Asymptotic formulas for a scalar linear delay differential equation

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7 Citations (Scopus)

Abstract

The linear delay differential equation x′(t) = p(t)x(t − r) is considered, where r > 0 and the coefficient p: [t0,∞) → ℝ is a continuous function such that p(t) → 0 as t → ∞. In a recent paper [M. Pituk, G. Röst, Bound. Value Probl. 2014:114] an asymptotic description of the solutions has been given in terms of a special solution of the associated formal adjoint equation and the initial data. In this paper, we give a representation of the special solution of the formal adjoint equation. Under some additional conditions, the representation theorem yields explicit asymptotic formulas for the solutions as t → ∞.

Original languageEnglish
Article number72
JournalElectronic Journal of Qualitative Theory of Differential Equations
Volume2016
DOIs
Publication statusPublished - 2016

Fingerprint

Adjoint Equation
Delay Differential Equations
Asymptotic Formula
Linear differential equation
Differential equations
Scalar
Representation Theorem
Explicit Formula
Continuous Function
Coefficient

Keywords

  • Asymptotic formulas
  • Delay differential equation
  • Formal adjoint equation

ASJC Scopus subject areas

  • Applied Mathematics

Cite this

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abstract = "The linear delay differential equation x′(t) = p(t)x(t − r) is considered, where r > 0 and the coefficient p: [t0,∞) → ℝ is a continuous function such that p(t) → 0 as t → ∞. In a recent paper [M. Pituk, G. R{\"o}st, Bound. Value Probl. 2014:114] an asymptotic description of the solutions has been given in terms of a special solution of the associated formal adjoint equation and the initial data. In this paper, we give a representation of the special solution of the formal adjoint equation. Under some additional conditions, the representation theorem yields explicit asymptotic formulas for the solutions as t → ∞.",
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AB - The linear delay differential equation x′(t) = p(t)x(t − r) is considered, where r > 0 and the coefficient p: [t0,∞) → ℝ is a continuous function such that p(t) → 0 as t → ∞. In a recent paper [M. Pituk, G. Röst, Bound. Value Probl. 2014:114] an asymptotic description of the solutions has been given in terms of a special solution of the associated formal adjoint equation and the initial data. In this paper, we give a representation of the special solution of the formal adjoint equation. Under some additional conditions, the representation theorem yields explicit asymptotic formulas for the solutions as t → ∞.

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