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* $y:[{t}_{0},\mathrm{\infty})\to \mathbb{R}$ *which is positive for all large* *t* *and such that* $\underset{t\to \mathrm{\infty}}{lim\hspace{0.17em}sup}\frac{y(t+r)}{y(t)}<\mathrm{\infty}.$

(3.1)

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

${p}_{+}$ and

${p}_{-}$ denote the positive part and the negative part of

*p*, respectively, defined by

${p}_{\pm}(t)=max\{0,\pm p(t)\},\phantom{\rule{1em}{0ex}}t\ge {t}_{0}.$

Since

$0\le {p}_{-}\le |p|$, by virtue of (1.6), there exists

${t}_{1}>{t}_{0}$ such that

${\int}_{t}^{t+r}{p}_{-}(s)\phantom{\rule{0.2em}{0ex}}ds\le \frac{1}{e},\phantom{\rule{1em}{0ex}}t\ge {t}_{1}.$

Let

$C([{t}_{1},\mathrm{\infty}),\mathbb{R})$ be the space of continuous functions mapping

$[{t}_{1},\mathrm{\infty})$ into ℝ with the topology of uniform convergence on compact subsets of

$[{t}_{1},\mathrm{\infty})$. Let Ω denote the set of functions from

$C([{t}_{1},\mathrm{\infty}),\mathbb{R})$ which satisfy the system of inequalities

$exp(-e{\int}_{{t}_{1}}^{t}{p}_{+}(s+r)\phantom{\rule{0.2em}{0ex}}ds)\le y(t)\le exp(e{\int}_{{t}_{1}}^{t}{p}_{-}(s+r)\phantom{\rule{0.2em}{0ex}}ds),\phantom{\rule{1em}{0ex}}t\ge {t}_{1},$

and

$\frac{y(t+r)}{y(t)}\le e,\phantom{\rule{1em}{0ex}}t\ge {t}_{1}.$

Clearly, Ω is a nonempty, closed and convex subset of

$C([{t}_{1},\mathrm{\infty}),\mathbb{R})$. Define the operator

$F:\mathrm{\Omega}\to C([{t}_{1},\mathrm{\infty}),\mathbb{R})$ by

$F(y)(t)=exp(-{\int}_{{t}_{1}}^{t}\frac{p(s+r)y(s+r)}{y(s)}\phantom{\rule{0.2em}{0ex}}ds),\phantom{\rule{1em}{0ex}}t\ge {t}_{1},y\in \mathrm{\Omega}.$

It is easily verified that *F* is continuous and $F(\mathrm{\Omega})\subset \mathrm{\Omega}$. Furthermore, the functions from $F(\mathrm{\Omega})$ are uniformly bounded and equicontinuous on each compact subinterval of $[{t}_{1},\mathrm{\infty})$. Therefore, by the Arzela-Ascoli theorem, the closure of $F(\mathrm{\Omega})$ is compact in $C([{t}_{1},\mathrm{\infty}),\mathbb{R})$. By the application of the Schauder-Tychonoff fixed point theorem, we conclude that there exists $y\in \mathrm{\Omega}$ such that $F(y)=y$. It is easily seen that this fixed point *y* is a solution of (1.7) on $[{t}_{1},\mathrm{\infty})$ with property (3.1). Clearly, the solution $y:[{t}_{1},\mathrm{\infty})\to \mathbb{R}$ can be extended backward to all $t\in [{t}_{0},{t}_{1})$ 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

${y}^{\prime}(t)=2t{e}^{-2t-1}y(t+1),\phantom{\rule{1em}{0ex}}t\ge 0,$

a special case of (1.7) when $r=1$, ${t}_{0}=0$ and $p(t)=-2(t-1){e}^{-2t+1}$, has the positive solution $y(t)={e}^{{t}^{2}}$ for which the ratio $y(t+1)/y(t)={e}^{2t+1}$ is unbounded as $t\to \mathrm{\infty}$.

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* ${y}_{1}$ *and* ${y}_{2}$ *be eventually positive solutions of* (1.7) *satisfying condition* (3.1). *Then* ${y}_{2}$ *is a constant multiple of* ${y}_{1}$.

*Proof* We begin with two simple observations. First, if

*y* is a solution of (1.7), then

$y(t)=y({t}_{1})-{\int}_{{t}_{1}}^{t}p(s+r)y(s+r)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{1em}{0ex}}\text{whenever}t\ge {t}_{1}\ge {t}_{0}.$

(3.2)

Second, if

*y* is a solution of (1.7) which is positive on

$[T,\mathrm{\infty})$ for some

$T>{t}_{0}$ and satisfies condition (3.1), then

$y(t)\le y({t}_{1})exp\left(M{\int}_{{t}_{1}}^{t}|p(s+r)|\phantom{\rule{0.2em}{0ex}}ds\right)\phantom{\rule{1em}{0ex}}\text{whenever}t\ge {t}_{1}\ge T,$

(3.3)

where

*M* is an arbitrary constant such that

$M\ge \underset{t\ge T}{sup}\frac{y(t+r)}{y(t)}.$

(3.4)

Indeed, from (1.7) we find for

$t\ge T$,

${y}^{\prime}(t)=-p(t+r)\frac{y(t+r)}{y(t)}y(t).$

Hence

$y(t)=y({t}_{1})exp(-{\int}_{{t}_{1}}^{t}p(s+r)\frac{y(s+r)}{y(s)}\phantom{\rule{0.2em}{0ex}}ds),\phantom{\rule{1em}{0ex}}t\ge {t}_{1}\ge T.$

This, together with (3.4), implies (3.3).

By assumptions, there exists

$T>{t}_{0}$ such that both solutions

${y}_{1}$ and

${y}_{2}$ are positive on

$[T,\mathrm{\infty})$ and satisfy condition (3.1). As noted before (see (3.3)), if

$M>1$ is sufficiently large, then

${y}_{j}(t)\le {y}_{j}({t}_{1})exp\left(M{\int}_{{t}_{1}}^{t}|p(s+r)|\phantom{\rule{0.2em}{0ex}}ds\right)\phantom{\rule{1em}{0ex}}\text{whenever}t\ge {t}_{1}\ge T,j=1,2.$

(3.5)

Since

$M>1$, if

$q>0$ is sufficiently small, then

${e}^{qM}<M$. By virtue of (1.6), there exists

${t}_{1}>T$ such that

${\int}_{t}^{t+r}|p(s)|\phantom{\rule{0.2em}{0ex}}ds<q,\phantom{\rule{1em}{0ex}}t\ge {t}_{1}.$

(3.6)

We will show that

${y}_{2}(t)=c{y}_{1}(t)$ for all

$t\ge {t}_{1}$, where

$c={y}_{2}({t}_{1})/{y}_{1}({t}_{1})$. In view of the linearity of (1.7), the function

${y}_{3}=c{y}_{1}$ is a solution of (1.7) and, by virtue of (3.5), the quantity

$S=\underset{t\ge {t}_{1}}{sup}[|{y}_{2}(t)-{y}_{3}(t)|exp(-M{\int}_{{t}_{1}}^{t}|p(s+r)|\phantom{\rule{0.2em}{0ex}}ds)]$

is finite. Applying (3.2) to both solutions

${y}_{2}$ and

${y}_{3}$ of (1.7) and taking into account that

${y}_{3}({t}_{1})={y}_{2}({t}_{1})$, we obtain, for

$t\ge {t}_{1}$,

$\begin{array}{rl}|{y}_{2}(t)-{y}_{3}(t)|& =|{\int}_{{t}_{1}}^{t}p(s+r)({y}_{2}(s+r)-{y}_{3}(s+r))\phantom{\rule{0.2em}{0ex}}ds|\\ \le {\int}_{{t}_{1}}^{t}|p(s+r)||{y}_{2}(s+r)-{y}_{3}(s+r)|\phantom{\rule{0.2em}{0ex}}ds\\ \le S{\int}_{{t}_{1}}^{t}|p(s+r)|exp\left(M{\int}_{{t}_{1}}^{s+r}|p(u+r)|\phantom{\rule{0.2em}{0ex}}du\right)\phantom{\rule{0.2em}{0ex}}ds\\ =S{\int}_{{t}_{1}}^{t}|p(s+r)|exp\left(M{\int}_{{t}_{1}}^{s}|p(u+r)|\phantom{\rule{0.2em}{0ex}}du\right)exp\left(M{\int}_{s}^{s+r}|p(u+r)|\phantom{\rule{0.2em}{0ex}}du\right)\phantom{\rule{0.2em}{0ex}}ds\\ \le S{e}^{qM}{\int}_{{t}_{1}}^{t}|p(s+r)|exp\left(M{\int}_{{t}_{1}}^{s}|p(u+r)|\phantom{\rule{0.2em}{0ex}}du\right)\phantom{\rule{0.2em}{0ex}}ds\\ =S{e}^{qM}{[\frac{1}{M}exp\left(M{\int}_{{t}_{1}}^{s}|p(u+r)|\phantom{\rule{0.2em}{0ex}}du\right)]}_{{t}_{1}}^{t}\\ \le S\frac{{e}^{qM}}{M}exp\left(M{\int}_{{t}_{1}}^{t}|p(u+r)|\phantom{\rule{0.2em}{0ex}}du\right),\end{array}$

where the last but one inequality is a consequence of (3.6). From the last inequality, we obtain

$|{y}_{2}(t)-{y}_{3}(t)|exp(-M{\int}_{{t}_{1}}^{t}|p(u+r)|\phantom{\rule{0.2em}{0ex}}du)\le S\frac{{e}^{qM}}{M},\phantom{\rule{1em}{0ex}}t\ge {t}_{1}.$

Hence

$S\le \frac{{e}^{qM}}{M}S.$

Since ${e}^{qM}<M$, this implies that $S=0$ and therefore ${y}_{2}(t)={y}_{3}(t)=c{y}_{1}(t)$ for all $t\in [{t}_{1},\mathrm{\infty})$. Finally, by the uniqueness of the backward continuation of the solutions of (1.7), we conclude that ${y}_{2}(t)=c{y}_{1}(t)$ for all $t\in [{t}_{0},\mathrm{\infty})$. □

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* ${x}^{\varphi}$ *denote the solution of* (1.5)

*with initial data* $x(t)=\varphi (t),\phantom{\rule{1em}{0ex}}{t}_{0}-r\le t\le {t}_{0},$

(3.7)

*where* $\varphi :[{t}_{0}-r,{t}_{0}]\to \mathbb{R}$ *is a continuous function*.

*Then* ${x}^{\varphi}(t)=\frac{1}{y(t)}(c(\varphi )+o(1)),\phantom{\rule{1em}{0ex}}t\to \mathrm{\infty},$

(3.8)

*where* *y* *is any eventually positive solution of* (1.7)

*satisfying* (3.1)

*and* $c(\varphi )$ *is a constant given by* $c(\varphi )=\varphi ({t}_{0})y({t}_{0})+{\int}_{{t}_{0}-r}^{{t}_{0}}p(s+r)\varphi (s)y(s+r)\phantom{\rule{0.2em}{0ex}}ds.$

(3.9)

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

$r=1$,

${t}_{0}=1$ and

$p(t)=-{t}^{-1}$; (1.5) reduces to the Dickman-de Bruijn equation (

1.1). Its formal adjoint equation

${y}^{\prime}(t)=\frac{y(t+1)}{t+1},\phantom{\rule{1em}{0ex}}t\ge 1,$

has the positive solution $y(t)=t$ 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 [[

7], Section 6.3]). Namely,

$x(t)y(t)+{\int}_{t}^{t+r}p(s)x(s-r)y(s)\phantom{\rule{0.2em}{0ex}}ds=\text{constant}$

(3.10)

for $t\ge {t}_{0}$ whenever *x* and *y* are solutions of (1.5) and (1.7), respectively. We will also need the following simple lemma.

**Lemma 3.4**
*Let*
$c\in \mathbb{R}$
*and suppose that*
$a:[{t}_{0},\mathrm{\infty})\to \mathbb{R}$
*is a continuous function such that*
${\int}_{t}^{t+r}|a(s)|\phantom{\rule{0.2em}{0ex}}ds\to 0,\phantom{\rule{1em}{0ex}}t\to \mathrm{\infty}.$

(3.11)

*Then every continuous solution of the integral equation*
$z(t)=c+{\int}_{t}^{t+r}a(s)z(s-r)\phantom{\rule{0.2em}{0ex}}ds,\phantom{\rule{1em}{0ex}}t\ge {t}_{0},$

(3.12)

*converges to* *c* *as* $t\to \mathrm{\infty}$.

*Proof* Let

$q\in (0,1)$. By virtue of (3.11), there exists

$T>{t}_{0}$ such that

${\int}_{t}^{t+r}|a(s)|\phantom{\rule{0.2em}{0ex}}ds<q,\phantom{\rule{1em}{0ex}}t\ge T.$

(3.13)

Define

$M=\underset{T-r\le t\le T}{max}|z(t)|.$

Choose a constant

*K* such that

$K>max\{M,|c|{(1-q)}^{-1}\}.$

(3.14)

Clearly,

$|z(t)|\le M<K$ for

$t\in [T-r,T]$ and we claim that

$|z(t)|<K\phantom{\rule{1em}{0ex}}\text{for all}t\ge T-r.$

(3.15)

Otherwise, there exists

${t}_{1}>T$ such that

$|z(t)|<K\phantom{\rule{1em}{0ex}}\text{for}t\in [T-r,{t}_{1})\text{and}|z({t}_{1})|=K.$

From this and (3.12), we find that

$\begin{array}{rl}K& =|z({t}_{1})|\le |c|+{\int}_{{t}_{1}}^{{t}_{1}+r}|a(s)||z(s-r)|\phantom{\rule{0.2em}{0ex}}ds\\ \le |c|+K{\int}_{{t}_{1}}^{{t}_{1}+r}|a(s)|\phantom{\rule{0.2em}{0ex}}ds\le |c|+Kq,\end{array}$

the last inequality being a consequence of (3.13). Hence $K\le |c|{(1-q)}^{-1}$, contradicting (3.14). Thus, (3.15) holds.

From (3.12) and (3.15), we find for

$t\ge T$,

$|z(t)-c|\le {\int}_{t}^{t+r}|a(s)||z(s-r)|\phantom{\rule{0.2em}{0ex}}ds\le K{\int}_{t}^{t+r}|a(s)|\phantom{\rule{0.2em}{0ex}}ds.$

Letting $t\to \mathrm{\infty}$ in the last inequality and using (3.11), we conclude that $z(t)\to c$ as $t\to \mathrm{\infty}$. □

Now we are in a position to give a proof of Theorem 3.3.

*Proof of Theorem 3.3* Write

${x}^{\varphi}=x$ for brevity and let

*y* be a solution of (1.7) which is positive on

$[{t}_{1},\mathrm{\infty})$ for some

${t}_{1}>{t}_{0}$ and satisfies condition (3.1). By virtue of (3.7) and (3.10), we have

$x(t)y(t)+{\int}_{t}^{t+r}p(s)x(s-r)y(s)\phantom{\rule{0.2em}{0ex}}ds=c(\varphi )$

(3.16)

for

$t\ge {t}_{0}$ with

$c(\varphi )$ as in (3.9). If we let

$z(t)=x(t)y(t),\phantom{\rule{1em}{0ex}}t\ge {t}_{0},$

then (3.16) can be written in the form (3.12) with

$c=c(\varphi ),\phantom{\rule{2em}{0ex}}a(t)=-p(t)\frac{y(t)}{y(t-r)},\phantom{\rule{1em}{0ex}}t\ge {t}_{1}+r,$

and

${t}_{0}$ replaced with

${t}_{1}+r$. 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

$\underset{t\to \mathrm{\infty}}{lim}z(t)=\underset{t\to \mathrm{\infty}}{lim}[x(t)y(t)]=c$

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 time-varying delays.