\documentclass[reqno]{amsart}
\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 10, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2005 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2005/10\hfil An asymptotic property for delay equations]
{An asymptotic property of solutions to linear nonautonomous delay
differential equations}
\author[J. G. Dix, Ch. G. Philos, \& I. K. Purnaras\hfil EJDE-2005/10\hfilneg]
{Julio G. Dix, Christos G. Philos, Ioannis K. Purnaras} % in alphabetical order
\address{Julio G. Dix \hfill\break
Department of Mathematics\\
Texas State University\\
601 University Drive, San Marcos, TX 78666, USA}
\email{julio@txstate.edu}
\address{Christos G. Philos \hfill\break
Department of Mathematics\\
University of Ioannina\\
P. O. Box 1186\\
451 10 Ioannina, Greece}
\email{cphilos@cc.uoi.gr}
\address{Ioannis K. Purnaras \hfill\break
Department of Mathematics\\
University of Ioannina\\
P. O. Box 1186\\
451 10 Ioannina, Greece}
\email{ipurnara@cc.uoi.gr}
\date{}
\thanks{Submitted July 9, 2004. Published January 12, 2005.}
\subjclass{34K25, 34C10, 35K15}
\keywords{Delay differential equation; asymptotic behavior; \hfill\break\indent
characteristic equation}
\begin{abstract}
We study first order linear delay differential equations with
variable coefficients and constant delays.
Using solutions to a characteristic equation, we show
asymptotic properties of solutions to the delay equation.
To illustrate the hypothesis of the main theorem, we present
an example.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\section{Introduction}
We study the asymptotic behavior of solutions to the delay differential
equation
\begin{equation}
x'(t)=a(t)x(t)+\sum_{j=1}^kb_j(t)x(t-\tau_j), \quad \text{for } t\geq 0 \label{E}\\
\end{equation}
with initial condition
\begin{equation} x(t)=\phi (t),\quad \text{for } \min_{1\leq j\leq k}\{-\tau_j\} \leq t \leq 0, \, \label{C}
\end{equation}
where the coefficients $a$ and $b_j$ are continuous real-valued
functions on $[0,\infty)$, the delays $\tau_j$
are positive real numbers ($j=1,2,\dots, k$), $k$ is a positive integer,
and $\phi$ is a given continuous function.
The work on this paper is motivated by the publication of the following
interesting results.
Driver, Sasser and Slater \cite{d5} obtained significant results on
asymptotic behavior, non-oscillation, and stability of
solutions to first order linear delay differential equations with constant
coefficients and one constant delay.
Driver \cite{d3} obtained similar results for first order linear autonomous
delay differential equations with infinitely many distributed delays.
The results in \cite{d5} have been improved and extended by Philos
\cite{p1} for first order linear delay differential equations with
coefficients that are periodic having a common period, and delays
that are constants multiples of this period. These results have been extended
and improved by Kordonis, Niyianni and Philos \cite{k1} for first order linear autonomous
neutral delay differential equations. The results in \cite{k1,p1}
have been extended and slightly improved by Philos and Purnaras in \cite{p2}.
There the authors study first order linear neutral delay differential
equations with periodic coefficients having a common period and constant
delays that are multiples of this period.
Graef and Qian \cite{g1} obtained results closely related to the ones above
for first order forced delay differential equations.
Driver \cite{d4} and Arino and Pituk \cite{a1} obtained important results
for linear differential systems with small delays.
In the present paper, we define a characteristic equation and then
utilize its solution to state asymptotic results for solutions of the
delay equation. Also we obtain a non-oscillation result, Remark \ref{rmk1}.
Our main result is stated as Theorem \ref{thm1}
and proved in the next section.
The limit obtained in Theorem \ref{thm1} is found explicitly
when the solution to the characteristic equation is a constant.
An application of Theorem \ref{thm1} provides a
necessary and sufficient condition for all solutions
of \eqref{E} to be bounded, and a necessary and sufficient condition for
all solutions of \eqref{E} to tend to zero at $\infty $.
The last section contains an example and discussions on the results
of the paper.
\section{Statement of Results}
We shall assume that the delays are positive and denote
\[
\tau =\max\{ \tau_j : 1\leq j \leq k\},\quad
\sigma= \min\{ \tau_j : 1\leq j \leq k\}\,.
\]
Let $C([-\tau ,0],\mathbb{R})$ denote the set of continuous real-valued
functions on $[-\tau ,0]$.
By a solution $x$ to the delay differential equation \eqref{E}, we mean
a continuous real-valued function, defined on $[-\tau,\infty )$,
which is continuously differentiable on $[0,\infty )$ and
satisfies \eqref{E}.
It is well-known that for each given $\phi \in C([-\tau ,0],\mathbb{R})$,
problem \eqref{E}-\eqref{C} has a unique solution; see for example
\cite{d1,h1,h2}.
With the delay equation \eqref{E}, we associate the integral equation
\begin{equation} \label{ast}
\begin{gathered}
\lambda (t)=a(t)+\sum_{j=1}^kb_j(t)
\exp\Big[ -\int_{t-\tau_j}^t\lambda (s)ds\Big],\quad \mbox{for } t\geq 0\,;\\
\lambda(t)= \lambda_0(t), \quad \mbox{for } -\tau\leq t \leq 0 \,
\end{gathered}
\end{equation}
which is called the (generalized) \emph{characteristic equation} of \eqref{E}.
This equation is obtained when looking for solutions of the form
\[
x(t)=\phi(0)\exp \Big[ \int_0^t\lambda (s)ds\Big]\,.
\]
Note that this solution can not change sign; i.e., $x(t)$ is either
positive or negative or identically zero.
By a solution $\lambda $ to the characteristic equation,
we mean a continuous real-valued function, defined on $[-\tau,\infty )$,
which satisfies \eqref{ast}.
\begin{lemma} \label{lem1}
For each $\lambda_0$ in $C([-\tau ,0],\mathbb{R})$, the characteristic equation has
a unique global solution.
\end{lemma}
\begin{proof}
Let $u(t)=\exp \big[ \int_0^t\lambda (s)ds\big]$ for $t\geq 0$,
and $w(t)=\exp \big[ \int_0^t\lambda_0(s)ds\big]$ for $ -\tau\leq t\leq 0$.
Then using the characteristic equation, for $0\leq t\leq \sigma$,
we obtain the linear differential equation
\[
u'(t)=\lambda(t)u(t)=a(t)u(t)+\sum_{j=1}^k b_j(t) w(t-\tau_j)
\]
with $u(0)=1$.
The solution to this equation is
\[
u(t)=\Big[1+\int_0^t \sum_{j=1}^k b_j(s)
\exp\Big[\int_0^{s-\tau_j}\lambda_0(r)\,dr -\int_0^s a(r)\,dr\Big] \,ds\Big]
\exp\Big[\int_0^t a(r)\,dr\Big]
\]
which allows defining $\lambda(t)=u'(t)/u(t)$ on $[0,\sigma]$.
For the next interval, let $w(t)=u(t)$ on $[-\tau,\sigma]$.
Then for $\sigma\leq t \leq 2\sigma$,
we obtain the differential equation
\[
u'(t)=a(t)u(t)+\sum_{j=1}^k b_j(t) w(t-\tau_j)
\]
whose solution is
\begin{equation} \label{linsol}
u(t)=\Big[1+\int_0^t \sum_{j=1}^k b_j(s)
\exp\Big[\int_0^{s-\tau_j}\lambda(r)\,dr -\int_0^s a(r)\,dr\Big] \,ds\Big]
\exp\Big[\int_0^t a(r)\,dr\Big]
\end{equation}
which allows defining $\lambda(t)$ on $[\sigma,2\sigma]$.
Proceeding in this manner,
we define $\lambda(t)$ for all $t\geq -\tau$,
which completes the proof.
\end{proof}
\begin{remark} \label{rmk1} \rm
If the solution to \eqref{E}-\eqref{C} does not have zeros on some
interval $[t^*-\tau,t^*]$, then the solution does not have zeros on
$[t^*,\infty)$; i.e., the solution can not change sign on
$[t^*,\infty)$. To show this claim let $t^*$ be the initial time
for the characteristic equation and $\lambda_0(t)$ be given
implicitly by
$x(t)=x(t^*)\exp\big[\int_{t^*}^t\lambda_0(s)\,ds\big]$, with
$t^*-\tau\leq t \leq t^*$. Then, by the uniqueness of solutions to
\eqref{E},
\[
x(t)=x(t^*)\exp\Big[\int_{t^*}^t\lambda(s)\,ds\Big],\quad \mbox{for } t\geq t^*\,.
\]
Therefore,
$x(t)$ can not have zeros on $[t^*,\infty)$.
\end{remark}
Our main result is the following theorem.
\begin{theorem} \label{thm1}
Assume that
\begin{equation} \label{H}
\sup_{t\geq \tau }\sum_{j=1}^k| b_j(t)|
\tau_j\exp \Big[ -\int_{t-\tau_j}^t\lambda(s)ds\Big]<1 \,.
\end{equation}
Then for each solution $x$ of \eqref{E}-\eqref{C}
there exists a constant $L_{\phi,\lambda_0}$ such that
\[
\lim_{t\to \infty } x(t)\exp \Big[ -\int_0^t\lambda(s)ds\Big] = L_{\phi,\lambda_0} \,,
\]
and
\[
\lim_{t\to \infty }\Big\{ x(t)\exp \Big[ -\int_0^t\lambda(s)ds\Big] \Big\} '=0.
\]
\end{theorem}
\begin{remark} \label{rmk2} \rm
Under the conditions of Theorem \ref{thm1}, a solution to \eqref{E}
can not grow faster than the exponential function determined by the
characteristic equation; i.e.,
there exists a constant $M$ such that
\[
\big| x(t)\big| \leq M
\exp \Big[ \int_0^t\lambda(s)ds\Big], \quad
\text{for }t\geq 0\,.
\]
\end{remark}
\begin{remark} \label{rmk3} \rm
When the solution to \eqref{ast} is a constant $\lambda_0$ satisfying \eqref{H},
\[
\lim_{t\to\infty} x(t)\exp (-t\lambda_0) =L_{\phi,\lambda_0}\,.
\]
In particular when zero is the solution to \eqref{ast},
$\lim_{t\to\infty} x(t)=L_{\phi,0}$.
\end{remark}
Note that if $\lambda$ is a solution of \eqref{ast}, then
\[
x(t)=\phi(0)\exp \Big[ \int_0^t\lambda(s)ds\Big]
\]
is a solution of \eqref{E} with initial function
$\phi(t)=\phi(0)\exp \big[ \int_0^t\lambda(s)ds\big]$.
Then we obtain easily the following results.
\begin{remark} \label{rmk4} \rm
Under the assumptions of Theorem \ref{thm1}, we have:
\begin{enumerate}
\item Every solution of \eqref{E} is bounded if and only if
$\limsup_{t\to \infty } \int_0^t\lambda(s)ds<\infty$.
\item Every solution of \eqref{E} tends to zero at $\infty $
if and only if \\
$\lim_{t\to \infty } \int_0^t\lambda(s)ds=-\infty$.
\end{enumerate}
\end{remark}
\section{Proof of main result}
\begin{proof}[Proof of Theorem \ref{thm1}]
For solutions $x$ of \eqref{E}-\eqref{C} and $\lambda$ of
\eqref{ast}, we define
\[
y(t)=x(t)\exp \Big[ -\int_0^t\lambda(s)ds\Big], \quad t\geq -\tau.
\]
Differentiating in this function, and using \eqref{E}, \eqref{ast},
we obtain
\begin{align*}
&y'(t)\\
&=\Big(x'(t)-x(t)\lambda(t)\Big) \exp\Big[-\int_0^t\lambda(s)ds\Big]\\
&= \Big(\sum_{j=1}^kb_j(t)x(t-\tau_j)
-x(t)\sum_{j=1}^kb_j(t)\exp \Big[ -\int_{t-\tau_j}^t\lambda(s)ds\Big]\Big)
\exp\Big[-\int_0^t\lambda(s)ds\Big]\,.
\end{align*}
Using that
$x(t-\tau_j)=y(t-\tau_j)\exp \big[ \int_0^{t-\tau_j}\lambda(s)ds\big]$,
the above equality yields
\[
y'(t)=-\sum_{j=1}^kb_j(t)[ y(t)-y(t-\tau_j)]
\exp \Big[ -\int_{t-\tau_j}^t\lambda(s)ds\Big]\quad
\mbox{for }t\geq0\,.
\]
From this equation and the fundamental theorem of calculus,
\begin{equation}
y'(t)=-\sum_{j=1}^kb_j(t) \Big[ \int_{t-\tau_j}^ty'(s)ds\Big]
\exp \Big[ -\int_{t-\tau_j}^t\lambda(s)ds\Big]\quad
\mbox{for }t\geq \tau\,.
\label{e3.1}
\end{equation}
If all $b_j$'s are identically zero on $[\tau,\infty)$, from \eqref{e3.1},
$y'=0$ and $y$ is constant on $[\tau ,\infty )$ which would complete the
proof.
Therefore, we assume that at least one $b_j$ is
not identically zero on $[\tau ,\infty )$ . Let
\[
\mu_{\lambda_0}=\sup_{t\geq \tau }\sum_{j=1}^k| b_j(t)|
\tau_j\exp \Big[ -\int_{t-\tau_j}^t\lambda(s)ds\Big] \,.
\]
Then, by \eqref{H},
\begin{equation}
0<\mu_{\lambda_0}<1\,. \label{e3.2}
\end{equation}
Note that the maximum of $|y'|$ on $[0,\tau]$ depends on $x$ and $\lambda $;
hence, on the initial functions $\phi $ and $\lambda_0$.
Let
\[
M_{\phi,\lambda_0} =\max\big\{| y'(t)|: 0\leq t\leq \tau \big\} \,.
\]
We shall show that $M_{\phi,\lambda_0} $ is also a bound of $|y'|$
on the whole interval $[0,\infty )$; i.e.,
\begin{equation}
|y'(t)| \leq M_{\phi,\lambda_0} \quad \mbox{for all } t\geq 0\,. \label{e3.3}
\end{equation}
On the contrary, assume that there exist $\epsilon>0$ and $t\geq 0$
such that $|y'(t)| > M_{\phi,\lambda_0}+\epsilon$.
Since $|y'(t)| \leq M_{\phi,\lambda_0}$ for $0\leq t\leq \tau$,
by the continuity of $y'$, there exists $t^{\ast }>\tau$ such that
\[
|y'(t)| < M_{\phi,\lambda_0}+\epsilon\,, \quad\mbox{for }0\leq t0$.
In view of \eqref{e3.1} and \eqref{e3.3},
\begin{align*}
| y'(t)| &\leq \sum_{j=1}^k| b_j(t)|
\Big[\int_{t-\tau_j}^t| y'(s)|ds\Big]
\exp \Big[ -\int_{t-\tau_j}^t\lambda(s)ds\Big] \\
&\leq M_{\phi,\lambda_0} \sum_{j=1}^k| b_j(t)| \tau_j
\exp \Big[-\int_{t-\tau_j}^t\lambda(s)ds\Big] \\
&\leq M_{\phi,\lambda_0} (\mu_{\lambda_0}) \quad
\mbox{for } t\geq \tau \,.
\end{align*}
Using this inequality, we can show by induction
that
\begin{equation}
| y'(t)| \leq M_{\phi,\lambda_0}(\mu_{\lambda_0})^n
\quad \text{for }t\geq n\tau \quad (n=0,1,\dots ).
\label{e3.6}
\end{equation}
For an arbitrary $t\geq 0$, we set $n=\lfloor t/\tau \rfloor$
(the greatest integer less than or equal to $t/\tau$).
Then $t\geq n\tau $ and $\frac{t}{\tau }-1T\geq0$, from \eqref{e3.7}, we have
\begin{align*}
| y(t)-y(T)|
&\leq \int_T^t| y'(s)| ds
\leq \int_T^t M_{\phi,\lambda_0}(\mu_{\lambda_0})^{\frac s{\tau} -1} \,ds\\
&=M_{\phi,\lambda_0}\frac \tau{\ln(\mu_{\lambda_0})}
\Big[ (\mu_{\lambda_0})^{\frac s{\tau} -1}\Big]_{s=T}^{s=t} \\
&=M_{\phi,\lambda_0}\frac \tau{\ln(\mu_{\lambda_0})}
\Big[ (\mu_{\lambda_0})^{\frac t{\tau} -1}-(\mu_{\lambda_0})^{\frac T{\tau} -1}\Big]\,.
\end{align*}
As $T\to \infty$, we have $t\to \infty$ and, by \eqref{e3.2}, the two right-most terms above
approach zero. Therefore,
$\lim_{T\to \infty }| y(t)-y(T)|=0$ which by the Cauchy convergence
criterion implies the existence of $\lim_{t\to \infty }y(t)$.
We call this limit $L_{\phi,\lambda_0} $ because it depends on $y$
which in turn depends on the initial functions $\phi$ and $\lambda_0$.
This shows the first limit in Theorem \ref{thm1} and completes the proof.
\end{proof}
\section{Discussion}
To illustrate the hypothesis in Theorem \ref{thm1}, we provide an example of
a non-autonomous (and non-periodic) delay differential equation of the
form \eqref{E}, for which \eqref{ast} has a explicit solution and
satisfies \eqref{H}. \medskip
\noindent\textbf{Example.} Let $k=1$, $\tau_1=2$, and
$a(t)=1/(2(t+3))$, $b_1(t)=1/(2(t+1))$ for $t\geq 0$.
It is easy to verify that
\[
\lambda (t)=\frac{1}{t+3}
\]
is a solution of \eqref{ast} and satisfies \eqref{H}.
Indeed, we can easily check that
\[
\sup_{t\geq \tau_1}| b_1(t)|\tau_1
\exp \Big[-\int_{t-\tau_1}^t\lambda(s)ds\Big]
=\sup_{t\geq 2}\frac{1}{t+3}=\frac{1}{5}<1 .
\]
\begin{remark} \label{rmk5} \rm
Finding conditions on $a$ and $b_j$ that guarantee hypothesis \eqref{H} remains
an open question. To imply this hypothesis, we can use for example
the stronger condition
\[
\sup_{t\geq \tau }\sum_{j=1}^k| b_j(t)|
\exp \Big[ -\int_{t-\tau_j}^t\lambda(s)ds\Big]<\frac{1}{\tau } \,.
\]
Furthermore, if, for each $t \geq 0$, it holds $b_j(t)\geq 0$ for all $j$'s or $b_j(t)\leq 0$ for all $j$'s, from the characteristic equation, it follows that
\[
|\lambda(t)-a(t)| = \sum_{j=1}^k| b_j(t)|
\exp \Big[ -\int_{t-\tau_j}^t\lambda(s)ds\Big]\,,\quad\mbox{for }t\geq0\,.
\]
Note that this equality is obvious when $k=1$.
Under the above assumptions, the condition \eqref{H} is implied by
$\sup_{t\geq 0} |\lambda(t)-a(t)|<1/\tau$ .
This is the strategy in the next lemma.
\end{remark}
\begin{lemma} \label{lem2}
Assume that the coefficients $a$, $b_j$ and the initial function of the
characteristic equation satisfy the following conditions for all $t\geq 0$:
$b_j(t)\geq 0$ and, for some $c$ with $0 \leq c<\frac{1}{\tau}$,
\begin{equation} \label{H2}
\begin{aligned}
&\sum_{j\in J(t)} b_j(t)\exp\Big[\int_0^{t-\tau_j}\lambda_0(s)\,ds
-\int_0^t a(s)\,ds\Big] \\
&+ \sum_{j\not\in J(t)} b_j(t)
\exp\Big[\int_0^{t-\tau_j}c\,ds- \int_{t-\tau_j}^t a(s)\,ds\Big]
\leq c,
\end{aligned}
\end{equation}
where $J(t)$ consists of those indices $j$ for which $t-\tau_j\leq 0$,
($j=1,2,\dots k$).
Then
\[
\sup_{t\geq 0} \sum_{j=1}^k| b_j(t)|
\tau_j\exp \Big[ -\int_{t-\tau_j}^t\lambda(s)ds\Big]<1 \,,
\]
which implies the hypothesis for Theorem \ref{thm1}.
\end{lemma}
\begin{proof}
Since $b_j(t)\geq 0$, the definitions of $\tau$ and of $\lambda$ imply
\[
\sum_{j=1}^k b_j(t)
\tau_j\exp \Big[ -\int_{t-\tau_j}^t\lambda(s)ds\Big]
\leq \tau \sum_{j=1}^k b_j(t)\exp \Big[ -\int_{t-\tau_j}^t\lambda(s)ds\Big]
=\tau |\lambda(t)-a(t)|\,.
\]
The statement of this lemma follows if we show that
$|\lambda(t)-a(t)|\leq c$ for $t\geq 0$.
As in the proof of Lemma \ref{lem1}, let
$u(t)= \exp\big[\int_0^t \lambda(s)\,ds\big]$ for $t\geq -\tau$,
with $\lambda$ defined by \eqref{ast}.
From the characteristic equation,
\[
\lambda(t)-a(t)=\sum_{j=1}^k b_j(t) \exp \Big[ -\int_{t-\tau_j}^t \lambda(s)\,ds\Big]
=\frac{1}{u(t)}\sum_{j=1}^k b_j(t) \exp \Big[ \int_0^{t-\tau_j} \lambda(s)\,ds
\Big]\,.
\]
Since $u(t)$ is the solution given by \eqref{linsol}, for $t\geq 0$,
\[
\lambda(t)-a(t)
=\frac{\sum_{j=1}^k b_j(t)\exp\big[\int_0^{t-\tau_j}\lambda(s)\,ds\big]
\exp\big[ -\int_0^t a(s)\,ds\big]}
{1+\int_0^t \sum_{j=1}^k b_j(s)
\exp\big[\int_0^{s-\tau_j}\lambda(r)\,dr -\int_0^{s} a(r)\,dr\big] \,ds}\,.
\]
Since $b_j(t)\geq 0$, the denominator in the above expression is greater than
or equal to 1 and
\begin{equation} \label{lu}
|\lambda(t)-a(t)|
\leq \sum_{j=1}^k b_j(t)\exp\Big[\int_0^{t-\tau_j}\lambda(s)\,ds
-\int_0^t a(s)\,ds\Big]\,.
\end{equation}
When $0\leq t \leq \sigma$, we have $t-\tau_j\leq 0$, then all $j$'s are in
the class $J(t)$ and $t-\tau_j\leq s\leq 0$. So we use
$\lambda(s)=\lambda_0(s)$ in \eqref{lu}.
Therefore, \eqref{H2} implies
\begin{equation} \label{part1}
|\lambda(t)-a(t)|\leq c \quad \mbox{for all $t$ in $[0,\sigma]$}\,.
\end{equation}
For each fixed $t$ in $[\sigma,2\sigma]$, we have two possible cases:\\
Case 1: $j\in J(t)$. Here $t-\tau_j\leq 0$ and $t-\tau_j\leq s\leq 0$;
so we use $\lambda(s)=\lambda_0(s)$ in \eqref{lu}.
Then, for this case, each summand in \eqref{lu} is equal to
\[
b_j(t)\exp\Big[\int_0^{t-\tau_j}\lambda_0(s)\,ds -\int_0^t a(s)\,ds\Big]\,.
\]
Case 2: $j\not\in J(t)$. Here $0\tau$, and the second summation
needs to be less than or equal to $c$ for all large $t$.
This is very restrictive. In particular, it requires
$\int_{t-\tau_j}^ta(s)\,ds\to \infty$ as $t\to\infty$.
Note that in the example above $a$, $b_j$ do not satisfy the conditions of
Lemma \ref{lem2}.
\smallskip
The real number $L_{\phi,\lambda_0} $, in Theorem \ref{thm1},
has been given explicitly in two special cases:
For linear autonomous delay differential equations
and for linear delay differential equations with periodic coefficients
having a common period and constant delays that are multiples of this period.
See \cite{d3,d5,g1,p1} (and \cite{k1,p2}
for linear neutral delay differential equations).
The proof of Theorem \ref{thm1} is based on an integral
representation of $y'$. Meanwhile, in the autonomous case,
and in the case where the coefficients are periodic with a common
period and the delays are multiples of this period, the proof is based
on an integral representation of $y$.
See \cite{d3,d5,g1,p1} (and \cite{k1,p2} for the neutral case).
We would be interested in generalizing our theorem to linear
delay differential equations with variable coefficients and variable delays.
For variable delays that are bounded, this seems easy to be achieved.
However, the general case of variable delays seems to be somewhat difficult.
Asymptotic behavior of solutions to differential equations
with variable delays and variable coefficients has been studied in \cite{d2},
using a method that does not use characteristic equations.
Furthermore, it would be interesting to generalize our theorem for
linear non-autonomous delay differential equations with infinitely many
distributed delays.
It will be the subject of a future work to extend the
present results to linear neutral delay differential equations
with variable coefficients and constant delays.
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