Abstract
The linear delay differential equation
MSC: 34K06, 34K25, 11A51.
Keywords:
delay differential equation; formal adjoint equation; Dickmande Bruijn equation; asymptotic behavior1 Introduction
The linear scalar delay differential equation
plays a significant role in analytic number theory. Its special solution
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 [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 Dickmande Bruijn equation (1.1) can be extended to the general linear equation
where
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 Dickmande Bruijn equation
In this section, we prove the existence of the limit (1.3) for the solutions of (1.1).
Theorem 2.1Suppose that
The proof of Theorem 2.1 will be based on the identity
where
Now we can give a simple short proof of Theorem 2.1.
Proof of Theorem 2.1 Write
Define
Then (2.2) can be written in the form
where
where
From the last inequality, by the application of Gronwall’s lemma (see, e.g., [[7], Chapter 1, Lemma 3.1]), we conclude that
Hence
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
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
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
Since
Let
and
Clearly, Ω is a nonempty, closed and convex subset of
It is easily verified that F is continuous and
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
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
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
where M is an arbitrary constant such that
Indeed, from (1.7) we find for
Hence
This, together with (3.4), implies (3.3).
By assumptions, there exists
Since
We will show that
is finite. Applying (3.2) to both solutions
where the last but one inequality is a consequence of (3.6). From the last inequality, we obtain
Hence
Since
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
where
whereyis any eventually positive solution of (1.7) satisfying (3.1) and
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
has the positive solution
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 wellknown duality between the solutions of a linear delay differential equation and its formal adjoint equation (see [[7], Section 6.3]). Namely,
for
Lemma 3.4Let
Then every continuous solution of the integral equation
converges tocas
Proof Let
Define
Choose a constant K such that
Clearly,
Otherwise, there exists
From this and (3.12), we find that
the last inequality being a consequence of (3.13). Hence
From (3.12) and (3.15), we find for
Letting
Now we are in a position to give a proof of Theorem 3.3.
Proof of Theorem 3.3 Write
for
then (3.16) can be written in the form (3.12) with
and
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 timevarying 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 cofunded by the European Social Fund under the project ‘Telemedicine focused research activities on the field of Mathematics, Informatics and Medical sciences’ of project number TÁMOP4.2.2.A11/1/KONV20120073, ERC Starting Grant No. 259559 and OTKA K109782.
References

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

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

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

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

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

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

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

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

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

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

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

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

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

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