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 3.1 Suppose condition (1.6) holds. Then (1.7) has a solution which is positive for all large t and 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 3.2 Suppose condition (1.6) holds. Let and be eventually positive solutions of (1.7) satisfying condition (3.1). Then is a constant multiple of .
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 3.3 Suppose condition (1.6) holds. Let denote the solution of (1.5) with initial data
where is a continuous function. Then
where y is 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).
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 [, 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.
and suppose that
is a continuous function such that
Then every continuous solution of the integral equation
converges to c as .
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). □
Finally, we remark that applying a transformation technique described in  and , Theorem 3.3 can possibly be extended to a class of equations with time-varying delays.