Behaviour of solutions of linear differential equations with delay

Josef Diblík

Archivum Mathematicum (1998)

  • Volume: 034, Issue: 1, page 31-47
  • ISSN: 0044-8753

Abstract

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This contribution is devoted to the problem of asymptotic behaviour of solutions of scalar linear differential equation with variable bounded delay of the form x ˙ ( t ) = - c ( t ) x ( t - τ ( t ) ) ( * ) with positive function c ( t ) . Results concerning the structure of its solutions are obtained with the aid of properties of solutions of auxiliary homogeneous equation y ˙ ( t ) = β ( t ) [ y ( t ) - y ( t - τ ( t ) ) ] where the function β ( t ) is positive. A result concerning the behaviour of solutions of Eq. (*) in critical case is given and, moreover, an analogy with behaviour of solutions of the second order ordinary differential equation x ' ' ( t ) + a ( t ) x ( t ) = 0 for positive function a ( t ) in critical case is considered.

How to cite

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Diblík, Josef. "Behaviour of solutions of linear differential equations with delay." Archivum Mathematicum 034.1 (1998): 31-47. <http://eudml.org/doc/248217>.

@article{Diblík1998,
abstract = {This contribution is devoted to the problem of asymptotic behaviour of solutions of scalar linear differential equation with variable bounded delay of the form \[ \dot\{x\}(t)= -c(t)x(t-\tau (t)) \qquad \mathrm \{\{(^*)\}\}\] with positive function $c(t).$ Results concerning the structure of its solutions are obtained with the aid of properties of solutions of auxiliary homogeneous equation \[ \dot\{y\}(t)=\beta (t)[y(t)-y(t-\tau (t))] \] where the function $\beta (t)$ is positive. A result concerning the behaviour of solutions of Eq. (*) in critical case is given and, moreover, an analogy with behaviour of solutions of the second order ordinary differential equation \[ x^\{\prime \prime \}(t)+a(t)x(t)=0 \] for positive function $a(t)$ in critical case is considered.},
author = {Diblík, Josef},
journal = {Archivum Mathematicum},
keywords = {Positive solution; oscillating solution; convergent solution; linear differential equation with delay; topological principle of Ważewski (Rybakowski’s approach); positive solutions; oscillating solutions; convergent solutions; linear differential equations with delay; topological principle of Wazewski (Rybakowski's approach)},
language = {eng},
number = {1},
pages = {31-47},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Behaviour of solutions of linear differential equations with delay},
url = {http://eudml.org/doc/248217},
volume = {034},
year = {1998},
}

TY - JOUR
AU - Diblík, Josef
TI - Behaviour of solutions of linear differential equations with delay
JO - Archivum Mathematicum
PY - 1998
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 034
IS - 1
SP - 31
EP - 47
AB - This contribution is devoted to the problem of asymptotic behaviour of solutions of scalar linear differential equation with variable bounded delay of the form \[ \dot{x}(t)= -c(t)x(t-\tau (t)) \qquad \mathrm {{(^*)}}\] with positive function $c(t).$ Results concerning the structure of its solutions are obtained with the aid of properties of solutions of auxiliary homogeneous equation \[ \dot{y}(t)=\beta (t)[y(t)-y(t-\tau (t))] \] where the function $\beta (t)$ is positive. A result concerning the behaviour of solutions of Eq. (*) in critical case is given and, moreover, an analogy with behaviour of solutions of the second order ordinary differential equation \[ x^{\prime \prime }(t)+a(t)x(t)=0 \] for positive function $a(t)$ in critical case is considered.
LA - eng
KW - Positive solution; oscillating solution; convergent solution; linear differential equation with delay; topological principle of Ważewski (Rybakowski’s approach); positive solutions; oscillating solutions; convergent solutions; linear differential equations with delay; topological principle of Wazewski (Rybakowski's approach)
UR - http://eudml.org/doc/248217
ER -

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