language = {eng},
number = {4},
pages = {729-748},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Some properties of third order differential operators},
url = {http://eudml.org/doc/30395},
volume = {47},
year = {1997},
}

TY - JOUR
AU - Cecchi, Mariella
AU - Došlá, Zuzana
AU - Marini, Mauro
TI - Some properties of third order differential operators
JO - Czechoslovak Mathematical Journal
PY - 1997
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 47
IS - 4
SP - 729
EP - 748
AB - Consider the third order differential operator $L$ given by \[L(\cdot )\equiv \,\frac{1}{a_3(t)}\frac{\mbox{d}}{\mbox{d} t}\frac{1}{a_2(t)}\frac{\mbox{d}}{\mbox{d} t} \frac{1}{a_1(t)}\frac{\mbox{d}}{\mbox{d} t}\,(\cdot ) \] and the related linear differential equation $L(x)(t)+x(t)=0$. We study the relations between $L$, its adjoint operator, the canonical representation of $L$, the operator obtained by a cyclic permutation of coefficients $a_i$, $ i=1,2,3$, in $L$ and the relations between the corresponding equations. We give the commutative diagrams for such equations and show some applications (oscillation, property A).
LA - eng
KW - Differential operators; linear differential equation of third order; canonical forms; adjoint equation; cyclic permutation; oscillatory solution; Kneser solution; property $\mathrm {A}$; differential operators; linear differential equations of third order; canonical forms; adjoint equation; cyclic permutation; oscillatory solution; Kneser solution; property A
UR - http://eudml.org/doc/30395
ER -

References

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