Asymptotic formulas for a scalar linear delay differential equation

István Győri,Mihály Pituk

doi:10.14232/ejqtde.2016.1.72

Abstract

The linear delay differential equation x 0 (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. Rost, 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 → ∞.

Highlights

Consider the delay differential equation x (t) = p(t)x(t − r), (1.1)where r > 0 and p : [t0, ∞) → R is a continuous function
Pituk we have given an asymptotic description of the solution of the initial value problem (1.1) and (1.2) in terms of a special solution of the formal adjoint equation y (t) = −p(t + r)y(t + r)
Letting n → ∞ in conclusion (2.10) of Lemma 2.3, we obtain that the special solution y of Eq (1.4) satisfies the ordinary differential equation y (t) = σ(t)y(t), t ≥ t1, where σ is defined by (2.6)

Summary

Introduction

Where r > 0 and p : [t0, ∞) → R is a continuous function. The initial value problem associated with (1.1) has the form x(t) = φ(t), t1 − r ≤ t ≤ t1,. Pituk we have given an asymptotic description of the solution of the initial value problem (1.1) and (1.2) in terms of a special solution of the formal adjoint equation y (t) = −p(t + r)y(t + r). A close look at the proof of Theorem 3.1 in [10] shows that the special solution of the adjoint equation y has the following additional properties: if t1 ≥ t0 is chosen such that t+r t p−(s) ds 1 e.

Representation of the special solution of the adjoint equation

Explicit asymptotic formulas