The linear delay differential equation , , is considered, where and the coefficient is continuous and small in the sense that , . 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, , , our result yields an explicit asymptotic representation of the solutions as .
MSC: 34K06, 34K25, 11A51.
Keywords:delay differential equation; formal adjoint equation; Dickman-de Bruijn equation; asymptotic behavior
The linear scalar delay differential equation
plays a significant role in analytic number theory. Its special solution with initial values for is called the Dickman-de Bruijn function. The Dickman-de Bruijn function was first studied by actuary Dickman  and later by de Bruijn [2,3] to estimate the proportion of smooth numbers up to a given bound. In  (see also ) van der Lune proposed some problems regarding the solutions of (1.1). The solutions to these problems, given by Bierkens, appeared in . Suppose that is a continuous function and let denote the unique solution of (1.1) with initial values
Among others, it was shown that if the limit
exists and is finite, then its value is given by
We emphasize that the solution presented in  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
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
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).
The proof of Theorem 2.1 will be based on the identity
where 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 . 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.
Then (2.2) can be written in the form
From the last inequality, by the application of Gronwall’s lemma (see, e.g., [, Chapter 1, Lemma 3.1]), we conclude that
In view of (2.4), this implies (2.3).
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 [, Theorem 3.3] and its proof) and by Diblík (see [, 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.
We will prove Theorem 3.1 by applying a technique known from the oscillation theory of delay differential equations (see [, Section 2.3]).
Let be the space of continuous functions mapping into ℝ with the topology of uniform convergence on compact subsets of . Let Ω denote the set of functions from which satisfy the system of inequalities
It is easily verified that F is continuous and . Furthermore, the functions from are uniformly bounded and equicontinuous on each compact subinterval of . Therefore, by the Arzela-Ascoli theorem, the closure of is compact in . By the application of the Schauder-Tychonoff fixed point theorem, we conclude that there exists such that . It is easily seen that this fixed point y is a solution of (1.7) on with property (3.1). Clearly, the solution can be extended backward to all 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
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.
Proof We begin with two simple observations. First, if y is a solution of (1.7), then
where M is an arbitrary constant such that
This, together with (3.4), implies (3.3).
where the last but one inequality is a consequence of (3.6). From the last inequality, we obtain
Now we can formulate our main result about the large time behavior of the solutions of (1.5).
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.
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 [, Section 6.3]). Namely,
Then every continuous solution of the integral equation
Choose a constant K such that
From this and (3.12), we find that
Now we are in a position to give a proof of Theorem 3.3.
then (3.16) can be written in the form (3.12) with
which is only a reformulation of the limit relation (3.8). □
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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.
Arino, O, Pituk, M: More on linear differential systems with small delays. J. Differ. Equ.. 170, 381–407 (2001). Publisher Full Text