Large time behavior of a linear delay differential equation with asymptotically small coefficient
© Pituk and Röst; licensee Springer. 2014
Received: 4 February 2014
Accepted: 15 April 2014
Published: 14 May 2014
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.
We emphasize that the solution presented in  does not imply the existence of the limit (1.3).
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).
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.
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]).
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. □
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 3.2 Suppose condition (1.6) holds. Let and be eventually positive solutions of (1.7) satisfying condition (3.1). Then is a constant multiple of .
This, together with (3.4), implies (3.3).
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).
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.
has the positive solution satisfying condition (3.1). Therefore, Theorem 3.3 applies and its conclusion reduces to the limit relation (1.3).
for whenever x and y are solutions of (1.5) and (1.7), respectively. We will also need the following simple lemma.
converges to c as .
the last inequality being a consequence of (3.13). Hence , contradicting (3.14). Thus, (3.15) holds.
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.
which is only a reformulation of the limit relation (3.8). □
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.
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