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
where , and is a continuous function such that
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).
Theorem 2.1Suppose that is a continuous function and let be the unique solution of the initial value problem (1.1) and (1.2). Then
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.
Proof of Theorem 2.1 Write for brevity. First we show that
Then (2.2) can be written in the form
where . From this, we find for ,
From the last inequality, by the application of Gronwall’s lemma (see, e.g., [, Chapter 1, Lemma 3.1]), we conclude that
From this and (2.5), we find for ,
In view of (2.4), this implies (2.3).
From (2.2) and (2.3), we obtain for ,
Letting in the last inequality, we obtain
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.
Theorem 3.1Suppose condition (1.6) holds. Then (1.7) has a solution which is positive for all largetand such that
We will prove Theorem 3.1 by applying a technique known from the oscillation theory of delay differential equations (see [, Section 2.3]).
Proof Let and denote the positive part and the negative part of p, respectively, defined by
Since , by virtue of (1.6), there exists such that
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
Clearly, Ω is a nonempty, closed and convex subset of . Define the operator by
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
a special case of (1.7) when , and , has the positive solution for which the ratio is unbounded as .
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 and be eventually positive solutions of (1.7) satisfying condition (3.1). Then is a constant multiple of .
Proof We begin with two simple observations. First, if y is a solution of (1.7), then
Second, if y is a solution of (1.7) which is positive on for some and satisfies condition (3.1), then
where M is an arbitrary constant such that
Indeed, from (1.7) we find for ,
This, together with (3.4), implies (3.3).
By assumptions, there exists such that both solutions and are positive on and satisfy condition (3.1). As noted before (see (3.3)), if is sufficiently large, then
Since , if is sufficiently small, then . By virtue of (1.6), there exists such that
We will show that for all , where . In view of the linearity of (1.7), the function is a solution of (1.7) and, by virtue of (3.5), the quantity
is finite. Applying (3.2) to both solutions and of (1.7) and taking into account that , we obtain, for ,
where the last but one inequality is a consequence of (3.6). From the last inequality, we obtain
Since , this implies that and therefore for all . Finally, by the uniqueness of the backward continuation of the solutions of (1.7), we conclude that for all . □
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 denote the solution of (1.5) with initial data
where is a continuous function. Then
whereyis any eventually positive solution of (1.7) satisfying (3.1) and is a constant given by
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 , and ; (1.5) reduces to the Dickman-de Bruijn equation (1.1). Its formal adjoint equation
has the positive solution satisfying condition (3.1). Therefore, Theorem 3.3 applies and its conclusion reduces to the limit relation (1.3).
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,
for whenever x and y are solutions of (1.5) and (1.7), respectively. We will also need the following simple lemma.
Lemma 3.4Let and suppose that is a continuous function such that
Then every continuous solution of the integral equation
converges tocas .
Proof Let . By virtue of (3.11), there exists such that
Choose a constant K such that
Clearly, for and we claim that
Otherwise, there exists such that
From this and (3.12), we find that
the last inequality being a consequence of (3.13). Hence , contradicting (3.14). Thus, (3.15) holds.
From (3.12) and (3.15), we find for ,
Letting in the last inequality and using (3.11), we conclude that as . □
Now we are in a position to give a proof of Theorem 3.3.
Proof of Theorem 3.3 Write for brevity and let y be a solution of (1.7) which is positive on for some and satisfies condition (3.1). By virtue of (3.7) and (3.10), we have
for with as in (3.9). If we let
then (3.16) can be written in the form (3.12) with
and replaced with . 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
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