On the oscillation of certain class of third-order nonlinear delay differential equations

S. H. Saker; J. Džurina

Mathematica Bohemica (2010)

  • Volume: 135, Issue: 3, page 225-237
  • ISSN: 0862-7959

Abstract

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In this paper we consider the third-order nonlinear delay differential equation (*) ( a ( t ) x ' ' ( t ) γ ) ' + q ( t ) x γ ( τ ( t ) ) = 0 , t ≥ t 0 , where a ( t ) , q ( t ) are positive functions, γ > 0 is a quotient of odd positive integers and the delay function τ ( t ) ≤ t satisfies lim t → i n f t y τ ( t ) = i n f t y . We establish some sufficient conditions which ensure that (*) is oscillatory or the solutions converge to zero. Our results in the nondelay case extend and improve some known results and in the delay case the results can be applied to new classes of equations which are not covered by the known criteria. Some examples are considered to illustrate the main results.

How to cite

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Saker, S. H., and Džurina, J.. "On the oscillation of certain class of third-order nonlinear delay differential equations." Mathematica Bohemica 135.3 (2010): 225-237. <http://eudml.org/doc/38127>.

@article{Saker2010,
abstract = {In this paper we consider the third-order nonlinear delay differential equation (*) \[ ( a(t)\left( x^\{\prime \prime \}(t)\right) ^\{\gamma \})^\{\prime \} +q(t)x^\{\gamma \}(\tau (t))=0,\quad t\ge t\_0, \] where $a(t)$, $q(t)$ are positive functions, $\gamma >0$ is a quotient of odd positive integers and the delay function $\tau (t)\le t$ satisfies $\lim _\{t\rightarrow infty \}\tau (t)=infty $. We establish some sufficient conditions which ensure that (*) is oscillatory or the solutions converge to zero. Our results in the nondelay case extend and improve some known results and in the delay case the results can be applied to new classes of equations which are not covered by the known criteria. Some examples are considered to illustrate the main results.},
author = {Saker, S. H., Džurina, J.},
journal = {Mathematica Bohemica},
keywords = {third-order differential equation; oscillation; nonoscillation; disconjugacy; third-order differential equation; oscillation; nonoscillation; disconjugacy},
language = {eng},
number = {3},
pages = {225-237},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the oscillation of certain class of third-order nonlinear delay differential equations},
url = {http://eudml.org/doc/38127},
volume = {135},
year = {2010},
}

TY - JOUR
AU - Saker, S. H.
AU - Džurina, J.
TI - On the oscillation of certain class of third-order nonlinear delay differential equations
JO - Mathematica Bohemica
PY - 2010
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 135
IS - 3
SP - 225
EP - 237
AB - In this paper we consider the third-order nonlinear delay differential equation (*) \[ ( a(t)\left( x^{\prime \prime }(t)\right) ^{\gamma })^{\prime } +q(t)x^{\gamma }(\tau (t))=0,\quad t\ge t_0, \] where $a(t)$, $q(t)$ are positive functions, $\gamma >0$ is a quotient of odd positive integers and the delay function $\tau (t)\le t$ satisfies $\lim _{t\rightarrow infty }\tau (t)=infty $. We establish some sufficient conditions which ensure that (*) is oscillatory or the solutions converge to zero. Our results in the nondelay case extend and improve some known results and in the delay case the results can be applied to new classes of equations which are not covered by the known criteria. Some examples are considered to illustrate the main results.
LA - eng
KW - third-order differential equation; oscillation; nonoscillation; disconjugacy; third-order differential equation; oscillation; nonoscillation; disconjugacy
UR - http://eudml.org/doc/38127
ER -

References

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