### Abstract

The linear delay differential equation x′(t) = p(t)x(t − r) is considered, where r > 0 and the coefficient p: [t_{0},∞) → 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 language | English |
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Article number | 72 |

Journal | Electronic Journal of Qualitative Theory of Differential Equations |

Volume | 2016 |

DOIs | |

Publication status | Published - 2016 |

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### ASJC Scopus subject areas

- Applied Mathematics

### Cite this

**Asymptotic formulas for a scalar linear delay differential equation.** / Győri, I.; Pituk, M.

Research output: Article

}

TY - JOUR

T1 - Asymptotic formulas for a scalar linear delay differential equation

AU - Győri, I.

AU - Pituk, M.

PY - 2016

Y1 - 2016

N2 - 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 → ∞.

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 → ∞.

KW - Asymptotic formulas

KW - Delay differential equation

KW - Formal adjoint equation

UR - http://www.scopus.com/inward/record.url?scp=84987802527&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84987802527&partnerID=8YFLogxK

U2 - 10.14232/ejqtde.2016.1.72

DO - 10.14232/ejqtde.2016.1.72

M3 - Article

AN - SCOPUS:84987802527

VL - 2016

JO - Electronic Journal of Qualitative Theory of Differential Equations

JF - Electronic Journal of Qualitative Theory of Differential Equations

SN - 1417-3875

M1 - 72

ER -