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This article is part of the series A Tribute to Professor Ivan Kiguradze.

Open Access Research

Large time behavior of a linear delay differential equation with asymptotically small coefficient

Mihály Pituk1* and Gergely Röst2

Author Affiliations

1 Department of Mathematics, University of Pannonia, P.O. Box 158, Veszprém, H-8201, Hungary

2 Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, Szeged, H-6720, Hungary

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Boundary Value Problems 2014, 2014:114  doi:10.1186/1687-2770-2014-114


Dedicated to Professor Ivan Kiguradze

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/114


Received:4 February 2014
Accepted:15 April 2014
Published:14 May 2014

© 2014 Pituk and Röst; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Abstract

The linear delay differential equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M1">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M2">View MathML</a>, is considered, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M3">View MathML</a> and the coefficient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M4">View MathML</a> is continuous and small in the sense that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M6">View MathML</a>. It is shown that the large time behavior of the solutions can be described in terms of a special solution of the associated formal adjoint equation and the initial data. In the special case of the Dickman-de Bruijn equation, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M8">View MathML</a>, our result yields an explicit asymptotic representation of the solutions as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M6">View MathML</a>.

MSC: 34K06, 34K25, 11A51.

Keywords:
delay differential equation; formal adjoint equation; Dickman-de Bruijn equation; asymptotic behavior

1 Introduction

The linear scalar delay differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M10">View MathML</a>

(1.1)

plays a significant role in analytic number theory. Its special solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M11">View MathML</a> with initial values <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M12">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M13">View MathML</a> is called the Dickman-de Bruijn function. The Dickman-de Bruijn function was first studied by actuary Dickman [1] and later by de Bruijn [2,3] to estimate the proportion of smooth numbers up to a given bound. In [4] (see also [5]) van der Lune proposed some problems regarding the solutions of (1.1). The solutions to these problems, given by Bierkens, appeared in [6]. Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M14">View MathML</a> is a continuous function and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M15">View MathML</a> denote the unique solution of (1.1) with initial values

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M16">View MathML</a>

(1.2)

Among others, it was shown that if the limit

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M17">View MathML</a>

(1.3)

exists and is finite, then its value is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M18">View MathML</a>

(1.4)

We emphasize that the solution presented in [6] does not imply the existence of the limit (1.3).

Our aim in this paper is twofold. First, we give an alternative proof of the limit relation (1.4) including the existence of the limit in (1.3). Second, we will show that the above result on the Dickman-de Bruijn equation (1.1) can be extended to the general linear equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M19">View MathML</a>

(1.5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M21">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M22">View MathML</a> is a continuous function such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M23">View MathML</a>

(1.6)

Our main result (see Theorem 3.3 below) provides an asymptotic description of the solutions of (1.5). The asymptotic formula is given in terms of the initial data and a special solution of the associated formal adjoint equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M24">View MathML</a>

(1.7)

The special solution of (1.7) is eventually positive, it has bounded growth and it is unique up to a constant multiple.

The large time behavior of the solutions of (1.1) is discussed in Section 2 and our general result on the asymptotic description of the solutions (1.5) is presented in Section 3.

2 Large time behavior of the Dickman-de Bruijn equation

In this section, we prove the existence of the limit (1.3) for the solutions of (1.1).

Theorem 2.1Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M14">View MathML</a>is a continuous function and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M26">View MathML</a>be the unique solution of the initial value problem (1.1) and (1.2). Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M27">View MathML</a>

(2.1)

The proof of Theorem 2.1 will be based on the identity

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M28">View MathML</a>

(2.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M29">View MathML</a> is the constant given by (1.4). Using (1.1), it is easily shown that the derivative of the function on the left-hand side of (2.2) is equal to 0 identically on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M30">View MathML</a>. This, together with (1.4), implies (2.2). Since the proof is straightforward, we omit it.

Now we can give a simple short proof of Theorem 2.1.

Proof of Theorem 2.1 Write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M31">View MathML</a> for brevity. First we show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M32">View MathML</a>

(2.3)

Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M33">View MathML</a>

(2.4)

Then (2.2) can be written in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M34">View MathML</a>

(2.5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M35">View MathML</a>. From this, we find for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M36">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M37">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M38">View MathML</a>

From the last inequality, by the application of Gronwall’s lemma (see, e.g., [[7], Chapter 1, Lemma 3.1]), we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M39">View MathML</a>

Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M40">View MathML</a>

From this and (2.5), we find for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M41">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M42">View MathML</a>

In view of (2.4), this implies (2.3).

From (2.2) and (2.3), we obtain for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M36">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M44">View MathML</a>

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M6">View MathML</a> in the last inequality, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M46">View MathML</a>

which is equivalent to the limit relation (2.1). □

We remark that the existence of the limit in (1.3) can also be deduced from the results by Győri and the first author (see [[8], Theorem 3.3] and its proof) and by Diblík (see [[9], Theorem 18 and Example 20]). However, the above results cannot be used to compute the value of the limit explicitly in terms of the initial data.

3 Main result

In this section, we present our main result on the large time behavior of the solutions of (1.5). First we show that under the smallness condition (1.6) the formal adjoint equation has an eventually positive solution with bounded growth.

Theorem 3.1Suppose condition (1.6) holds. Then (1.7) has a solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M47">View MathML</a>which is positive for all largetand such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M48">View MathML</a>

(3.1)

We will prove Theorem 3.1 by applying a technique known from the oscillation theory of delay differential equations (see [[10], Section 2.3]).

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M49">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M50">View MathML</a> denote the positive part and the negative part of p, respectively, defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M51">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M52">View MathML</a>, by virtue of (1.6), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M53">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M54">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M55">View MathML</a> be the space of continuous functions mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M56">View MathML</a> into ℝ with the topology of uniform convergence on compact subsets of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M56">View MathML</a>. Let Ω denote the set of functions from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M55">View MathML</a> which satisfy the system of inequalities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M59">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M60">View MathML</a>

Clearly, Ω is a nonempty, closed and convex subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M55">View MathML</a>. Define the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M62">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M63">View MathML</a>

It is easily verified that F is continuous and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M64">View MathML</a>. Furthermore, the functions from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M65">View MathML</a> are uniformly bounded and equicontinuous on each compact subinterval of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M56">View MathML</a>. Therefore, by the Arzela-Ascoli theorem, the closure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M65">View MathML</a> is compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M55">View MathML</a>. By the application of the Schauder-Tychonoff fixed point theorem, we conclude that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M69">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M70">View MathML</a>. It is easily seen that this fixed point y is a solution of (1.7) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M56">View MathML</a> with property (3.1). Clearly, the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M72">View MathML</a> can be extended backward to all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M73">View MathML</a> by the method of steps. □

It should be noted that under the smallness condition (1.6), (1.7) may have a positive solution which does not satisfy condition (3.1). Indeed, the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M74">View MathML</a>

a special case of (1.7) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M75">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M76">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M77">View MathML</a>, has the positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M78">View MathML</a> for which the ratio <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M79">View MathML</a> is unbounded as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M6">View MathML</a>.

In the next theorem, we show that up to a constant multiple the special solution of (1.7) described in Theorem 3.1 is unique.

Theorem 3.2Suppose condition (1.6) holds. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M81">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M82">View MathML</a>be eventually positive solutions of (1.7) satisfying condition (3.1). Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M82">View MathML</a>is a constant multiple of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M81">View MathML</a>.

Proof We begin with two simple observations. First, if y is a solution of (1.7), then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M85">View MathML</a>

(3.2)

Second, if y is a solution of (1.7) which is positive on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M86">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M87">View MathML</a> and satisfies condition (3.1), then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M88">View MathML</a>

(3.3)

where M is an arbitrary constant such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M89">View MathML</a>

(3.4)

Indeed, from (1.7) we find for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M90">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M91">View MathML</a>

Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M92">View MathML</a>

This, together with (3.4), implies (3.3).

By assumptions, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M87">View MathML</a> such that both solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M81">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M82">View MathML</a> are positive on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M86">View MathML</a> and satisfy condition (3.1). As noted before (see (3.3)), if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M97">View MathML</a> is sufficiently large, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M98">View MathML</a>

(3.5)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M97">View MathML</a>, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M100">View MathML</a> is sufficiently small, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M101">View MathML</a>. By virtue of (1.6), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M102">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M103">View MathML</a>

(3.6)

We will show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M104">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M105">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M106">View MathML</a>. In view of the linearity of (1.7), the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M107">View MathML</a> is a solution of (1.7) and, by virtue of (3.5), the quantity

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M108">View MathML</a>

is finite. Applying (3.2) to both solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M82">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M110">View MathML</a> of (1.7) and taking into account that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M111">View MathML</a>, we obtain, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M105">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M113">View MathML</a>

where the last but one inequality is a consequence of (3.6). From the last inequality, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M114">View MathML</a>

Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M115">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M101">View MathML</a>, this implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M117">View MathML</a> and therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M118">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M119">View MathML</a>. Finally, by the uniqueness of the backward continuation of the solutions of (1.7), we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M104">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M121">View MathML</a>. □

Now we can formulate our main result about the large time behavior of the solutions of (1.5).

Theorem 3.3Suppose condition (1.6) holds. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M26">View MathML</a>denote the solution of (1.5) with initial data

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M123">View MathML</a>

(3.7)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M124">View MathML</a>is a continuous function. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M125">View MathML</a>

(3.8)

whereyis any eventually positive solution of (1.7) satisfying (3.1) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M29">View MathML</a>is a constant given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M127">View MathML</a>

(3.9)

As shown in Theorem 3.2, the special solution y of (1.7) in the asymptotic relation (3.8) is unique up to a constant multiple. Thus, (3.8) gives the same asymptotic representation independently of the choice of y.

Theorem 3.3 is a generalization of Theorem 2.1 to (1.5). Indeed, in the special case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M75">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M129">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M130">View MathML</a>; (1.5) reduces to the Dickman-de Bruijn equation (1.1). Its formal adjoint equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M131">View MathML</a>

has the positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M132">View MathML</a> satisfying condition (3.1). Therefore, Theorem 3.3 applies and its conclusion reduces to the limit relation (1.3).

For qualitative results similar to Theorem 3.3, see [8,9,11,12] and the references therein.

The proof of Theorem 3.3 will be based on the well-known duality between the solutions of a linear delay differential equation and its formal adjoint equation (see [[7], Section 6.3]). Namely,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M133">View MathML</a>

(3.10)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M2">View MathML</a> whenever x and y are solutions of (1.5) and (1.7), respectively. We will also need the following simple lemma.

Lemma 3.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M135">View MathML</a>and suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M136">View MathML</a>is a continuous function such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M137">View MathML</a>

(3.11)

Then every continuous solution of the integral equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M138">View MathML</a>

(3.12)

converges tocas<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M6">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M140">View MathML</a>. By virtue of (3.11), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M87">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M142">View MathML</a>

(3.13)

Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M143">View MathML</a>

Choose a constant K such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M144">View MathML</a>

(3.14)

Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M145">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M146">View MathML</a> and we claim that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M147">View MathML</a>

(3.15)

Otherwise, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M102">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M149">View MathML</a>

From this and (3.12), we find that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M150">View MathML</a>

the last inequality being a consequence of (3.13). Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M151">View MathML</a>, contradicting (3.14). Thus, (3.15) holds.

From (3.12) and (3.15), we find for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M90">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M153">View MathML</a>

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M6">View MathML</a> in the last inequality and using (3.11), we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M155">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M6">View MathML</a>. □

Now we are in a position to give a proof of Theorem 3.3.

Proof of Theorem 3.3 Write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M31">View MathML</a> for brevity and let y be a solution of (1.7) which is positive on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M56">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M53">View MathML</a> and satisfies condition (3.1). By virtue of (3.7) and (3.10), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M160">View MathML</a>

(3.16)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M2">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M29">View MathML</a> as in (3.9). If we let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M163">View MathML</a>

then (3.16) can be written in the form (3.12) with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M164">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M165">View MathML</a> replaced with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M166">View MathML</a>. Clearly, conditions (1.6) and (3.1) imply that assumption (3.11) of Lemma 3.4 is satisfied. By the application of Lemma 3.4, we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M167">View MathML</a>

which is only a reformulation of the limit relation (3.8). □

Finally, we remark that applying a transformation technique described in [13] and [14], Theorem 3.3 can possibly be extended to a class of equations with time-varying delays.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgements

M Pituk was supported in part by the Hungarian National Foundation for Scientific Research (OTKA) Grant No. K101217. G Röst was supported in part by European Union and co-funded by the European Social Fund under the project ‘Telemedicine focused research activities on the field of Mathematics, Informatics and Medical sciences’ of project number TÁMOP-4.2.2.A-11/1/KONV-2012-0073, ERC Starting Grant No. 259559 and OTKA K109782.

References

  1. Dickman, K: On the frequency of numbers containing prime factors of a certain relative magnitude. Ark. Mat. Astron. Fys.. 22A(10), 1–14 (1930)

  2. de Bruijn, NG: On the number of positive integers ≤x and free of prime factors >y. I. Indag. Math.. 13, 50–60 (1951)

  3. de Bruijn, NG: On the number of positive integers ≤x and free of prime factors >y. II. Indag. Math.. 2, 239–247 (1966)

  4. Nieuw Archief voor Wiskunde, Problem Section, 4/14 no. 3 Nov. 1996, p.429

  5. Nieuw Archief voor Wiskunde, Problem Section, 5/9 no. 2 June 2008, p.232

  6. Nieuw Archief voor Wiskunde, Problem Section, 5/11 no. 1 March 2010, p.76

  7. Hale, JK: Theory of Functional Differential Equations, Springer, New York (1977)

  8. Győri, I, Pituk, M: Stability criteria for linear delay differential equations. Differ. Integral Equ.. 10, 841–852 (1997)

  9. Diblík, J: Behaviour of solutions of linear differential equations with delay. Arch. Math.. 34, 31–47 (1998)

  10. Győri, I, Ladas, G: Oscillation Theory of Delay Differential Equations with Applications, Oxford University Press, New York (1991)

  11. Arino, O, Pituk, M: More on linear differential systems with small delays. J. Differ. Equ.. 170, 381–407 (2001). Publisher Full Text OpenURL

  12. Diblík, J, Růžičková, M: Asymptotic behavior of solutions and positive solutions of differential delayed equations. Funct. Differ. Equ.. 14, 83–105 (2007)

  13. Čermák, J: Note on canonical forms for functional differential equations. Math. Pannon.. 11, 29–39 (2000)

  14. Neuman, F: On transformations of differential equations and systems with deviating argument. Czechoslov. Math. J.. 31, 87–90 (1981)