On the oscillation of a class of linear homogeneous third order differential equations

N. Parhi; P. Das

Archivum Mathematicum (1998)

  • Volume: 034, Issue: 4, page 435-443
  • ISSN: 0044-8753

Abstract

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In this paper we have considered completely the equation y ' ' ' + a ( t ) y ' ' + b ( t ) y ' + c ( t ) y = 0 , ( * ) where a ∈ C 2 ( [ σ , ∞ ) , R ) , b ∈ C 1 ( [ σ , ∞ ) , R ) , c ∈ C ( [ σ , ∞ ) , R ) and σ ∈ R such that a ( t ) ≤ 0 , b ( t ) ≤ 0 and c ( t ) ≤ 0 . It has been shown that the set of all oscillatory solutions of (*) forms a two-dimensional subspace of the solution space of (*) provided that (*) has an oscillatory solution. This answers a question raised by S. Ahmad and A.  C. Lazer earlier.

How to cite

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Parhi, N., and Das, P.. "On the oscillation of a class of linear homogeneous third order differential equations." Archivum Mathematicum 034.4 (1998): 435-443. <http://eudml.org/doc/248215>.

@article{Parhi1998,
abstract = {In this paper we have considered completely the equation \[ y^\{\prime \prime \prime \}+ a(t)y^\{\prime \prime \}+ b(t)y^\prime + c(t)y=0\,, \qquad \mathrm \{(*)\}\] where $a\in C^2([\sigma , \infty ), R)$, $b\in C^1([\sigma , \infty ),R)$, $c\in C([\sigma , \infty ), R)$ and $\sigma \in R$ such that $a(t)\le 0$, $b(t)\le 0$ and $c(t)\le 0$. It has been shown that the set of all oscillatory solutions of (*) forms a two-dimensional subspace of the solution space of (*) provided that (*) has an oscillatory solution. This answers a question raised by S. Ahmad and A.  C. Lazer earlier.},
author = {Parhi, N., Das, P.},
journal = {Archivum Mathematicum},
keywords = {third order differential equations; oscillation; nonoscillation; asymptotic behaviour of solutions; third-order differential equations; oscillation; nonoscillation; asymptotic behaviour of solutions},
language = {eng},
number = {4},
pages = {435-443},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On the oscillation of a class of linear homogeneous third order differential equations},
url = {http://eudml.org/doc/248215},
volume = {034},
year = {1998},
}

TY - JOUR
AU - Parhi, N.
AU - Das, P.
TI - On the oscillation of a class of linear homogeneous third order differential equations
JO - Archivum Mathematicum
PY - 1998
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 034
IS - 4
SP - 435
EP - 443
AB - In this paper we have considered completely the equation \[ y^{\prime \prime \prime }+ a(t)y^{\prime \prime }+ b(t)y^\prime + c(t)y=0\,, \qquad \mathrm {(*)}\] where $a\in C^2([\sigma , \infty ), R)$, $b\in C^1([\sigma , \infty ),R)$, $c\in C([\sigma , \infty ), R)$ and $\sigma \in R$ such that $a(t)\le 0$, $b(t)\le 0$ and $c(t)\le 0$. It has been shown that the set of all oscillatory solutions of (*) forms a two-dimensional subspace of the solution space of (*) provided that (*) has an oscillatory solution. This answers a question raised by S. Ahmad and A.  C. Lazer earlier.
LA - eng
KW - third order differential equations; oscillation; nonoscillation; asymptotic behaviour of solutions; third-order differential equations; oscillation; nonoscillation; asymptotic behaviour of solutions
UR - http://eudml.org/doc/248215
ER -

References

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  1. On the oscillatory behaviour of a class of linear third order differential equations, J. Math. Anal. Appl. 28(1970), 681-689, MR 40#1646. MR0248394
  2. Oscillation criteria for a third order linear differential equations, Pacific J. Math. 11(1961), 919-944, MR 26# 2695. MR0145160
  3. Properties of solutions of a class of third order differential equations, J. Math. Anal. Appl. 48(1974), 165-169. Zbl0289.34046MR0352608
  4. The behaviour of solutions of the differential equation y ' ' ' + p ( x ) y ' + q ( x ) y = 0 , Pacific J. Math. 17(1966), 435-466, MR 33#1552. MR0193332
  5. On the oscillation of self adjoint linear differential equations of the fourth order, Trans. Amer. Math. Soc. 89(1958), 325-377. MR0102639
  6. On asymptotic property of solutions of a class of third order differential equations, Proc. Amer. Math. Soc. 110(1990), 387-393. MR1019279

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