On property (B) of higher order delay differential equations

Blanka Baculíková; Jozef Džurina

Archivum Mathematicum (2012)

  • Volume: 048, Issue: 4, page 301-309
  • ISSN: 0044-8753

Abstract

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In this paper we offer criteria for property (B) and additional asymptotic behavior of solutions of the n -th order delay differential equations ( r ( t ) [ x ( n - 1 ) ( t ) ] γ ) ' = q ( t ) f ( x ( τ ( t ) ) ) . Obtained results essentially use new comparison theorems, that permit to reduce the problem of the oscillation of the n-th order equation to the the oscillation of a set of certain the first order equations. So that established comparison principles essentially simplify the examination of studied equations. Both cases ∫ ∞ r - 1 / γ ( t ) t = ∞ and ∫ ∞ r - 1 / γ ( t ) t < ∞ are discussed.

How to cite

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Baculíková, Blanka, and Džurina, Jozef. "On property (B) of higher order delay differential equations." Archivum Mathematicum 048.4 (2012): 301-309. <http://eudml.org/doc/251365>.

@article{Baculíková2012,
abstract = {In this paper we offer criteria for property (B) and additional asymptotic behavior of solutions of the $n$-th order delay differential equations \begin\{equation*\} \big (r(t)\big [x^\{(n-1)\}(t)\big ]^\{\gamma \}\big )^\{\prime \}=q(t)f\big (x(\tau (t))\big )\,. \end\{equation*\} Obtained results essentially use new comparison theorems, that permit to reduce the problem of the oscillation of the n-th order equation to the the oscillation of a set of certain the first order equations. So that established comparison principles essentially simplify the examination of studied equations. Both cases $\int ^\{\infty \} r^\{-1/\gamma \}(t)\,\{t\}=\infty $ and $\int ^\{\infty \} r^\{-1/\gamma \}(t)\,\{t\}<\infty $ are discussed.},
author = {Baculíková, Blanka, Džurina, Jozef},
journal = {Archivum Mathematicum},
keywords = {$n$-th order differential equations; comparison theorem; oscillation; property (B); -th order differential equation; comparison theorem; oscillation; property (B)},
language = {eng},
number = {4},
pages = {301-309},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On property (B) of higher order delay differential equations},
url = {http://eudml.org/doc/251365},
volume = {048},
year = {2012},
}

TY - JOUR
AU - Baculíková, Blanka
AU - Džurina, Jozef
TI - On property (B) of higher order delay differential equations
JO - Archivum Mathematicum
PY - 2012
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 048
IS - 4
SP - 301
EP - 309
AB - In this paper we offer criteria for property (B) and additional asymptotic behavior of solutions of the $n$-th order delay differential equations \begin{equation*} \big (r(t)\big [x^{(n-1)}(t)\big ]^{\gamma }\big )^{\prime }=q(t)f\big (x(\tau (t))\big )\,. \end{equation*} Obtained results essentially use new comparison theorems, that permit to reduce the problem of the oscillation of the n-th order equation to the the oscillation of a set of certain the first order equations. So that established comparison principles essentially simplify the examination of studied equations. Both cases $\int ^{\infty } r^{-1/\gamma }(t)\,{t}=\infty $ and $\int ^{\infty } r^{-1/\gamma }(t)\,{t}<\infty $ are discussed.
LA - eng
KW - $n$-th order differential equations; comparison theorem; oscillation; property (B); -th order differential equation; comparison theorem; oscillation; property (B)
UR - http://eudml.org/doc/251365
ER -

References

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