\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 85, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2011/85\hfil Asymptotic behavior of solutions]
{Asymptotic behavior of solutions to a class of linear non-autonomous
 neutral delay differential equations}

\author[G. Chen\hfil EJDE-2011/85\hfilneg]
{Guiling Chen}

\address{Guiling Chen \newline
Mathematical Institute,  Leiden University,
P.O. Box 9512, 2300 RA, Leiden, The Netherlands}
\email{guiling@math.leidenuniv.nl}

\thanks{Submitted May 24, 2011. Published June 29, 2011.}
\subjclass[2000]{34K11, 34K40, 34K25}
\keywords{Neutral delay differential equation;
  characteristic equation; \hfill\break\indent
  asymptotic behavior}

\begin{abstract}
 We study a class of linear non-autonomous neutral delay differential
 equations, and establish a criterion for the asymptotic behavior of
 their solutions, by using the corresponding characteristic equation.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}

\section{Introduction}

Let $\mathbb{C}$ be the complex numbers with norm $|\cdot|$. For
$r\geq 0 $, let $\mathcal{C} = \mathcal{C}([-r,0],\mathbb{C})$ be
the space of continuous functions taking $[-r,0]$ into $
\mathbb{C} $ with norm defined by
$\|\varphi\|=\max_{-r\leq\theta\leq 0}|\varphi|$. A functional
differential equation of neutral type, or shortly a neutral
equation, is a system of the form
\begin{equation}\label{e1.1}
\frac{d}{dt}M x_t=L(t) x_t, \quad t\geqslant t_0\in\mathbb{R},
\end{equation}
where $ x_t\in \mathcal{C}$ is defined by $x_t(\theta)=x(t+\theta)$,
$ -r\leq\theta\leq 0 $,
$M: {\mathcal{C}}\to \mathbb{C}$ is continuous,
linear and atomic at zero,
(see \cite[page 255]{HVL}  for the concept of atomic at zero),
\begin{equation}\label{e1.2}
M\varphi=\varphi(0)-\int_{-r}^{0}\varphi(\theta)\,d\mu(\theta),
\end{equation}
where $\operatorname{Var}_{[s, 0]}\mu \to 0$, as $ s\to 0 $.

For \eqref{e1.1}, $L(t)$ denote a family of bounded linear
functionals on $\mathcal{C}$. By the Riesz representation theorem,
for each $t\geqslant t_0$, there exists a complex valued function
of bounded variation $\eta(t,\cdot)$ on $[-r,0]$, normalized so
that $\eta(t,0)=0$ and $\eta(t,\cdot)$ is
continuous from the left in $(-r,0)$ such that
\begin{equation}\label{e1.3}
L(t) \varphi = \int_{-r}^{0}\varphi(\theta)\,d_\theta\eta(t,\theta) .
\end{equation}

For any $ \varphi\in\mathcal{C}$, $ \sigma\in [t_0, \infty) $,
a function $ x=x(\sigma, \varphi) $
defined on $[\sigma-r, \sigma+A)  $ is said to be a solution
of \eqref{e1.1} on $ (\sigma,\sigma+A)$ with initial
$ \varphi $ at $ \sigma $ if $ x $ is continuous on
$[\sigma-r, \sigma+A)  $,   $x_{\sigma}=\varphi $,
$ M x_t $ is continuously differentiable on $  (\sigma,\sigma+A)$,
and relation \eqref{e1.1} is satisfied on
$  (\sigma,\sigma+A)$. For more information on this type of equations,
see \cite{HVL}.

The initial-value problem (IVP) is stated as
\begin{equation}\label{e1.4}
\begin{gathered}
\frac{d}{dt} Mx_t=L(t) x_t \quad t\geqslant \sigma,\\
x_{\sigma}=\varphi.
\end{gathered}
\end{equation}

For $ \mu=0 $ in \eqref{e1.2}, $ M\varphi=\varphi(0) $ and
equation \eqref{e1.1} becomes the
retarded functional differential equation
\begin{equation}\label{e1.5}
  x'(t) = L(t) x_t.
\end{equation}
Consider the \textit{characteristic equation} associated with
 \eqref{e1.5},
\begin{equation}\label{e1.6}
  \lambda(t) = \int_0^r\exp\Big({-\int_{t-\theta}^t} \lambda(s)ds\Big)\,d_\theta \eta(t,\theta)
\end{equation}
which is obtained by looking for solutions to \eqref{e1.5} of
the form
\begin{equation}\label{e1.7}
x(t) = \exp\Big[\int_{0}^t \lambda(s)\,ds\Big].
\end{equation}
The solutions of \eqref{e1.6} are continuous functions
$\lambda(\cdot)$ defined in
$[t_0-r,\infty)$ which satisfy \eqref{e1.5}.

Cuevas and Frasson  \cite{CF} studied
the asymptotic behavior of solutions of \eqref{e1.5} with initial
condition $ x_{\sigma}=\varphi $,
and obtained the following result.

\begin{theorem}\label{thm1.1}
Assume that $\lambda(t)$ is a solution of \eqref{e1.6}
such that
\[
 \limsup_{t\to\infty} \int_0^r \theta
 |e^{-\int_{t-\theta}^t \lambda(s)ds}|
  d_\theta|\eta|(t,\theta) < 1.
\]
Then for each solution $x$ of \eqref{e1.5}, we have that
the limit
\[
\lim_{t\to\infty} x(t) e^{-\int_{t_0}^t \lambda(s)ds}
\]
exists, and
\[
\lim_{t\to\infty} \Big[ x(t) e^{-\int_{t_0}^t \lambda(s)ds}\Big]' =0.
\]
Furthermore,
\[
  \lim_{t\to\infty} x'(t) e^{-\int_{t_0}^t \lambda(s)ds} =
  \lim_{t\to\infty} \lambda(t) x(t) e^{-\int_{t_0}^t \lambda(s)ds},
\]
if $ \lim_{t\to\infty} \lambda(t) x(t) e^{-\int_{t_0}^t \lambda(s)ds} $
exists.
\end{theorem}

Motivated by the work in \cite{CF}, we provide a generalization of
\cite{CF}, and consider the asymptotic behavior of solutions to
\eqref{e1.4}. The method for the proving our main result is
similar to the one in \cite{CF, DixPP06}.
In Section 2, we state  the main results.
In Section 3, some examples will be shown as
applications of the main results of this paper.

\section{Main results}

For equation \eqref{e1.1}, the characteristic equation is
\begin{equation}\label{e2.1}
  \lambda(t) = \int_{-r}^{0} d\mu(\theta)\lambda(t+\theta)
  \exp\Big({-\int_{t+\theta}^t} \lambda(s)ds\Big)+\int_{-r}^{0} d_\theta \eta(t,\theta)
  \exp\Big({-\int_{t+\theta}^t} \lambda(s)ds\Big),
\end{equation}
which is obtained by looking for solutions of \eqref{e1.1} of the
form \eqref{e1.7} and the solutions of \eqref{e2.1} are continuous
functions defined in $ [\sigma-r, \infty) $ satisfying
\eqref{e2.1}. For autonomous neutral functional differential
equations (NFDEs), the constant solutions of \eqref{e2.1} are the
roots of the so called characteristic equation, for detailed
discussion of this type, refer to \cite{M, MVL, HVL}.

\begin{theorem}\label{thm2.1}
Assume that $\lambda(t)$ is a solution of \eqref{e2.1}
such that
\begin{equation}  \label{e2.2}
 \limsup_{t\to \infty}\chi_{\lambda, t}< 1,
\end{equation}
where
\begin{align*}
\chi_{\lambda, t}
&=\int_{-r}^{0}
 |e^{-\int_{t+\theta}^t \lambda(s)\,ds}|\,  d|\mu|(\theta)\\
&\quad + \int_{-r}^{0}(-\theta)|e^{-\int_{t+\theta}^t \lambda(s)\,ds}|
  \Big(|\lambda(t+\theta)|\,d|\mu|(\theta)
+\,d_\theta|\eta|(t,\theta)\Big).
\end{align*}
Then for each solution $x$ of  \eqref{e1.4}, we have that
the limit
\begin{equation}
  \label{e2.3}
   \lim_{t\to\infty} x(t) e^{-\int_{t_0}^t \lambda(s)\,ds}
\end{equation}
exists, and
\begin{equation}
  \label{e2.4}
  \lim_{t\to\infty} \Big[ x(t) e^{-\int_{t_0}^t \lambda(s)\,ds}\Big]' =
  0.
\end{equation}
Furthermore,
\begin{equation}
  \label{e2.5}
  \lim_{t\to\infty} x'(t) e^{-\int_{t_0}^t \lambda(s)\,ds} =
  \lim_{t\to\infty} \lambda(t) x(t) e^{-\int_{t_0}^t \lambda(s)\,ds}
\end{equation}
if the limit in the right-hand side exists.
\end{theorem}

\begin{proof}
From \eqref{e2.2}, there exists $ t_{1}\geq t_{0} $, such that
\[
\sup_{t\geq t_{1}}\chi_{\lambda, t}< 1.
\]
Hence without loss of generality, we assume that $t_{0}=0 $ and define
\[
\Gamma_{\lambda}:=\sup_{t\geq 0}\chi_{\lambda, t}< 1.
\]
For solutions $ x $ of  \eqref{e1.4}, we set
\[
  y(t) = x(t) e^{-\int_{0}^t \lambda(s)\,ds}, \quad t\geqslant -r.
\]
Then  \eqref{e1.4} becomes
\begin{equation}
  \begin{aligned}
   & y'(t)+\lambda(t)y(t)-\int_{-r}^{0}\,d\mu(\theta)y'(t+\theta)e^{-\int_{t+\theta}^t
        \lambda(s)\,ds} \\
=&\int_{-r}^{0}y(t+\theta)e^{-\int_{t+\theta}^t
        \lambda(s)\,ds}\Big(\lambda(t+\theta)\,d\mu(\theta)+\,d_\theta \eta(t,\theta)\Big)
 \label{e2.6}
  \end{aligned}
\end{equation}
and the initial condition is equivalent to
\begin{equation}\label{e2.7}
y(t)=\varphi(t)e^{-\int_{0}^t \lambda(s)\,ds}, \quad     -r\leq t\leq 0.
\end{equation}
Combining \eqref{e2.7} with \eqref{e2.1}, for $ t\geq -r $, we have
\begin{equation}
  \begin{aligned}
   y'(t)&=\int_{-r}^{0}\,d\mu(\theta)y'(t+\theta)e^{-\int_{t+\theta}^t
        \lambda(s)\,ds} \\
&-\int_{-r}^{0}e^{-\int_{t+\theta}^t\lambda(s)\,ds}\int_{-r}^{0}y'(s)\,ds\Big(\lambda(t+\theta)\,d\mu(\theta)+\,d_\theta \eta(t,\theta)\Big).
 \label{e2.8}
  \end{aligned}
\end{equation}
From the definition of the solutions to \eqref{e1.4}, we know that
$ y'(t) $ is continuous, Let
\[
M_{\varphi, \lambda_{0}}=\max\{|\varphi'(t)e^{-\int_{0}^t
        \lambda(s)\,ds}-\lambda(t)\varphi(t)e^{-\int_{0}^t
        \lambda(s)\,ds}|:-r\leq t\leq 0 \}.
\]
We shall show that $M_{\varphi}  $ is also a bound of $ y' $
on the whole interval $ [-r, \infty) $; i.e.,
\begin{equation}\label{e2.9}
|y'(t)|\leq M_{\varphi, \lambda_{0}}, \quad t\geq -r.
\end{equation}
For this purpose, let us consider an arbitrary number $ \varepsilon >0 $. Then
\begin{equation} \label{e2.10}
|y'(t)|< M_{\varphi, \lambda_{0}} + \varepsilon  \quad
\text{for } t\geq -r.
\end{equation}
Indeed, in the opposite case, we suppose there exists a point
$ t^*>0 $ such that
\begin{equation} \label{e2.11}
\begin{gathered}
|y'(t)|< M_{\varphi, \lambda_{0}} + \varepsilon  \quad\text{for }
 -r\leq t<t^*, \\
|y(t^*)|= M(\lambda_{0}, \mu_{0};\phi)+ \varepsilon.
\end{gathered}
\end{equation}
Then combining \eqref{e2.8} and \eqref{e2.11},  we obtain
\begin{equation}
  \begin{aligned}
&M(\lambda_{0}, \mu_{0};\phi)+ \varepsilon \\
&= y'(t^*)\\
&\leq\Big|\int_{-r}^{0}y'(t^*+\theta)e^{-\int_{t^*+\theta}^{t^*}
        \lambda(s)\,ds}\,d\mu(\theta)\Big| \\
&\quad +\Big|\int_{-r}^{0}e^{-\int_{t^*+\theta}^{t^*}\lambda(s)\,ds}\int_{-r}^{0}y'(s)\,ds
 \Big(\lambda(t^*+\theta)\,d\mu(\theta)+\,d_\theta \eta(t^*,\theta)\Big)\Big|\\
&\leq (M_{\varphi, \lambda_{0}} + \varepsilon)\Big\{\int_{-r}^{0}
 |e^{-\int_{t^*+\theta}^{t^*} \lambda(s)\,ds}|
  \,d|\mu|(\theta)\\
&\quad + \int_{-r}^{0}(-\theta)|e^{-\int_{t^*+\theta}^{t^*} \lambda(s)\,ds}|
  \Big(|\lambda(t^*+\theta)|\,d|\mu |(\theta)
 +d_\theta|\eta|(t^*,\theta)\Big)\Big\}\\
&=(M_{\varphi, \lambda_{0}} + \varepsilon)\Gamma_{\lambda}\\
&<(M_{\varphi, \lambda_{0}} + \varepsilon),
  \end{aligned}  \label{e2.12}
\end{equation}
which is a contradiction, so \eqref{e2.10} holds. Since \eqref{e2.10} holds for every $ \varepsilon>0 $, it follows that
$ |y'(t)|\leq M_{\varphi, \lambda_{0}}$, for all $t\geq -r $.
By using \eqref{e2.8} and \eqref{e2.9},  for $t\geq 0 $  we have
\begin{equation}
  \begin{aligned}
 |y'(t)|
&\leq\Big|\int_{-r}^{0}y'(t+\theta)e^{-\int_{t+\theta}^t
        \lambda(s)ds}\,d\mu(\theta)\Big| \\
&\quad +\Big|\int_{-r}^{0}e^{-\int_{t+\theta}^t\lambda(s)\,ds}\int_{-r}^{0}y'(s)\,ds
\Big(\lambda(t+\theta)\,d\mu(\theta)+\,d_\theta \eta(t,\theta)\Big)\Big|\\
&\leq M_{\varphi, \lambda_{0}}\Big\{\int_{-r}^{0}
 |e^{-\int_{t+\theta}^t \lambda(s)\,ds}|
  \,d|\mu|(\theta)\\
&\quad + \int_{-r}^{0}(-\theta)|e^{-\int_{t+\theta}^t \lambda(s)\,ds}|
  \Big(|\lambda(t+\theta)|\,d|\mu|(\theta)+\,d_\theta|\eta|(t,\theta)\Big)\Big\}\\
&=M_{\varphi, \lambda_{0}}\Gamma_{\lambda}, \\
 \label{e2.13}
  \end{aligned}
\end{equation}
which means, for $t\geq 0 $,
\[
|y'(t)|\leq M_{\varphi, \lambda_{0}}\Gamma_{\lambda_{0}}.
\]
One can show by induction, that $y'(t) $ satisfies
\begin{equation} \label{e2.14}
|y'(t)|\leq M_{\varphi, \lambda_{0}}(\Gamma_{\lambda})^{n}
\quad\text{for } t \geq nr-r,\quad (n=0,1,2,3,\dots).
\end{equation}
Since $0\leq \chi_{\lambda, t}< 1$, it follows
that $ y'(t) $ tends to zero as $ t\to \infty $.
So we proved \eqref{e2.4}. In the following, we will
show \eqref{e2.3} holds.

To prove that $ \lim_{t\to \infty}y(t) $ exists, we
consider \eqref{e2.14}.
For an arbitrary $ t\geq 0 $, we set $ n=[t/r]+1 $
(the greatest integer less than or equal to $ t/r+1 $), then
from $ n=[t/r]+1\leq t/r+1\leq[t/r]+2=n+1 $, we have $t/r\leq n$.
From  \eqref{e2.14},
\begin{equation} \label{e2.15}
|y'(t)|\leq M_{\varphi, \lambda_{0}}(\Gamma_{\lambda})^{n}
\leq M_{\varphi, \lambda_{0}}(\Gamma_{\lambda})^{t/r}\quad\text{for }
 t \geq nr-r.
\end{equation}
Now we use the Cauchy convergence criterion, for
$  t>T\geq 0$, from \eqref{e2.15}, we have
\begin{equation}
  \begin{aligned}
|y(t)-y(T)|
&\leq \int_{T}^{t}|y'(s)|\,ds\leq \int_{T}^{t}M_{\varphi, \lambda_{0}}(\Gamma_{\lambda})^{s/r}\,ds\\
&=M_{\varphi, \lambda_{0}}\frac{r}{\ln\Gamma_{\lambda}}\Big[(\Gamma_{\lambda})^{s/r}\Big]_{s=T}^{s=t}\\
&=M_{\varphi, \lambda_{0}}\frac{r}{\ln\Gamma_{\lambda}}\Big[(\Gamma_{\lambda})^{t/r}-(\Gamma_{\lambda})^{T/r}\Big].
\end{aligned} \label{e2.16}
\end{equation}
Let $ T\to \infty $, we have $ t\to \infty $, and by \eqref{e2.16},
we have
\[
M_{\varphi, \lambda}\frac{r}{\ln\Gamma_{\lambda}}
\Big[(\Gamma_{\lambda})^{t/r}-(\Gamma_{\lambda})^{T/r}\Big]\to 0;
\]
and $ \lim_{T\to \infty}|y(t)-y(T)|=0 $.
The Cauchy convergence criterion implies the existence of
$\lim_{t\to \infty} y(t) $. We obtain \eqref{e2.5} by a straight
forward application of \eqref{e2.4}.
\end{proof}

\begin{remark}\label{rem2.1} \rm
Under the conditions of Theorem \ref{thm2.1}, a solution of \eqref{e1.4}
can not grow faster than the exponential function; i.e.,
there exists a constant $ M>0 $, such that
\begin{equation} \label{e2.17}
|x(t)|\leq M e^{\int_{0}^t \lambda(s)\,ds}, \quad\text{for } t\geq 0.
\end{equation}
 From \eqref{e2.17}, it is not difficult to show that:
\begin{itemize}
\item Every solution of \eqref{e1.4} is bounded if and only if
 $\limsup_{t\to \infty}\int_{0}^t \lambda(s)\,ds<\infty$;
\item Every solution of \eqref{e1.4} tends to zero if and only if
$\limsup_{t\to \infty}\int_{0}^t \lambda(s)\,ds\to -\infty$.
\end{itemize}
\end{remark}

\begin{remark} \label{rem2.2} \rm
If the characteristic equation \eqref{e2.1} has a constant
solution $ \lambda(t)=\lambda_{0} $, then from Theorem \ref{thm2.1},
$\lim_{t\to \infty}x(t)e^{-\lambda_{0}t}$  exists.
\end{remark}

\section{Examples}

\begin{example}\label{exmp3.1}\rm
  Consider the linear differential equation with distributed delay
  \begin{equation}
    \label{e3.1}
    x'(t)-\frac{1}{2}x'(t-1)
= \int_{-1}^0 \frac{x(t+\theta)}{2(t+\theta)}\,d\theta,\quad t>1.
  \end{equation}
This equation can be written in the form
\eqref{e1.1} by setting $ \mu(\theta)=-1/2 $ for
$ \theta\leq -1 $, $ \mu(\theta)=0$ for $ \theta >-1 $,
$\eta(t,\theta) = \ln t + \frac{1}{2}\ln(t+\theta)$ for
  $t>1$ and $\theta\in[-1,0]$.
Since  both $\theta\mapsto \eta(t,\theta)$ and
$\theta\mapsto \mu(\theta)$  are increasing functions,
$|\mu|=\mu, |\eta| = \eta$.

  The characteristic equation associated
with \eqref{e3.1} is
\begin{equation}\label{e3.2}
    \lambda(t) = \frac{\lambda(t-1)}{2}\exp\Big[{-\int_{t-1}^t}
      \lambda(s)\,ds\Big]+
    \int_{-1}^0 \frac{1}{2(t+\theta)}
\exp\Big[{-\int_{t+\theta}^t}
      \lambda(s)\,ds\Big] \,d\theta,
\end{equation}
  which has a solution
  \begin{equation}
    \label{eq:lambda ex distrib}
    \lambda(t) = 1/t.
  \end{equation}
  For this $\lambda(t)$ and for $t>1$, using the expression of $ \chi_{\lambda, t} $, we have
  \[
\frac{1}{2}\big(1-\frac{1}{t}\big)+\frac{1}{4t}+
  \int_{-1}^0 \frac{-\theta}{2(t+\theta)} \exp\Big[{-\int_{t+\theta}^t}
      \frac{ds}{s}\Big]\, d\theta =\frac{1}{2}<1\quad
    \text{as } t\to\infty.
  \]
Hence the hypothesis \eqref{e2.2} of Theorem \ref{thm2.1} is fulfilled.
So we obtain that
  \begin{equation}
    \label{eq:ex var distrib - resultados teo}
    \lim_{t\to\infty} \frac{x(t)}{t}\text{ exists},\quad
    \lim_{t\to\infty} \Big[\frac{x(t)}{t}\Big]' =0\quad
    \text{and}\quad
    \lim_{t\to\infty} \frac{x'(t)}{t} =0.
  \end{equation}
\end{example}

\begin{example}\label{exmp3.2} \rm
Consider the equation with variable delay
  \begin{equation}
    \label{e3.5}
    x'(t)-\frac{2}{3}x'(t-1)= \frac{x(t-\tau(t))}{3(t+c-\tau(t))},
\quad t\geqslant t_0.
  \end{equation}
where $c\in\mathbb{R}$ and $\tau:[0,\infty)\to [-1,0]$ is a
continuous function such that $t+c-\tau(t)>0$ for $ t\geqslant t_0$.
Equation \eqref{e3.5} can be written in the form \eqref{e1.1} by
letting $ \mu(\theta)=-2/3$ for $ \theta\leq -1 $,
$ \mu(\theta)=0$ for $ \theta >-1 $, $\eta(t,\theta)=0$ for
$\theta<\tau(t)$, $\eta(t,\theta)=(t+c-\tau(t))/3$ for
$\theta\geqslant\tau(t)$.  Since both
$\theta\mapsto \eta(t,\theta)$ and $\theta\mapsto \mu(\theta)$
are increasing functions, we have that $|\mu|=\mu, |\eta| = \eta$.

The characteristic equation associated with
\eqref{e3.5} is
  \begin{equation}\label{e3.6}
    \lambda(t) = \frac{2\lambda(t-1)}{3}\exp\Big[{-\int_{t-1}^t}
      \lambda(s)ds\Big]+
    \frac{1}{3(t+c-\tau(t))}
\exp\Big[{-\int_{t-\tau(t)}^t}
      \lambda(s)ds\Big]
  \end{equation}
and we have that a solution of \eqref{e3.6} is
 \begin{equation}\label{e3.7}
    \lambda(t) = \frac{1}{t+c}.
  \end{equation}
For \eqref{e3.7}, the left hand side of~\eqref{e2.2}
reads as
  \begin{align*}
&\limsup_{t\to\infty}\Big[\frac{2}{3}\Big(1-\frac{1}{t+c}\Big)
+\frac{1}{6(t+c)}+\int_{-1}^0 (-\theta)
  |e^{-\int_{t-\theta}^t \lambda(s)ds}|
  d_\theta|\eta|(t,\theta)\Big]\\
&=\limsup_{t\to\infty} \Big[\frac{2}{3}-\frac{\tau(t)}{3(t+c)}\Big]=\frac{2}{3}<1.
\end{align*}
and hence hypothesis \eqref{e2.2} of
 Theorem \ref{thm2.1} is fulfilled and therefore,
  for all solutions
  $x(t)$ of \eqref{e3.5}, we have that
\begin{equation}\label{e3.8}
   \lim_{t\to\infty} \frac{x(t)}{t+c}\text{ exists, and }
  \lim_{t\to\infty} \Big[\frac{x(t)}{t+c}\Big]' =0.
\end{equation}
Manipulating further the limits in \eqref{e3.5}, we are able
to establish that
  $x(t) = O(t)$ and $x'(t) =o(t)$ as $t\to\infty$.
\end{example}

\subsection*{Acknowledgements}
I express my thanks to my supervisors Sjoerd Verduyn Lunel and Onno van Gaans who have provided me with
valuable guidance in every stage of my research. Also, I would like to show my deepest gratitude to
Chinese Scholarship Council.

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\newblock On the dominance of roots of characteristic equations for
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\newblock {\em Journal of Mathematical Analysis and Applications 360\/}
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\bibitem{MVL} {Frasson, M. V.~S., and Verduyn~Lunel, S.~M.}
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\bibitem{HVL} {Hale, J.~K., and Verduyn~Lunel, S.~M.}
\newblock {\em Introduction to functional-differential equations}, vol.~99 of
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\newblock Springer-Verlag, New York, 1993.

\end{thebibliography}

\end{document}