Solutions of an advance-delay differential equation and their asymptotic behaviour

Gabriela Vážanová

Archivum Mathematicum (2023)

  • Issue: 1, page 141-149
  • ISSN: 0044-8753

Abstract

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The paper considers a scalar differential equation of an advance-delay type y ˙ ( t ) = - a 0 + a 1 t y ( t - τ ) + b 0 + b 1 t y ( t + σ ) , where constants a 0 , b 0 , τ and σ are positive, and a 1 and b 1 are arbitrary. The behavior of its solutions for t → ∞ is analyzed provided that the transcendental equation λ = - a 0 e - λ τ + b 0 e λ σ has a positive real root. An exponential-type function approximating the solution is searched for to be used in proving the existence of a semi-global solution. Moreover, the lower and upper estimates are given for such a solution.

How to cite

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Vážanová, Gabriela. "Solutions of an advance-delay differential equation and their asymptotic behaviour." Archivum Mathematicum (2023): 141-149. <http://eudml.org/doc/298997>.

@article{Vážanová2023,
abstract = {The paper considers a scalar differential equation of an advance-delay type \begin\{equation*\} \dot\{y\}(t)= -\left(a\_0+\frac\{a\_1\}\{t\}\right)y(t-\tau )+\left(b\_0+\frac\{b\_1\}\{t\}\right)y(t+\sigma )\,, \end\{equation*\} where constants $a_0$, $b_0$, $\tau $ and $\sigma $ are positive, and $a_1$ and $b_1$ are arbitrary. The behavior of its solutions for $t\rightarrow \infty $ is analyzed provided that the transcendental equation \begin\{equation*\} \lambda = -a\_0\mathrm \{e\}^\{-\lambda \tau \}+b\_0\mathrm \{e\}^\{\lambda \sigma \} \end\{equation*\} has a positive real root. An exponential-type function approximating the solution is searched for to be used in proving the existence of a semi-global solution. Moreover, the lower and upper estimates are given for such a solution.},
author = {Vážanová, Gabriela},
journal = {Archivum Mathematicum},
keywords = {advance-delay differential equation; mixed-type differential equation; asymptotic behaviour; existence of solutions},
language = {eng},
number = {1},
pages = {141-149},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Solutions of an advance-delay differential equation and their asymptotic behaviour},
url = {http://eudml.org/doc/298997},
year = {2023},
}

TY - JOUR
AU - Vážanová, Gabriela
TI - Solutions of an advance-delay differential equation and their asymptotic behaviour
JO - Archivum Mathematicum
PY - 2023
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
IS - 1
SP - 141
EP - 149
AB - The paper considers a scalar differential equation of an advance-delay type \begin{equation*} \dot{y}(t)= -\left(a_0+\frac{a_1}{t}\right)y(t-\tau )+\left(b_0+\frac{b_1}{t}\right)y(t+\sigma )\,, \end{equation*} where constants $a_0$, $b_0$, $\tau $ and $\sigma $ are positive, and $a_1$ and $b_1$ are arbitrary. The behavior of its solutions for $t\rightarrow \infty $ is analyzed provided that the transcendental equation \begin{equation*} \lambda = -a_0\mathrm {e}^{-\lambda \tau }+b_0\mathrm {e}^{\lambda \sigma } \end{equation*} has a positive real root. An exponential-type function approximating the solution is searched for to be used in proving the existence of a semi-global solution. Moreover, the lower and upper estimates are given for such a solution.
LA - eng
KW - advance-delay differential equation; mixed-type differential equation; asymptotic behaviour; existence of solutions
UR - http://eudml.org/doc/298997
ER -

References

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  7. Györi, I., Ladas, G., Oscillation Theory of Delay Differential Equations, Clarendon Press, Oxford, 1991. (1991) 
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