# Asymptotic formulas for a scalar linear delay differential equation

Research output: Article

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 language English 72 Electronic Journal of Qualitative Theory of Differential Equations 2016 https://doi.org/10.14232/ejqtde.2016.1.72 Published - 2016

### Fingerprint

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

### ASJC Scopus subject areas

• Applied Mathematics

### Cite this

In: Electronic Journal of Qualitative Theory of Differential Equations, Vol. 2016, 72, 2016.

Research output: Article

title = "Asymptotic formulas for a scalar linear delay differential equation",
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 → ∞.",
keywords = "Asymptotic formulas, Delay differential equation, Formal adjoint equation",
author = "I. Győri and M. Pituk",
year = "2016",
doi = "10.14232/ejqtde.2016.1.72",
language = "English",
volume = "2016",
journal = "Electronic Journal of Qualitative Theory of Differential Equations",
issn = "1417-3875",
publisher = "University of Szeged",

}

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

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 -