\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 172, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/172\hfil Stability for integro-differential equations]
{Stability for linear neutral integro-differential equations
with variable delays}

\author[A. Ardjouni,  A. Djoudi, I. Soualhia \hfil EJDE-2012/172\hfilneg]
{Abdelouaheb Ardjouni, Ahcene Djoudi, Imene Soualhia}  % in alphabetical order

\address{Abdelouaheb Ardjouni \newline
Applied Mathematics Lab.,
Facullty of Sciences, Department of Mathematics,
Badji Mokhtar University of Annaba\\
P.O. Box 12, Annaba 23000, Algeria}
\email{abd\_ardjouni@yahoo.fr}

\address{Ahcene Djoudi \newline
Applied Mathematics Lab.,
Facullty of Sciences, Department of Mathematics,
Badji Mokhtar University of Annaba\\
P.O. Box 12, Annaba 23000, Algeria}
\email{adjoudi@yahoo.com}

\address{Imene Soualhia \newline
Applied Mathematics Lab.,
Facullty of Sciences, Department of Mathematics,
Badji Mokhtar University of Annaba\\
P.O. Box 12, Annaba 23000, Algeria}

\thanks{Submitted June 20, 2012. Published October 12, 2012.}
\subjclass[2000]{34K20, 34K30, 34K40}
\keywords{Fixed point; stability; integro-differential equation;
variable delay}

\begin{abstract}
 In this article we study a linear neutral integro-differential equation with
 variable delays and give suitable conditions to obtain asymptotic stability
 of the zero solution, by means of fixed point technique.
 An asymptotic stability theorem with a necessary and sufficient condition is
 proved, which improves and generalizes previous results due to Burton
 \cite{b2}, Becker and Burton \cite{b1} and Jin and Luo \cite{j1}.
 We provide an example that illustrates our results.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

Without doubt, Lyapunov's direct method has been, for more than 100 years,
the main tool for investigating the stability properties of a wide variety
of ordinary, functional, partial differential and integro-differential
equations. Nevertheless, the application of this method to problems of
stability in differential and integro-differential equations with delays has
encountered serious obstacles if the delays are unbounded or if the equation
has unbounded terms \cite{b3}--\cite{b5}. In recent years, several
investigators have tried stability by using a new technique. Particularly,
Burton, Furumochi, Becker and others began a study in which they noticed
that some of these difficulties vanish or might be overcome by means of
fixed point theory (see \cite{a1}--\cite{r}, \cite{z}). The fixed point
theory does not only solve the problem on stability but has other
significant advantage over Lyapunov's. The conditions of the former are
often averages but those of the latter are usually pointwise (see \cite{b3}).

In this article we consider the linear neutral integro-differential equation
with variable delays
\begin{equation}
x'(t)=-\sum_{j=1}^N\int_{t-\tau _j(t)}^t a_j(t,s)x(s)
ds+\sum_{j=1}^Nc_j(t)x'(t-\tau _j(t)),  \label{1.1}
\end{equation}
with the initial condition
\[
x(t)=\psi (t)\quad \text{for }t\in [m(0),0] ,
\]
where $\psi \in C([m(0),0] ,\mathbb{R})$ and
\[
m_j(0)=\inf \{ t-\tau _j(t),\; t\geq0\} ,\quad
m(0)=\min \{ m_j(0),\; 1\leq j\leq N\} .
\]
Here $C(S_1,S_2)$ denotes the set of all continuous
functions $\varphi :S_1\to S_2$ with the supremum norm
$\| \cdot \| $. Throughout this paper we assume that
$a_j\in C(\mathbb{R}^{+}\times [m(0),\infty ),\mathbb{R})$,
$c_j\in C^{1}(\mathbb{R}^{+},\mathbb{R})$, and
$\tau _j\in C(\mathbb{R}^{+},\mathbb{R}^{+})$ with
$t-\tau _j(t)\to \infty $ as $t\to \infty $ for $j=1,2,\dots,N$.

Special cases of equation \eqref{1.1} have been investigated by many
authors. For example, Burton in \cite{b2}, Becker and Burton in \cite{b1},
Jin and Luo in \cite{j1} studied the equation
\begin{equation}
x'(t)=-\int_{t-\tau _1(t)}^t a_1(t,s)x(s)ds,  \label{1.2}
\end{equation}
and proved the following theorems, respectively,

\begin{theorem}[\cite{b2}] \label{thmA}
 Suppose that $\tau _1(t)=r$  and there exists a constant
$\alpha<1 $ such that
\begin{gather}
2\int_{t-r}^t | A(t,s)| ds\leq \alpha \quad \text{for all }t\geq 0,  \label{1.3}
\\
\int_0^t A(s,s)ds\to \infty \quad \text{as }t\to \infty ,  \label{1.4}
\end{gather}
where
\[
A(t,s)=\int_{t-s}^{r}a_1(u+s,s)du\quad
\text{with }A(t,t)=\int_0^{r}a_1(u+t,t)du.
\]
Then the zero solution of \eqref{1.2} is asymptotically stable.
\end{theorem}

\begin{theorem}[\cite{b1}] \label{thmB}
Suppose that $\tau _1$ is differentiable, $t-\tau _1(t)$
is strictly increasing, and there exist constants
$k\geq 0$, $\alpha \in (0,1)$ such that for $t\geq 0$,
\begin{gather}
-\int_0^t G(s,s)ds\leq k,  \label{1.5}
\\
\int_{t-\tau _1(t)}^t | G(t,s)
| ds+\int_0^t e^{-\int_{s}^t G(u,u)du}|
G(s,s)| \Big(\int_{s-\tau _1(s) }^{s}| G(s,u)| du\Big)ds\leq \alpha ,
\label{1.6}
\end{gather}
with
\[
G(t,s)=\int_{t}^{f(s)} a_1(u,s)du,\quad
G(t,t)=\int_{t}^{f(t)}a_1(u,t)du,
\]
where $f$ is the inverse function of $t-\tau _1(t)$.
Then for each continuous initial function
$\psi :[m_1(0),0] \to\mathbb{R}$,
 there is a unique continuous function
$x:[m_1(0),\infty )\to\mathbb{R}$ with
$x(t)=\psi (t)$  on $[m_1(0),0]$ that satisfies \eqref{1.2} on $[0,\infty )$.
Moreover, $x$ is bounded on $[m_1(0),\infty )$. Furthermore, the
zero solution of \eqref{1.2} is stable at $t=0$. If, in addition,
\begin{equation}
\int_0^t G(s,s)ds\to \infty \quad \text{as }t\to \infty ,  \label{1.7}
\end{equation}
then $x(t)\to 0$  as $t\to \infty $.
\end{theorem}

\begin{theorem}[\cite{j1}] \label{thmC}
 Let $\tau_1$ be differentiable. Suppose that there exist constants
$k\geq0 $, $\alpha \in (0,1)$ and a function
$h_1\in C(\mathbb{R}^{+},\mathbb{R})$ such that for $t\geq 0$,
\begin{gather}
-\int_0^t h_1(s)ds\leq k,  \label{1.8}
\\
\begin{aligned}
&\int_{t-\tau _1(t)}^t | h_1(s) +B_1(t,s)|ds\\
&+\int_0^t e^{-\int_{s}^t h_1(u)du}| h_1(s)| \Big(\int_{s-\tau _1(s)
}^{s}| h_1(u)+B_1(s,u)| du\Big)ds   \\
& +\int_0^t e^{-\int_{s}^t h_1(u)du}|
h_1(s-\tau _1(s))+B_1(s,s-\tau_1(s))| | 1-\tau _1'(s)|
\leq \alpha,
\end{aligned} \label{1.9}
\end{gather}
where
\[
B_1(t,s)=\int_{t}^{s}a_1(u,s)du\quad\text{with }
B_1(t,t-\tau _1(t))=\int_{t}^{t-\tau _1(t)}a_1(u,t-\tau _1(t))du.
\]
Then for each continuous initial function
$\psi :[m_1(0),0] \to\mathbb{R}$,
 there is an unique continuous function
$x:[m_1(0),\infty )\to\mathbb{R}$ with
$x(t)=\psi (t)$ on $[m_1(0),0] $ that satisfies \eqref{1.2}
on $[0,\infty )$. Moreover, $x$ is bounded on $[m_1(0),\infty )$.
 Furthermore, the zero solution of \eqref{1.2} is stable at $t=0$.
If, in addition,
\begin{equation}
\int_0^t h_1(s)ds\to \infty \quad \text{as }t\to \infty ,  \label{1.10}
\end{equation}
then $x(t)\to 0$ as $t\to\infty $.
\end{theorem}

\begin{remark} \label{rmk1} \rm
The result by Becker and Burton in Theorem \ref{thmB} requires
that $t-\tau _1(t)$ be  strictly increasing.
In Theorem \ref{thmC}, this condition is clearly removed.
Also, the conditions of stability in Theorem \ref{thmC} are less restrictive than
Theorem \ref{thmB}. Thus, Theorem \ref{thmC} improves Theorems \ref{thmA} 
and \ref{thmB}.
\end{remark}

Our objective here is to improve Theorem \ref{thmC} and extend it to investigate a
wide class of linear integro-differential equation with variable delays of
neutral type presented in \eqref{1.1}. To do this we define a suitable
continuous function $H$ (see Theorem \ref{thm1} below) and find conditions for $H$,
with no need of further assumptions on the inverse of delays
$t-\tau _j(t)$, so that for a given continuous initial function
$\psi$ a mapping $P$ for \eqref{1.1} is constructed in such a way to map a
complete metric space $S_{\psi }$ in itself and in which $P$ possesses a
fixed point. This procedure will enable us to establish and prove an
asymptotic stability theorem for the zero solution of \eqref{1.1} with a
necessary and sufficient condition and with less restrictive conditions. The
results obtained in this paper improve and generalize the main results in
\cite{b1,b2,j1}. We provide an example to illustrate our claim.

\section{Main results}

For each $\psi \in C([m(0),0],\mathbb{R})$, a solution of \eqref{1.1}
through $(0,\psi )$ is a continuous function
$x:[m(0),\sigma )\to\mathbb{R}$ for some positive constant $\sigma >0$
such that $x$ satisfies \eqref{1.1} on $[0,\sigma )$ and $x=\psi $ on
 $[m(0),0] $. We denote such a solution by
$x(t)=x(t,0,\psi)$. From the existence theory we can conclude that for each
$\psi \in C([m(0),0],\mathbb{R})$, there exists a unique solution
 $x(t)=x(t,0,\psi )$ of \eqref{1.1} defined on $[0,\infty )$. We
define $\| \psi \| =\max \{ | \psi (t)| :m(0)\leq t\leq 0\} $. Stability
definitions may be found in \cite{b3}, for example.

Our aim here is to generalize Theorem \ref{thmC} to equation \eqref{1.1} by giving
a necessary and sufficient condition for asymptotic stability of the zero
solution.

It is known that studying the stability of
an equation using a fixed point technic involves the construction of a
suitable fixed point mapping. This can be an arduous task.
So, to construct our mapping, we begin by transforming \eqref{1.1} to
a more tractable, but equivalent, equation, which we then invert to obtain
an equivalent integral equation from which we derive a fixed point mapping.
After that, we define  a suitable complete space, depending on the
initial condition, so that the mapping is a contraction. Using Banach's
contraction mapping principle, we obtain a solution for this mapping, and
hence a solution for \eqref{1.1}, which is asymptotically stable.

First, we have to transform \eqref{1.1} into an equivalent equation that
possesses the same basic structure and properties to which we apply the
variation of parameters to define a fixed point mapping.

\begin{lemma} \label{lem1}
Equation \eqref{1.1} is equivalent to
\begin{equation}
\begin{aligned}
x'(t)& =\sum_{j=1}^NB_j(t,t-\tau_j(t))(1-\tau _j'(t))x(t-\tau _j(t)) \\
&\quad +\sum_{j=1}^N\frac{d}{dt}\int_{t-\tau _j(t)}^t B_j(t,s)x(s)ds
+\sum_{j=1}^Nc_j(t)x'(t-\tau_j(t)),  \label{00}
\end{aligned}
\end{equation}
where
\[
B_j(t,s)=\int_{t}^{s}a_j(u,s)du\quad\text{and}\quad
B_j(t,t-\tau _j(t))=\int_{t}^{t-\tau
_j(t)}a_j(u,t-\tau _j(t))du.
\]
\end{lemma}

\begin{proof}
Differentiating the integral term in \eqref{00}, we obtain
\begin{align*}
& \frac{d}{dt}\int_{t-\tau _j(t)}^t B_j(t,s) x(s)ds \\
& =B_j(t,t)x(t)-B_j(t,t-\tau _j(t))(1-\tau _j'(t))x(t-\tau _j(t))
 +\int_{t-\tau _j(t)}^t \frac{\partial }{\partial t}B_j(t,s)x(s)ds.
\end{align*}
Substituting this into \eqref{00}, it follows that \eqref{00} is equivalent
to \eqref{1.1} provided $B_j$ satisfies the following conditions
\begin{equation}
B_j(t,t)=0\quad \text{and}\quad
\frac{\partial }{\partial t}B_j(t,s)=-a_j(t,s).  \label{01}
\end{equation}
This euqality implies
\begin{equation}
B_j(t,s)=-\int_0^t a_j(u,s)du+\phi (s),  \label{02}
\end{equation}
for some function $\phi $, and $B_j(t,s)$ must satisfy
\[
B_j(t,t)=-\int_0^t a_j(u,t)du+\phi (t)=0.
\]
Consequently,
\[
\phi (t)=\int_0^t a_j(u,t)du.
\]
Substituting this into \eqref{02}, we obtain
\[
B_j(t,s) =-\int_0^t a_j(u,s) du+\int_0^{s}a_j(u,s)du \\
 =\int_{t}^{s}a_j(u,s)du.
\]
This definition of $B_j$ satisfies \eqref{01}. Consequently, \eqref{1.1}
is equivalent to \eqref{00}.
\end{proof}

\begin{theorem} \label{thm1}
Suppose that $\tau _j$ is twice differentiable and $\tau _j'(t)\neq 1$
for all $t\in\mathbb{R}^{+}$, and there exist continuous functions
$h_j:[m_j(0),\infty )\to\mathbb{R}$ for $j=1,2,\dots,N$ and a constant
$\alpha \in (0,1)$ such that for $t\geq 0$
\begin{equation}
\liminf_{t\to \infty } \int_0^t H(s) ds>-\infty ,  \label{2.1}
\end{equation}
and
\begin{equation}
\begin{aligned}
& \sum_{j=1}^N| \frac{c_j(t)}{1-\tau_j'(t)}|
+\sum_{j=1}^N\int_{t-\tau _j(t)}^t |
h_j(s)+B_j(t,s)| ds   \\
& +\sum_{j=1}^N\int_0^t e^{-\int_{s}^t H(u)
du}| [h_j(s-\tau _j(s))+B(
s,s-\tau _j(s))] (1-\tau _j'(s))-r_j( s)| ds   \\
& +\sum_{j=1}^N\int_0^t e^{-\int_{s}^t H(u)
du}| H(s)| (\int_{s-\tau _j(s)}^{s}| h_j(u)+B_j(s,u)| du)ds
\leq \alpha,
\end{aligned} \label{2.2}
\end{equation}
where
\[
H(t)=\sum_{j=1}^Nh_j(t),\quad
r_j(t)=\frac{[c_j(t)H(t) +c_j'(t)] (1-\tau _j'(
t))+c_j(t)\tau _j''(t)}{(1-\tau _j'(t))^2},
\]
and
\[
B_j(t,s)=\int_{t}^{s}a_j(u,s)du\quad\text{with}\quad
B_j(t,t-\tau _j(t))=\int_{t}^{t-\tau_j(t)}a_j(u,t-\tau _j(t))du.
\]
Then the zero solution of \eqref{1.1} is asymptotically stable
if and only if
\begin{equation}
\int_0^t H(s)ds\to \infty \quad \text{as }t\to
\infty .  \label{2.3}
\end{equation}
\end{theorem}

\begin{proof}
First, suppose that \eqref{2.3} holds. We set
\begin{equation}
K=\underset{t\geq 0}{\sup }\{ e^{-\int_0^t H(s)ds}\} .  \label{2.4}
\end{equation}
Let $\psi \in C([m(0),0] ,\mathbb{R})$ be fixed and define
\[
S_{\psi }:=\{ \varphi \in C([m(0),\infty),\mathbb{R}):
\varphi (t)=\psi (t)\ \text{for}\ t\in
[m(0),0] \text{ and }\varphi (t)
\to 0\ \text{as}\ t\to \infty \} .
\]
Endowed with the supremum norm $\| \cdot \| $; that is,
for $\phi \in S_{\psi }$,
\[
\| \phi \| :=\sup \{ | \phi (t)| :t\in [m(0),\infty )\} .
\]
In other words, we carry out our investigations in the complete metric
space $(S_{\psi },\rho )$ where $\rho $\ is supremum metric
\[
\rho (x,y):=\underset{t\geq m(0)}{\sup }| x(
t)-y(t)| =\| x-y\| ,\quad \text{for }x,y\in S_{\psi }.
\]
Rewrite \eqref{1.1} in the following equivalent form
\begin{equation}
\begin{aligned}
x'(t)& =\sum_{j=1}^NB_j(t,t-\tau
_j(t))(1-\tau _j'(t))x(t-\tau _j(t)) \\
&\quad +\sum_{j=1}^N\frac{d}{dt}\int_{t-\tau _j(t)}^t B_j(t,s)x(s)ds
+\sum_{j=1}^Nc_j(t)x'(t-\tau_j(t))\label{2.5}
\end{aligned}
\end{equation}
Multiplying both sides of \eqref{2.5} by $\exp\big(\int_0^t H(u)du\big)$ and
integrating with respect to $s$ from $0$ to $t$, we obtain
\begin{align*}
x(t)& =\psi (0)e^{-\int_0^t H(u)
du}+\int_0^t e^{-\int_{s}^t H(u)
du}\sum_{j=1}^Nh_j(s)x(s)ds \\
&\quad +\int_0^t e^{-\int_{s}^t H(u)du}\sum_{j=1}^N\frac{d}{ds}
\int_{s-\tau _j(s)}^{s}B_j(s,u)x(
u)du \\
&\quad +\int_0^t e^{-\int_{s}^t H(u)du}\sum_{j=1}^NB_j(
s,s-\tau _j(s))(1-\tau _j'(
s))x(s-\tau _j(s))ds \\
&\quad +\int_0^t e^{-\int_{s}^t H(u)du}\sum_{j=1}^Nc_j(
s)x'(s-\tau _j(s))ds.
\end{align*}
Thus,
\begin{align*}
x(t)& =\psi (0)e^{-\int_0^t H(u)
du}+\sum_{j=1}^N\int_0^t e^{-\int_{s}^t H(u)
du}h_j(s)x(s)ds \\
&\quad +\sum_{j=1}^N\int_0^t e^{-\int_{s}^t H(u)du}\frac{d}{ds}
\int_{s-\tau _j(s)}^{s}B_j(s,u)x(
u)du \\
&\quad +\sum_{j=1}^N\int_0^t e^{-\int_{s}^t H(u)du}B_j(
s,s-\tau _j(s))(1-\tau _j'(
s))x(s-\tau _j(s))ds \\
&\quad +\sum_{j=1}^N\int_0^t e^{-\int_{s}^t H(u)du}c_j(
s)x'(s-\tau _j(s))ds.
\end{align*}
Performing an integration by parts, we obtain
\begin{align*}
x(t)& =\psi (0)e^{-\int_0^t H(u)
du}+\sum_{j=1}^N\int_0^t e^{-\int_{s}^t H(u)
du}d\Big(\int_{s-\tau _j(s)}^{s}[h_j(u)
+B_j(s,u)] x(u)du\Big)
 \\
&\quad +\sum_{j=1}^N\int_0^t e^{-\int_{s}^t H(u)du}[
h_j(s-\tau _j(s))+B_j(s,s-\tau_j(s))]\\
&\quad\times  (1-\tau _j'(s))x(s-\tau _j(s))ds
 +\sum_{j=1}^N\int_0^t \frac{c_j(s)}{1-\tau
_j'(s)}e^{-\int_{s}^t H(u)du}dx(s-\tau _j(s))
\\
& =\Big(\psi (0)-\sum_{j=1}^N\frac{c_j(0)}{
1-\tau _j'(0)}\psi (-\tau _j(0)
)-\sum_{j=1}^N\int_{-\tau _j(0)}^{0}[
h_j(s)+B_j(0,s)] \psi (s)ds\Big)\\
&\quad\times e^{-\int_0^t H(u)du}
\\
& +\sum_{j=1}^N\frac{c_j(t)}{1-\tau _j'(t)}x(t-\tau _j(t))
+\sum_{j=1}^N\int_{t-\tau _j(t)}^t [h_j(s)+B_j(t,s)] x(s)ds
 \\
&\quad +\sum_{j=1}^N\int_0^t e^{-\int_{s}^t H(u)du}\{
[h_j(s-\tau _j(s))+B_j(s,s-\tau _j(s))]\\
&\quad\times  (1-\tau _j'(s))-r_j(s)\} x(s-\tau _j(s))ds   \\
& -\sum_{j=1}^N\int_0^t e^{-\int_{s}^t H(u)du}H(s)
\Big(\int_{s-\tau _j(s)}^{s}[
h_j(u)+B_j(s,u)] x(u)du\Big)ds.
\end{align*} %\label{2.6}
Now use this equality to define the operator $P:S_{\psi }\to S_{\psi }$
by $(P\varphi )(t)=\psi (t)$ if
$t\in [m(0),0] $ and for $t\geq 0$ we let
\begin{equation}
\begin{aligned}
(P\varphi )(t)
& =\Big(\psi (0)
-\sum_{j=1}^N\frac{c_j(0)}{1-\tau _j'(0)}\psi (-\tau _j(0))\\
&\quad -\sum_{j=1}^N\int_{-\tau _j(0)}^{0}
[h_j(s)+B_j(0,s)] \psi (s)ds\Big)e^{-\int_0^t H(u)du}
 \\
&\quad +\sum_{j=1}^N\frac{c_j(t)}{1-\tau _j'(
t)}\varphi (t-\tau _j(t))
+\sum_{j=1}^N\int_{t-\tau _j(t)}^t [
h_j(s)+B_j(t,s)] \varphi (s) ds   \\
&\quad +\sum_{j=1}^N\int_0^t e^{-\int_{s}^t H(u)du}\{
[h_j(s-\tau _j(s))+B_j(s,s-\tau_j(s))] \\
&\quad \times (1-\tau _j'(s))-r_j(s)\} \varphi (s-\tau _j(s))ds   \\
&\quad -\sum_{j=1}^N\int_0^t e^{-\int_{s}^t H(u)du}H(s)
\Big(\int_{s-\tau _j(s)}^{s}[
h_j(u)+B_j(s,u)] \varphi (u) du\Big)ds.
\end{aligned}  \label{2.7}
\end{equation}
It is clear that $(P\varphi )\in C([m(0),\infty ),\mathbb{R})$.
We will show that $(P\varphi )(t) \to 0$ as $t\to \infty $.
To this end, denote the five terms on the right hand side of \eqref{2.7}
by $I_1,I_2,\dots I_5$,
respectively. It is obvious that the first term $I_1$ tends to zero as $
t\to \infty $, by condition \eqref{2.3}. Also, due to the facts that
$\varphi (t)\to 0$ and $t-\tau _j(t)
\to \infty $ for $j=1,2,\dots,N$ as $t\to \infty $, the second
term $I_2$ in \eqref{2.7} tends to zero as $t\to \infty $. What is
left to show is that each of the remaining terms in \eqref{2.7} go to zero
at infinity.

Let $\varphi \in S_{\psi }$ be fixed. For a given $\varepsilon >0$, we
choose $T_0>0$ large enough such that $t-\tau _j(t)\geq
T_0$, $j=1,2,\dots,N$, implies $| \varphi (s)
| <\varepsilon $ if $s\geq t-\tau _j(t)$.
Therefore, the third term $I_3$ in \eqref{2.7} satisfies
\begin{align*}
| I_3| & =| \sum_{j=1}^N\int_{t-\tau
_j(t)}^t [h_j(s)+B_j(t,s)
] \varphi (s)ds|  \\
& \leq \sum_{j=1}^N\int_{t-\tau _j(t)}^t |
h_j(s)+B_j(t,s)| | \varphi
(s)| ds \\
& \leq \varepsilon \sum_{j=1}^N\int_{t-\tau _j(t)
}^t | h_j(s)+B_j(t,s)|
ds\leq \alpha \varepsilon <\varepsilon .
\end{align*}
Thus, $I_3\to 0$ as $t\to \infty $. Now consider $I_4$.\
For the given $\varepsilon >0$, there exists a $T_1>0$ such that $s\geq
T_1$ implies $| \varphi (s-\tau _j(s))
| <\varepsilon $ for $j=1,2,\dots,N$. Thus, for $t\geq T_1$, the
term $I_4$ in \eqref{2.7} satisfies
\begin{align*}
| I_4| &
=\Big|
\sum_{j=1}^N\int_0^t e^{-\int_{s}^t H(u)du}\big\{ [
h_j(s-\tau _j(s))+B_j(s,s-\tau
_j(s))] (1-\tau _j'(s))-r_j(s)\big\} \\
&\quad \times \varphi (s-\tau_j(s))ds\Big|
\\
& \leq \sum_{j=1}^N\int_0^{T_1}e^{-\int_{s}^t H(u)
du}\big| [h_j(s-\tau _j(s))+B_j(s,s-\tau _j(s))] (1-\tau
_j'(s))-r_j(s)\big|
\\
&\quad | \varphi (s-\tau _j(s))| ds
\\
&\quad +\sum_{j=1}^N\int_{T_1}^t e^{-\int_{s}^t H(u)du}| [h_j(s-\tau _j(s))
+B_j(s,s-\tau _j(s))] (1-\tau_j'(s))-r_j(s)\big| \\
&\quad | \varphi (s-\tau _j(s))| ds
\\
& \leq \underset{\sigma \geq m(0)}{\sup }| \varphi (\sigma )|
\sum_{j=1}^N\int_0^{T_1}e^{-\int_{s}^t H(u)du}
\big|[h_j(s-\tau _j(s))+B_j(s,s-\tau _j(s))]\\
&\quad\times  (1-\tau _j'(s))-r_j(s)\big| ds
\\
& +\varepsilon \sum_{j=1}^N\int_{T_1}^t e^{-\int_{s}^t H(
u)du}| [h_j(s-\tau _j(s))
+B_j(s,s-\tau _j(s))] (1-\tau_j'(s))-r_j(s)| ds.
\end{align*}
By \eqref{2.3}, we can find $T_2>T_1$ such that $t\geq T_2$ implies
\begin{align*}
& \sup_{\sigma \geq m(0)} | \varphi (\sigma )|
\sum_{j=1}^N\int_0^{T_1}e^{-\int_{s}^t H(u)du}|
[h_j(s-\tau _j(s))+B_j(s,s-\tau _j(s))] \\
&\times (1-\tau _j'(s))-r_j(s)| ds \\
& =\sup_{\sigma \geq m(0)} | \varphi (\sigma )| e^{-\int_{T_2}^t H(u)du}
 \sum_{j=1}^N\int_0^{T_1}e^{-\int_{s}^{T_2}H(u)
du}| [h_j(s-\tau _j(s))
+B_j(s,s-\tau _j(s))] \\
&\quad\times (1-\tau_j'(s))-r_j(s)| ds
 <\varepsilon .
\end{align*}
Now, apply \eqref{2.2} to have $| I_4| <\varepsilon
+\alpha \varepsilon <2\varepsilon $. Thus, $I_4\to 0$ as $
t\to \infty $. Similarly, by using \eqref{2.2}, then, if $t\geq T_2
$ then term $I_5$ in \eqref{2.7} satisfies
\begin{align*}
| I_5|
 & =\big|\sum_{j=1}^N\int_0^t e^{-\int_{s}^t H(u)du}H(
s)\Big(\int_{s-\tau _j(s)}^{s}[h_j(u)+B_j(s,u)] \varphi (u)du\Big)
ds\big|  \\
& \leq \sup_{\sigma \geq m(0)} | \varphi (\sigma )| e^{-\int_{T_2}^t H(u)du}
\sum_{j=1}^N\int_0^{T_1}e^{-\int_{s}^{T_2}H(u) du}  | H(s)| \\
&\quad\times \Big(\int_{s-\tau _j(
s)}^{s}| h_j(u)+B_j(s,u) | du\Big)ds \\
&\quad +\varepsilon \sum_{j=1}^N\int_{T_1}^t e^{-\int_{s}^t H(
u)du}| H(s)| \Big(\int_{s-\tau
_j(s)}^{s}| h_j(u)+B_j(
s,u)| du\Big)ds \\
& <\varepsilon +\alpha \varepsilon <2\varepsilon .
\end{align*}
Thus, $I_5\to 0$ as $t\to \infty $. In conclusion
$(P\varphi )(t)\to 0$ as $t\to \infty $,
as required. Hence $P$ maps $S_{\psi }$ into $S_{\psi }$. Also, by condition
\eqref{2.2}, $P$ is a contraction mapping with contraction constant
$\alpha $. Indeed, for $\phi ,\eta \in S_{\psi }$ and $t>0$
\begin{align*}
& | (P\varphi )(t)-(P\eta )(t)|  \\
& \leq \sum_{j=1}^N| \frac{c_j(t)}{1-\tau
_j'(t)}| | \varphi (t-\tau
_j(t))-\eta (t-\tau _j(t))| \\
&\quad +\sum_{j=1}^N\int_{t-\tau _j(t)}^t | h_j(s)+B_j(t,s)|
| \varphi (s)-\eta (s)| ds
\\
& +\sum_{j=1}^N\int_0^t e^{-\int_{s}^t H(u)du}|
[h_j(s-\tau _j(s))+B_j(s,s-\tau _j(s))] \\
&\quad\times  (1-\tau _j'(s))-r_j(s)| | \varphi (
s-\tau _j(s))-\eta (s-\tau _j(s) )| ds
\\
&\quad +\sum_{j=1}^N\int_0^t e^{-\int_{s}^t H(u)
du}H(s)\Big(\int_{s-\tau _j(s)}^{s}|
h_j(u)+B_j(s,u)| | \varphi (u)-\eta (u)| du\Big)ds
\\
&\leq \bigg(\sum_{j=1}^N| \frac{c_j(t)}{1-\tau_j'(t)}|
+\sum_{j=1}^N\int_{t-\tau _j(t)}^t |
h_j(s)+B_j(t,s)| ds  \\
&\quad +\sum_{j=1}^N\int_0^t e^{-\int_{s}^t H(u)du}|
[h_j(s-\tau _j(s))+B_j(s,s-\tau_j(s))] (1-\tau _j'(
s))-r_j(s)| ds \\
&\quad  +\sum_{j=1}^N\int_0^t e^{-\int_{s}^t H(u)
du}H(s)\Big(\int_{s-\tau _j(s)}^{s}|
h_j(u)+B_j(s,u)| du\Big)ds\bigg)
\| \varphi -\eta \| .
\end{align*}

By  condition \eqref{2.2}, $P$ is a contraction mapping with constant $
\alpha $. By the contraction mapping principle (Smart \cite[p. 2]{s}), $P$
has a unique fixed point $x$ in $S_{\psi }$ which is a solution of \eqref{1.1}
 with $x(t)=\psi (t)$ on $[m(0),0] $ and $x(t)=x(t,0,\psi )\to 0$
as $t\to \infty $.

To obtain the asymptotic stability, we need to show that the zero solution
of \eqref{1.1} is stable. Let $\varepsilon >0$ be given and choose
 $\delta>0 $ $(\delta <\varepsilon )$ satisfying $2\delta K+\alpha
\varepsilon <\varepsilon $. If $x(t)=x(t,0,\psi )$
is a solution of \eqref{1.1} with $\| \psi \| <\delta $,
then $x(t)=(Px)(t)$ defined in (\eqref
{2.7}. We claim that $| x(t)| <\varepsilon $
for all $t\geq t_0$. Notice that $| x(s)|
<\varepsilon $ on $[m(0),0] $. If there exists $
t^{\ast }>0$ such that $| x(t^{\ast })|
=\varepsilon $ and $| x(s)| <\varepsilon $
for $m(0)\leq s<t^{\ast }$, then it follows from \eqref{2.7}
that
\begin{align*}
&| x(t^{\ast })| \\
& \leq \| \psi \|
\Big(1+\sum_{j=1}^N| \frac{c_j(
0)}{1-\tau _j'(0)}|
+\sum_{j=1}^N\int_{-\tau _j(0)}^{0}|
h_j(s)+B_j(0,s)| ds\Big)
e^{-\int_0^{t^{\ast }}H(u)du}
\\
&\quad +\varepsilon \sum_{j=1}^N| \frac{c_j(t^{\ast
})}{1-\tau _j'(t^{\ast })}|
+\varepsilon \sum_{j=1}^N\int_{t^{\ast }-\tau _j(t^{\ast
})}^{t^{\ast }}| h_j(s)+B_j(t^{\ast},s)| ds
\\
&\quad +\varepsilon \int_{t_0}^{t^{\ast }}e^{-\int_{s}^{t^{\ast }}H(
u)du}| [h_j(s-\tau _j(s))
+B_j(s,s-\tau _j(s))] (1-\tau _j'(s))-r_j(s)| ds
\\
&\quad +\varepsilon \sum_{j=1}^N\int_{t_0}^{t^{\ast
}}e^{-\int_{s}^{t^{\ast }}H(u)du}| H(s)
| (\int_{s-\tau _j(s)}^{s}|h_j(u)+B_j(s,u)| du)ds
\\
& \leq 2\delta K+\alpha \varepsilon <\varepsilon ,
\end{align*}
which contradicts the definition of $t^{\ast }$. Thus,
$| x(t)| <\varepsilon $ for all $t\geq 0$, and the zero solution
of $(1.1)$ is stable. This shows that the zero solution of
\eqref{1.1} is asymptotically stable if \eqref{2.3} holds.

Conversely, suppose \eqref{2.3} fails. Then, by \eqref{2.1} there exists a
sequence $\{ t_{n}\} $, $t_{n}\to \infty $ as $
n\to \infty $ such that $\underset{n\to \infty }{\lim }
\int_0^{t_{n}}H(u)du=l$ for some $l\in\mathbb{R}$.
 We may also choose a positive constant $J$ satisfying
\[
-J\leq \int_0^{t_{n}}H(u)du\leq J,
\]
for all $n\geq 1$. To simplify our expressions, we define
\begin{align*}
\omega (s)
& =\sum_{j=1}^N| [h_j(s-\tau_j(s))+B_j(s,s-\tau _j(s)
)] (1-\tau _j'(s))-r_j(s)| \\
&\quad +\sum_{j=1}^N| H(s)| (\int_{s-\tau
_j(s)}^{s}| h_j(u)+B(s,u)| du),
\end{align*}
for all $s\geq 0$. By \eqref{2.2}, we have
\[
\int_0^{t_{n}}e^{-\int_{s}^{t_{n}}H(u)du}\omega (
s)ds\leq \alpha .
\]
This yields
\[
\int_0^{t_{n}}e^{\int_0^{s}H(u)du}\omega (s)
ds\leq \alpha e^{\int_0^{t_{n}}H(u)du}\leq J.
\]
The sequence $\{ \int_0^{t_{n}}e^{\int_0^{s}H(u)
du}\omega (s)ds\} $ is bounded, so there exists a
convergent subsequence. For brevity of notation, we may assume that
\[
\lim_{n\to \infty } \int_0^{t_{n}}e^{\int_0^{s}H(u)du}\omega (s)ds=\gamma ,
\]
for some $\gamma \in\mathbb{R}^{+}$ and choose a positive integer $m$
so large that
\[
\int_{t_{m}}^{t_{n}}e^{\int_0^{s}H(u)du}\omega (
s)ds<\delta _0/4K,
\]
for all $n\geq m$, where $\delta _0>0$ satisfies $2\delta_0Ke^{J}+\alpha \leq 1$.

 By \eqref{2.1}, $K$ in \eqref{2.4} is well defined. We now
consider the solution $x(t)=x(t,t_{m},\psi )$ of
\eqref{1.1} with $\psi (t_{m})=\delta _0$ and $| \psi (s)| \leq \delta _0$
for $s\leq t_{m}$. We may
choose $\psi $ so that $| x(t)| \leq 1$ for
$ t\geq t_{m}$ and
\begin{align*}
&\psi (t_{m})-\sum_{j=1}^N[\frac{c_j(
t_{m})}{1-\tau _j'(t_{m})}\psi (
t_{m}-\tau _j(t_{m}))\\
&+\int_{t_{m}-\tau _j(t_{m})}^{t_{m}}[h_j(s)+B_j(
t_{m},s)] \psi (s)ds] \\
&\geq \frac{1}{2}\delta _0.
\end{align*}
It follows from \eqref{2.7} with $x(t)=(Px)(t)$ that for
 $n\geq m$
\begin{equation}
\begin{aligned}
& \Big| x(t_{n})-\sum_{j=1}^N \Big[\frac{c_j(
t_{n})}{1-\tau _j'(t_{n})}x(t_{n}-\tau
_j(t_{n}))+\int_{t_{n}-\tau _j(t_{n})
}^{t_{n}}[h_j(s)+B_j(t_{n},s)]
x(s)ds\Big] \Big|   \\
& \geq \frac{1}{2}\delta _0e^{-\int_{t_{m}}^{t_{n}}H(u)
du}-\int_{t_{m}}^{t_{n}}e^{-\int_{s}^{t_{n}}H(u)du}\omega
(s)ds   \\
& =\frac{1}{2}\delta _0e^{-\int_{t_{m}}^{t_{n}}H(u)
du}-e^{-\int_0^{t_{n}}H(u)
du}\int_{t_{m}}^{t_{n}}e^{\int_0^{s}H(u)du}\omega (
s)ds   \\
& =e^{-\int_{t_{m}}^{t_{n}}H(u)du}\Big(\frac{1}{2}\delta
_0-e^{-\int_0^{t_{m}}H(u)
du}\int_{t_{m}}^{t_{n}}e^{\int_0^{s}H(u)du}\omega (
s)ds\Big) \\
& \geq e^{-\int_{t_{m}}^{t_{n}}H(u)du}\Big(\frac{1}{2}\delta
_0-K\int_{t_{m}}^{t_{n}}e^{\int_0^{s}H(u)du}\omega (
s)ds\Big) \\
& \geq \frac{1}{4}\delta _0e^{-\int_{t_{m}}^{t_{n}}H(u)
du}\geq \frac{1}{4}\delta _0e^{-2J}>0.
\end{aligned} \label{2.8}
\end{equation}
 On the other hand, if the zero solution of \eqref{1.1} is
asymptotically stable, then $x(t)=x(t,t_{m},\psi )
\to 0$ as $t\to \infty $. Since $t_{n}-\tau _j(
t_{n})\to \infty $ as $n\to \infty $ and \eqref{2.2}
holds, we have
\[
x(t_{n})-\sum_{j=1}^N\Big[\frac{c_j(t_{n})}{
1-\tau _j'(t_{n})}x(t_{n}-\tau _j(
t_{n}))+\int_{t_{n}-\tau _j(t_{n})}^{t_{n}}
[h_j(s)+B_j(t_{n},s)] x(
s)ds\Big] \to 0
\]
 as $n\to \infty$,
which contradicts \eqref{2.8}. Hence condition \eqref{2.3} is necessary for
the asymptotic stability of the zero solution of \eqref{1.1}. The proof is
complete.
\end{proof}


\begin{remark} \label{rmk2} \rm
It follows from the first part of the proof of Theorem \ref{thm1} that the zero
solution of \eqref{1.1} is stable under \eqref{2.1} and \eqref{2.2}.
Moreover, Theorem \ref{thm1} still holds if \eqref{2.2} is satisfied for
 $t\geq t_{\sigma }$ for some $t_{\sigma }\in \mathbb{R}^{+}$.
\end{remark}

For the special case $N=1$ and $c_1=0$, we have the following result.

\begin{corollary} \label{coro1}
Suppose that $\tau _1$ is differentiable and there exist continuous
function $h_1:[m_1(0),\infty )\to \mathbb{R}$ and a constant $\alpha \in (0,1)$
 such that for $t\geq 0$
\begin{equation}
\lim_{t\to \infty } \inf \int_0^t h_1(s) ds>-\infty ,  \label{2.9}
\end{equation}
and
\begin{equation}
\begin{aligned}
& \int_{t-\tau _1(t)}^t | h_1(s)
+B_1(t,s)|ds\\
&+\int_0^t e^{-\int_{s}^t h_1(u)du}|
h_1(s-\tau _1(s))+B_1(s,s-\tau
_1(s))| | 1-\tau _1'(s)| ds   \\
& +\int_0^t e^{-\int_{s}^t h_1(u)du}|
h_1(s)| (\int_{s-\tau _1(s)}^{s}| h_1(u)+B_1(s,u)|du)ds
\leq \alpha.
\end{aligned}\label{2.10}
\end{equation}
Then the zero solution of \eqref{1.2} is asymptotically stable if and only if
\begin{equation}
\int_0^t h_1(s)ds\to \infty \quad \text{as }t\to \infty .  \label{2.11}
\end{equation}
\end{corollary}

Obviously, Corollary \ref{coro1} extends Theorem \ref{thmC}. Thus, Theorem \ref{thm1}
generalizes Theorem \ref{thmC}.

\section{An Example}

In this section, we give an example to illustrate the applications of
Theorem \ref{thm1}.

\begin{example} \label{example1}
Consider the  linear neutral integro-differential equation with
variable delays
\begin{equation}
x'(t)=-\sum_{j=1}^2\int_{t-\tau _j(t)
}^t a_j(t,s)x(s)ds+\sum_{j=1}^2 c_j(t)x'(t-\tau _j(t)),  \label{3.1}
\end{equation}
where $\tau _1(t)=0.489t$, $\tau _2(t)=0.478t$,
$a_1(t,s)=0.48/(s^2+1)$, $a_2(t,s)=0.52/(s^2+1)$, $c_1(t)=0.015$,
$c_2(t)=0.017$. Then the zero solution of \eqref{3.1} is
asymptotically stable.
\end{example}

\begin{proof}
We have
\[
B_1(t,s)=\int_{t}^{s}\frac{0.48}{s^2+1}du=\frac{0.48(
s-t)}{s^2+1},\quad
B_2(t,s)=\int_{t}^{s}\frac{
0.52}{s^2+1}du=\frac{0.52(s-t)}{s^2+1}.
\]
Choosing $h_1(t)=0.52t/(t^2+1)$ and $
h_2(t)=0.48t/(t^2+1)$ in Theorem \ref{thm1}, we have $
H(t)=t/(t^2+1)$ and
\[
\sum_{j=1}^2| \frac{c_j(t)}{1-\tau _j'(t)}| =| \frac{0.015}{1-0.489}|
+| \frac{0.017}{1-0.478}| <0.062,
\]
\begin{align*}
& \sum_{j=1}^2\int_{t-\tau _j(t)}^t | h_j(
s)+B_j(t,s)|\,ds \\
& =\int_{0.511t}^t | \frac{s-0.48t}{s^2+1}|
ds+\int_{0.522t}^t | \frac{s-0.52t}{s^2+1}| ds \\
& =\int_{0.511t}^t \frac{s-0.48t}{s^2+1}ds+\int_{0.522t}^t \frac{s-0.52t
}{s^2+1}ds \\
& =t[0.48\arctan 0.511t+0.52\arctan 0.522t-\arctan t] +\ln
(t^2+1)\\
&\quad -\frac{1}{2}[\ln (0.511^2t^2+1)+\ln (
0.522^2t^2+1)] \\
& =\omega (t).
\end{align*}
Since the function $\omega $ is increasing in $[0,\infty )$ and
\[
\lim_{t\to \infty }\omega (t)
=1-0.48/0.511-0.52/0.522-\ln (0.511\times 0.522)\simeq 0.386,
\]
then
\[
\sum_{j=1}^2\int_{t-\tau _j(t)}^t | h_j(s)+B_j(t,s)|\,ds <0.386,
\]
\[
\sum_{j=1}^2\int_0^t e^{-\int_{s}^t H(u)du}|
H(s)| \Big(\int_{s-\tau _j(s)
}^{s}| h_j(u)+B_j(s,u)| du\Big)ds<0.386,
\]
and
\begin{align*}
& \sum_{j=1}^2\int_0^t e^{-\int_{s}^t H(u)du}|
[h_j(s-\tau _j(s))+B(s,s-\tau
_j(s))] (1-\tau _j'(
s))-r_j(s)| ds \\
& =\int_0^t e^{-\int_{s}^t \frac{u}{u^2+1}du}| 0.511(
\frac{0.52\times 0.511s}{0.511^2s^2+1}+\frac{0.48(0.511s-s)
}{0.511^2s^2+1})-\frac{0.015s}{0.511(s^2+1)}
| ds \\
& +\int_0^t e^{-\int_{s}^t \frac{u}{u^2+1}du}| 0.522(
\frac{0.48\times 0.522s}{0.522^2s^2+1}+\frac{0.52(0.522s-s)
}{0.522^2s^2+1})-\frac{0.017s}{0.522(s^2+1)}
| ds \\
& \leq (1-\frac{0.48}{0.511})\int_0^t e^{-\int_{s}^t \frac{u
}{u^2+1}du}\frac{s}{s^2+1/0.511^2}ds+\frac{0.015}{0.511}
\int_0^t e^{-\int_{s}^t \frac{u}{u^2+1}du}\frac{s}{s^2+1}ds \\
& +(1-\frac{0.52}{0.522})\int_0^t e^{-\int_{s}^t \frac{u}{
u^2+1}du}\frac{s}{s^2+1/0.522^2}ds+\frac{0.017}{0.522}
\int_0^t e^{-\int_{s}^t \frac{u}{u^2+1}du}\frac{s}{s^2+1}ds \\
& <1-\frac{0.48}{0.511}+\frac{0.015}{0.511}+1-\frac{0.52}{0.522}+\frac{0.017
}{0.522}<0.127.
\end{align*}
It is easy to see that all the conditions of Theorem \ref{thm1} hold for
$\alpha =0.062+0.386+0.386+0.127=0.961<1$.
Thus, Theorem \ref{thm1} implies that
the zero solution of \eqref{3.1} is asymptotically stable.
\end{proof}

\subsection*{Acknowledgements}
The authors would like to express their sincere thanks to the anonymous referee for
his/her careful reading of our manuscript and the helpful report.

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\end{document}