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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 118, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/118\hfil 
Stability of  neutral delay differential equations]
{Stability of nonlinear neutral delay differential equations with variable delays}

\author[G. Chen,D. Li, O. van Gaans, S. Verduyn Lunel \hfil EJDE-2017/118\hfilneg]
{Guiling Chen, Dingshi Li, Onno van Gaans, Sjoerd Verduyn Lunel}


\address{Guiling Chen \newline
Department of Mathematics,
Southwest Jiaotong University,
Chengdu 610031, China}
\email{guiling@home.swjtu.edu.cn}

\address{Dingshi Li (corresponding author)\newline
Department of Mathematics,
Southwest Jiaotong University,
Chengdu 610031,  China}
\email{lidingshi2006@163.com}

\address{Onno van Gaans \newline
Mathematical Institute,  Leiden University,
 P.O. Box 9512, 2300 RA,
 Leiden, The Netherlands}
\email{vangaans@math.leidenuniv.nl}

\address{Sjoerd Verduyn Lunel \newline
Mathematical Institute, Utrecht University,
P.O. Box 80010, 3508 TA,
Utrecht, The Netherlands}
\email{S.M.VerduynLunel@uu.nl}

\thanks{Submitted December 16, 2016. Published May 3, 2017.}
\subjclass[2010]{34K20, 34K25, 34K40}
\keywords{Asymptotic stability; fixed point theory; variable delay;
\hfill\break\indent nonlinear neutral differential equations}

\begin{abstract}
 We present new criteria for asymptotic stability of two classes of 
 nonlinear neutral  delay differential equations. 
 By using two auxiliary functions on a contraction condition, we
 extend the results in \cite{AR}. Also we give two examples that
 illustrate our results.
\end{abstract}

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

\section{Introduction}

In recent years there has been an increasing interest in stability results
for neutral delay differential equations involving terms of the form
$c(t)x(t-r(t))x'(t-r(t))$ (see \cite{AA2, YN}). In this article,
we consider the following two classes of nonlinear neutral delay
differential equations
\begin{gather}\label{1}
x'(t)-c(t)x(t-r(t))x'(t-r(t))=-a(t)x(t)+b(t)g(x(t-r(t))), \\
\label{2}
x'(t)-c(t)x(t-r(t))x'(t-r(t))=-a(t)x(t)+\int_{t-r(t)}^{t}K(t,s)g(x(s))\,ds,
\end{gather}
where $ a, b: [0, \infty)\to \mathbb{R} $ are continuous functions,
 $ c: [0, \infty)\to \mathbb{R} $ is continuously differentiable
function and $ r: [0, \infty)\to (0, \infty)$ is a continuous function, 
$ K(t, s): [0, \infty)\times [r_0, \infty)\to \mathbb{R} $ is a continuous 
function, $ r_0=\inf\{t-r(t): t\geq 0\} $, $ g(x)=|x|^{\gamma} $, $ \gamma\geq 1 $
is a constant, then $ g $ satisfies a locally Lipschitz condition; 
that is, there exists $ L>0 $ and $ l>0 $ such that $ g $ satisfies
\begin{equation}\label{two constants}
|g(x)-g(y)|\leq L|x-y| \quad \text{for } x, y\in [-l, l].
\end{equation}
Ahcene and Rabah \cite{AR} studied the special case of
\eqref{1} and \eqref{2} when $ g(x)=x^{2} $. The results in \cite{AR} mainly 
dependent on the constraint $|\frac{c(t)}{1-r'(t)}|<1$. However, 
there are interesting examples where the constraint is not satisfied.
 It is our aim in this paper to remove this constraint condition and study 
the stability properties of \eqref{1} and \eqref{2}.

Recent work of Burton and many others 
\cite{AA,AA1,AA2,LTA,TA1,TA2,chen,chen1,DL,AR,M,YN,DLZ} has shown the power 
of the fixed point method in studying stability properties of functional 
differential equations. The idea of using fixed point method to study 
properties of solutions seems to have emerged independently several times
 by different schools of authors. In addition to Burton's work, 
we would like to mention Corduneann \cite{CC} and Azbelev and his 
co-workers \cite{AMR}. In this paper, we will use the fixed point method 
to study stability properties of \eqref{1} and \eqref{2}. In particular, 
we introduce two auxiliary continuous functions $ v(t) $ and $ p(t) $ to 
define an appropriate mapping, and present new criteria for asymptotic stability 
of equations \eqref{1} and \eqref{2} which can be applied in the case 
$ |\frac{c(t)}{1-r'(t)} |\geq 1$ as well.

An initial condition for the differential equation \eqref{1} is defined by
\begin{equation}\label{initial}
x(t)=\phi(t) \quad \text{for }  t\in [r_0, 0],
\end{equation}
where $ \phi\in C([r_0, 0], \mathbb{R}) $. Here $ C([r_0, 0], \mathbb{R}) $ 
denotes the set of all continuous functions 
$ \varphi: [r_0, 0]\to \mathbb{R} $ with the supremum norm $ \|\cdot\| $. 
For $ \phi\in C([r_0, 0], \mathbb{R}) $, we call a continuous function 
$  x(t, \phi)$ to be a solution of \eqref{1} with initial condition 
\eqref{initial} if $ x:[r_0, a)\to \mathbb{R} $ for some positive constant 
$ a>0 $ satisfies
\begin{equation}\label{1'}
\begin{aligned}
&\frac{d}{dt}\Big(x(t)-\frac{c(t)}{2(1-r'(t))}x(t-r(t))^{2}\Big)\\ 
&=-a(t)x(t)+b(t)g(x(t-r(t)))-\frac{d}{dt}\Big(\frac{c(t)}{2(1-r'(t))}\Big)
x^{2}(t-r(t))
\end{aligned}
\end{equation}
on $ [0, a) $ and $ x=\phi $ on $ [r_0, 0] $. We denote such a solution by
 $ x(t):=x(t, \phi) $. Note that equation \eqref{1'} is in the standard form
 $ \frac{d}{dt}(D(t, x_{t}))=f(t, x_{t}) $ as studied in \cite{Hale-VL}. 
According to \cite[Theorems  8.1  and 8.3]{Hale-VL}, for each 
$ \phi\in C([r_0, 0], \mathbb{R}) $, there exists a unique solution 
$ x(t)=x(t, \phi) $ of \eqref{1} defined on $[0, \infty)$.

\begin{definition} \rm
The zero solution of \eqref{1} is said to be stable, if for every $ \epsilon>0 $, 
there exists a $ \delta=\delta(\epsilon)>0 $ such that
 $ \phi:[r_0, 0]\to (-\delta, \delta) $ implies that $ |x(t)|<\epsilon $ for 
$ t\geq 0$.
\end{definition}

\begin{definition} \rm
The zero solution of \eqref{1} is said to be asymptotically stable, if it
 is stable and there exists a $ \delta>0 $ such that for any initial function 
$ \phi:[r_0, 0]\to (-\delta, \delta) $, the solution $ x(t)$ with $ x(t)=\phi(t) $ 
on $[r_0, 0] $ tends to zero as $ t\to \infty $.
\end{definition}

By introducing two auxiliary functions $ v(t) $ and $ p(t) $ to construct a 
contraction mapping on a complete metric space, we obtain Theorem \ref{thm1} 
and Theorem \ref{thm2}, which will be proved in Section \ref{sec2} and 
Section \ref{sec3}, respectively.

\begin{theorem}\label{thm1}
Consider the neutral delay differential equation \eqref{1} and suppose the 
following conditions are satisfied:

{\rm (i)} $ r(t) $ is twice differentiable with $ r'(t)\neq 1 $ and 
$t-r(t)\to \infty$ as $t\to \infty$;

{\rm (ii)} there exists a bounded function $ p:[r_0, \infty)\to (0, \infty)  $ 
with $ p(t)=1 $ for $ t\in [r_0, 0] $ such that $ p'(t) $ exists on 
$ [r_0, \infty) $ and there exists a constant $ \alpha \in (0,1) $ 
and an arbitrary continuous functions $ v: [r_0, \infty)\to \mathbb{R} $ such that
\begin{equation}\label{condition1}
\begin{aligned}
&l\Big\{\big|\frac{c(t)p^{2}(t-r(t))}{p(t)(1-r'(t))}\big|
 +\int_0^{t}|\overline{k}(s)-2b_{1}(s)|e^{-\int_{s}^{t}v(u)\,du}\,ds\Big\} \\
&+L\int_0^{t}e^{-\int_{s}^{t}v(u)\,du}\frac{|b(s)|p(s-r(s))^{\gamma}}{p(s)}\,ds \\
&+\int_{t-r(t)}^{t}\big|v(s)-a(s)-\frac{p'(s)}{p(s)}\big|\,ds\\
&+\int_0^{t}e^{-\int_{s}^{t}v(u)\,du}|v(s)|
 \int_{s-r(s)}^{s}\big|v(u)-a(u)-\frac{p'(u)}{p(u)}\big|\,du\,ds \\
&+\int_0^{t}e^{-\int_{s}^{t}v(u)\,du}
\big|v(s-r(s))-a(s-r(s))-\frac{p'(s-r(s))}{p(s-r(s))}\big||1-r'(s)|\,ds \\
&\leq \alpha,
\end{aligned}
\end{equation}
where
\begin{equation}\label{overline}
\overline{k}(s)=\frac{[\overline{c}(s)v(s)+\overline{c}'(s)](1-r'(s))
+\overline{c}(s)r''(s)}{(1-r'(s))^{2}}, \quad \overline{c}(s)
=\frac{c(s)p^{2}(s-r(s))}{p(s)},
\end{equation}
\begin{equation}\label{b}
b_{1}(s)=\frac{c(s)p(s-r(s))p'(s-r(s))}{p(s)},
\end{equation}
and the constants $ l, L$ are defined as in \eqref{two constants};

{\rm (iii)} and such that
\[
\liminf_{t\to \infty}\int_0^{t}v(s)\,ds>-\infty.
\]
Then the zero solution of \eqref{1} is asymptotically stable if and only if
\begin{equation} \label{thm1iv}
\int_0^{t}v(s)\,ds\to \infty \quad \text{as }  t\to \infty.
\end{equation}
\end{theorem}

\begin{theorem}\label{thm2}
Consider the neutral Voterra integro-differential equation \eqref{2}
 and suppose the following conditions are satisfied:

{\rm (i)} $ r(t) $ is twice differentiable, $ r'(t)\neq 1 $, 
$t-r(t)\to \infty$ as $t\to \infty$;

{\rm (ii)} There exists a bounded function $ p:[r_0, \infty)\to (0, \infty)  $ 
with $ p(0)=1 $ such that $ p'(t) $ exists on $ [r_0, \infty) $ and there exists 
a constant $ \alpha \in (0,1) $ and a continuous functions 
$ v: [r_0, \infty)\to \mathbb{R} $ such that
\begin{equation}\label{alpha2}
\begin{aligned}
&l\Big\{\big|\frac{c(t)p^{2}(t-r(t))}{p(t)(1-r'(t))}\big|
 +\int_0^{t}|\overline{k}(s)-2b_{1}(s)|e^{-\int_{s}^{t}v(u)\,du}\,ds\Big\} \\
&+L\int_0^{t}e^{-\int_{s}^{t}v(u)\,du}\int_{s-r(s)}^{s}
 \frac{|K(s,u)|p^{\gamma}(u)}{p(s)}\,du  \\
&+\int_{t-r(t)}^{t}\big|v(s)-a(s)-\frac{p'(s)}{p(s)}\big|\,ds\\
&+\int_0^{t}e^{-\int_{s}^{t}v(u)\,du}|v(s)|\int_{s-r(s)}^{s}
 \big|v(u)-a(u)-\frac{p'(u)}{p(u)}\big|\,du\,ds \\
&+\int_0^{t}e^{-\int_{s}^{t}v(u)\,du}
 \big|v(s-r(s))-a(s-r(s))-\frac{p'(s-r(s))}{p(s-r(s))}\big||1-r'(s)|\,ds\\
&\leq \alpha,
\end{aligned}
\end{equation}
where $\overline{k}(s)$ and $ b_{1}(s) $ are defined as \eqref{overline}
 and \eqref{b}, respectively, the constants $ l, L$ are defined as in 
\eqref{two constants};

{\rm (iii)} and such that
\[
\liminf_{t\to \infty}\int_0^{t}v(s)\,ds>-\infty.
\]
Then the zero solution of \eqref{2} is asymptotically stable if and only if
\begin{equation} \label{thm2iv}
\int_0^{t}v(s)\,ds\to \infty \quad\text{as }  t\to \infty.
\end{equation}
\end{theorem}

The technique for constructing a contraction mapping comes from an idea 
in \cite{DLZ}. Our work extends and improves the results in \cite{AR, DLZ}.

\begin{remark} \rm
The method applied in this paper can be used to treat more general equations 
such as
\[
\frac{d}{dt}x(t)=-a(t)h(t-r(t))+\frac{d}{dt}Q(t,x(t-r(t)))+G(t,x(t),x(t-r(t)))
\]
(studied by Mesmouli, Ardjouni and Djoudi in \cite{M}) to give more general results.
\end{remark}


\section{Proof of Theorem \ref{thm1}} \label{sec2}

We start with some preparations. Define
\begin{align*}
S_{\phi}^{l}
&=\Big\{\varphi\in C([r_0, \infty), \mathbb{R}):
 \|\varphi\|=\sup_{t \geq r_0}|\varphi(t)|\leq l, \;
 \varphi(t)=\phi(t)\\
&\quad \text{ for } t\in[r_0, 0],\; \varphi(t)\to 0  \text{ as } 
  t\to \infty \Big\}.
\end{align*}
Then $ S_{\phi}^{l} $ is a complete metric space with metric 
$\rho(x,y)=\sup_{t\geq r_0}\{|x(t)-y(t)|\}$.

Let $ z(t)=\phi(t)$ on $ [r_0, 0] $, and let $x(t)=p(t)z(t)$ for $t\geq 0$. 
If $ z $ satisfies
\begin{equation}\label{transformed}
\begin{aligned}
z'(t)
&=-\Big(a(t)+\frac{p'(t)}{p(t)}\Big)z(t)
 +\frac{c(t)p(t-r(t))p'(t-r(t))}{p(t)}z^{2}(t-r(t))\\
&\quad +\frac{c(t)p^{2}(t-r(t))}{p(t)}z(t-r(t))z'(t-r(t)) \\
&\quad +\frac{b(t)p(t-r(t))^{\gamma}}{p(t)}g(z(t-r(t))),
\end{aligned}
\end{equation}
then it can be verified that $ x $ satisfies \eqref{1'}. 
Since $ p(t) $ is a positive bounded function, we only have to prove that 
the zero solution of \eqref{transformed} is asymptotically stable.

If we multiply both sides of \eqref{transformed} by $ e^{\int_0^{t}v(s)\,ds} $ 
and then integrate from $ 0$ to $ t $, we obtain
\begin{equation}\label{step-0}
\begin{aligned}
z(t)
&=\phi(0)e^{-\int_0^{t}v(s)\,ds}
 +\int_0^{t}\Big(v(s)-a(s)-\frac{p'(s)}{p(s)}\Big)e^{-\int_{s}^{t}v(u)\,du}z(s)\,ds \\
&\quad +\int_0^{t}e^{-\int_{s}^{t}v(u)\,du}\frac{c(s)p(s-r(s))
 p'(s-r(s))}{p(s)}z^{2}(s-r(s))\,ds \\
&\quad +\int_0^{t}e^{-\int_{s}^{t}v(u)\,du}\frac{c(s)p^{2}
 (s-r(s))}{p(s)}z(s-r(s))z'(s-r(s))\,ds \\
&\quad +\int_0^{t}e^{-\int_{s}^{t}v(u)\,du}
 \frac{b(s)p(s-r(s))^{\gamma}}{p(s)}g(z(s-r(s)))\,ds.
\end{aligned}
\end{equation}
Taking
\begin{equation}\label{step-1}
\begin{aligned}
&\int_0^{t}e^{-\int_{s}^{t}v(u)\,du}
 \frac{c(s)p^{2}(s-r(s))}{p(s)}z(s-r(s))z'(s-r(s))\,ds\\
&=\int_0^{t}e^{-\int_{s}^{t}v(u)\,du}\frac{c(s)p^{2}(s-r(s))}
 {p(s)}z(s-r(s))z'(s-r(s)) \\
&\quad\times (1-r'(s))\frac{1}{1-r'(s)}\,ds
\end{aligned}
\end{equation}
and integrating by parts the right-hand side of \eqref{step-1}, we obtain
\begin{equation}\label{step-2}
\begin{aligned}
&\int_0^{t}e^{-\int_{s}^{t}v(u)\,du}\frac{c(s)p^{2}(s-r(s))}{p(s)}
 z(s-r(s))z'(s-r(s))\,ds \\
&=\frac{p^{2}(t-r(t))}{2p(t)}\frac{c(t)}{1-r'(t)}z^{2}(t-r(t))
 -\frac{p^{2}(-r(0))}{2p(0)}\frac{c(0)}{1-r'(0)}\phi^{2}(-r(0)) \\
&\quad \times e^{-\int_0^{t}v(s)\,ds}
-\frac{1}{2}\int_0^{t}e^{-\int_{s}^{t}v(u)\,du}\overline{k}(s)z^{2}(s-r(s))\,ds
\end{aligned}
\end{equation}
where $ \overline{k}(s) $ is given by \eqref{overline}.

Integrating by parts, we obtain
\begin{equation}\label{step-3}
\begin{aligned}
&\int_0^{t}\Big(v(s)-a(s)-\frac{p'(s)}{p(s)}\Big)e^{-\int_{s}^{t}v(u)\,du}z(s)\,ds \\
&=\int_0^{t}e^{-\int_{s}^{t}v(u)\,du}\,d
\Big(\int_{s-r(s)}^{s}\Big(v(u)-a(u)-\frac{p'(u)}{p(u)}\Big)z(u)\,du\Big) \\
&\quad +\int_0^{t}e^{-\int_{s}^{t}v(u)\,du}
 \Big(v(s-r(s))-a(s-r(s))-\frac{p'(s-r(s))}{p(s-r(s))}\Big) \\
&\quad \times (1-r'(s))z(s-r(s))\,ds \\
& =\int_{t-r(t)}^{t}\Big(v(s)-a(s)-\frac{p'(s)}{p(s)}\Big)z(s)\,ds \\
&\quad-\int_0^{t}e^{-\int_{s}^{t}v(u)\,du}v(s)\int_{s-r(s)}^{s}
 \Big(v(u)-a(u)-\frac{p'(u)}{p(u)}\Big)z(u)\,du\,ds \\
&\quad +\int_0^{t}e^{-\int_{s}^{t}v(u)\,du}\Big(v(s-r(s))-a(s-r(s))
 -\frac{p'(s-r(s))}{p(s-r(s))}\Big) \\
&\quad \times (1-r'(s))z(s-r(s))\,ds
\end{aligned}
\end{equation}
Combining \eqref{step-0}, \eqref{step-2} and \eqref{step-3}, we obtain 
that a solution of \eqref{transformed} has the form
\begin{align*}
z(t)
&=\Big[\phi(0)-\int_{-r(0)}^{0}\Big(v(s)-a(s)-\frac{p'(s)}{p(s)}\Big)\phi(s)\,ds\\
&\quad-\frac{p^{2}(-r(0))}{2p(0)}\frac{c(0)}{1-r'(0)}\phi^{2}(-r(0))\Big]
  e^{-\int_0^{t}v(s)\,ds}\\
&\quad +\frac{p^{2}(t-r(t))}{2p(t)}\frac{c(t)}{1-r'(t)}z^{2}(t-r(t))\\
&\quad +\int_{t-r(t)}^{t}\Big(v(s)-a(s)-\frac{p'(s)}{p(s)}\Big)z(s)\,ds\\
&\quad -\int_0^{t}e^{-\int_{s}^{t}v(u)\,du}v(s)\int_{s-r(s)}^{s}
 \Big(v(u)-a(u)-\frac{p'(u)}{p(u)}\Big)z(u)\,du\,ds\\
&\quad +\int_0^{t}e^{-\int_{s}^{t}v(u)\,du}
 \Big(v(s-r(s))-a(s-r(s))-\frac{p'(s-r(s))}{p(s-r(s))}\Big)\\
&\quad \times (1-r'(s))z(s-r(s))\,ds\\
&\quad -\frac{1}{2}\int_0^{t}e^{-\int_{s}^{t}v(u)\,du}
\left(\overline{k}(s)-2b_{1}(s)\right)z^{2}(s-r(s))\,ds\\
&\quad +\int_0^{t}e^{-\int_{s}^{t}v(u)\,du}
 \frac{b(s)p(s-r(s))^{\gamma}}{p(s)}g(z(s-r(s)))\,ds
:=\sum_{i=1}^{7}I_{i}(t),
\end{align*}
where $\overline{k}(s)$ and $ b_{1}(s) $ are defined in
 \eqref{overline} and \eqref{b}, respectively.

\begin{lemma}\label{lemma}
Let $ z\in S_{\phi}^{l} $ and define an operator by $ (Pz)(t)=\phi(t)$ 
for $ t\in [r_0, 0] $ and for $ t\geq 0$, $(Pz)(t)=\sum_{i=1}^{7}I_{i}(t)$.
 If conditions {\rm (i)--(ii)} and \ref{thm1iv}  in Theorem \ref{thm1} are satisfied, 
then there exists $ \delta>0 $ such that for any initial function
 $ \phi:[r_0, 0]\to (-\delta, \delta) $, we have that 
$ P: S_{\phi}^{l}\to S_{\phi}^{l} $ and $ P $ is a contraction with 
respect to the metric $ \rho $ defined on $ S_{\phi}^{l} $.
\end{lemma}

\begin{proof}
Set $J=\sup_{t\geq 0}\{e^{-\int_0^{t}v(s)\,ds}\}$, by 
(iii), $J$ is well defined.
Suppose that \ref{thm1iv} holds. We choose $ \delta>0 $ such that
\begin{align*}
\Big[\delta+\delta \int_{-r(0)}^{0}\big|v(s)-a(s)
-\frac{p'(s)}{p(s)}\big|\,ds+\frac{p^{2}(-r(0))}{2p(0)}
\frac{c(0)}{1-r'(0)}\delta^{2}\Big]J\leq(1-\alpha)l.
\end{align*}
Let $ \phi $ be a given small bounded initial function with $ \|\phi\|<\delta $, 
and let $ \varphi\in S_{\phi}^{l} $, then $ \|\varphi\|\leq l $. 
Since $ g $ satisfies a locally Lipschitz condition, from 
\eqref{condition1} in Theorem \ref{thm1}, we have
\begin{align*}
|P\varphi(t)|\leq\Big[\delta+\delta \int_{-r(0)}^{0}
\big|v(s)-a(s)-\frac{p'(s)}{p(s)}\big|\,ds
+\frac{p^{2}(-r(0))}{2p(0)}\frac{c(0)}{1-r'(0)}\delta^{2}\Big]J+\alpha l\leq l.
\end{align*}
Thus, $ \|P\varphi\|\leq l $.

Next, we show that $ P\varphi\to 0 $ as $ t\to \infty $.
It is clear that $ I_{i}(t)\to 0 $ for $ i=1, 2, 3, 4, 5, 7 $, since 
$ e^{\int_0^{t}v(s)\,ds}\to \infty $, $ t-r(t)\to \infty $ and 
$\varphi\to 0 $ as $t\to \infty$. Now, we prove that $ I_{6}(t)\to 0 $ as 
$ t\to \infty $. For $ t-r(t)\to \infty $ and $\varphi\to 0 $, we obtain
 that for any $ \epsilon>0 $, there is a positive number $ T_{1}>0 $ such that 
for $ t\geq T_{1} $, $ \varphi(t-r(t))<\epsilon $, so we have
\begin{align*}
|I_{6}(t)|
&=  \big|\frac{1}{2}\int_0^{t}e^{-\int_{s}^{t}v(u)\,du}\left(\overline{k}(s)
 -2b_{1}(s)\right)\varphi^{2}(s-r(s))\,ds\big|\\
&\leq  \frac{1}{2}e^{-\int_{T_{1}}^{t}v(u)\,du}
 \int_0^{T_{1}}e^{-\int_{s}^{T_{1}}v(u)\,du}
 |\overline{k}(s)-2b_{1}(s)|\varphi^{2}(s-r(s))\,ds\\
&\quad  +\frac{1}{2}\int_{T_{1}}^{t}e^{-\int_{s}^{t}v(u)\,du}
 \left|\overline{k}(s)-2b_{1}(s)\right|\varphi^{2}(s-r(s))\,ds\\
&\leq  \frac{1}{2}\Big(\sup_{t\geq r_0}|\varphi(t)|\Big)^{2}
 e^{-\int_{T_{1}}^{t}v(u)\,du}\int_0^{T_{1}}
 e^{-\int_{s}^{T_{1}}v(u)\,du}|\overline{k}(s)-2b_{1}(s)|\,ds\\
&\quad  +\frac{1}{2}\epsilon^{2}\int_{T_{1}}^{t}e^{-\int_{s}^{t}v(u)\,du}
 |\overline{k}(s)-2b_{1}(s)|\,ds\\
&\leq \frac{\alpha}{2}l^{2}e^{-\int_{T_{1}}^{t}v(u)\,du}+\alpha\epsilon.
\end{align*}
By \ref{thm1iv}, there exists $ T_{2}>T_{1} $ such that $ t>T_{2} $ implies 
$ \frac{\alpha}{2}l^{2}e^{-\int_{T_{1}}^{t}v(u)\,du}<\epsilon $, which 
implies $I_{6}(t)\to 0  $ as $ t\to \infty $. Hence, we have 
$ (P\varphi)(t)\to 0 $ as $t\to \infty$.

Finally, we show that $ P $ is a contraction mapping. In fact,
for $ \varphi, \eta \in S_{\phi}^{l} $, using condition 
\eqref{condition1} in Theorem \ref{thm1}, we obtain
\begin{align*}
&\big|(P\varphi)(t)-(P\eta)(t)\big|\\
&\leq  2l\left|\frac{p^{2}(t-r(t))}{2p(t)}\frac{c(t)}{1-r'(t)}\right|\,
\left\|\varphi-\eta \right\|
+\int_{t-r(t)}^{t}\big|v(s)-a(s)-\frac{p'(s)}{p(s)}\big|\|\varphi-\eta \|\,ds\\
&\quad  +\int_0^{t}e^{-\int_{s}^{t}v(u)\,du}|v(s)|\int_{s-r(s)}^{s}
 \big|v(u)-a(u)-\frac{p'(u)}{p(u)}\big|\, \|\varphi-\eta \|\,du\,ds\\
&\quad  +\int_0^{t}e^{-\int_{s}^{t}v(u)\,du}\left|v(s-r(s))-a(s-r(s))-\frac{p'(s-r(s))}{p(s-r(s))}\right|\\
&\quad\times |1-r'(s)|\|\varphi-\eta \|\,ds\\
&\quad  +l\int_0^{t}e^{-\int_{s}^{t}v(u)\,du}
 |\overline{k}(s)-2b_{1}(s)|\,\|\varphi-\eta \|\,ds\\
&\quad  +L\int_0^{t}e^{-\int_{s}^{t}v(u)\,du}
\frac{|b(s)|p(s-r(s))^{\gamma}}{p(s)}\|\varphi-\eta \|\,ds\\
&\quad  \leq \alpha \|\varphi-\eta \|.
\end{align*}
Therefore, $ P: S_{\phi}^{l}\to S_{\phi}^{l} $ is a contraction mapping.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1}]
Let $ P $ be defined as in Lemma \ref{lemma}. By the contraction mapping 
principle, $ P $ has a unique fixed point $ z $ in $ S_{\phi}^{l} $ 
which is a solution of \eqref{transformed} with $ z(t)=\phi(t) $ on 
$ [r_0, 0] $ and $ z(t)\to 0 $ as $ t\to \infty $.

To prove stability, let $ \epsilon>0 $ be given, then we choose $ m>0 $ so 
that $ m<\min\{L, \epsilon\} $.
Replacing $ l $ with $ m $ in $  S_{\phi}^{l}  $, we obtain that there is 
a $ \delta>0 $ such that $ \|\phi\|<\delta $ implies that the unique solution 
of \eqref{transformed} with $ z(t)=\phi(t)  $ on $ [r_0, 0] $ satisfies 
$ |z(t)|\leq m<\epsilon $ for all $ t\geq r_0 $. 
This shows that the zero solution of \eqref{transformed} is 
asymptotically stable if \ref{thm1iv} holds.

Conversely, we suppose that \ref{thm1iv} fails. Then by (iii), there exists 
a sequence  $ \{t_{n}\} $, $ t_{n}\to \infty $ as $ n\to \infty $ such that 
$ \lim_{n\to \infty} \int_0^{t_{n}}v(s)\,ds=v $ for some $ v\in\mathbb{R} $. 
We may choose a positive constant $ M $ such that
\begin{equation}\label{chapter3-6-6}
-M \leq \int_0^{t_{n}}v(s)\,ds\leq M
\end{equation}
for all $ n\geq 1 $. To simplify our expressions, we define
\begin{align*}
w(s)
&= l|\overline{k}(s)-2b_{1}(s)|+|v(s)|\int_{s-r(s)}^{s}
\big|v(u)-a(u)-\frac{p'(u)}{p(u)}\big|\,du\\
&\quad +\frac{L|b(s)|p(s-r(s))^{\gamma}}{p(s)}\\
&\quad +\big|v(s-r(s))-a(s-r(s))-\frac{p'(s-r(s))}{p(s-r(s))}\big||1-r'(s)|
\end{align*}
for all $ s\geq 0 $. By $\rm{(ii)}$ we have
\begin{equation}\label{chapter3-6-7}
\int_0^{t_{n}}e^{-\int_{s}^{t_{n}}v(u)\,du}w(s)\,ds\leq \alpha.
\end{equation}
Combining \eqref{chapter3-6-6} and \eqref{chapter3-6-7}, we have 
$\int_0^{t_{n}}e^{\int_0^{s}v(u)\,du}w(s)\,ds
\leq \alpha e^{\int_0^{t_{n}}v(u)\,du}\leq \alpha e^{M}$, 
which yields that $ \int_0^{t_{n}}e^{\int_0^{s}v(u)\,du}w(s)\,ds $ 
is bounded. Hence, there exists a convergent subsequence, we assume that 
$\lim_{k\to \infty}\int_0^{t_{n_{k}}}e^{\int_0^{s}v(u)\,du}w(s)\,ds=\gamma$ 
for some $ \gamma\in \mathbb{R^+} $.
We choose a positive integer $k_{1}$ so large that 
$\lim_{k\to \infty}\int_{t_{n_{k_{1}}}}^{t_{n_{k}}}
e^{\int_0^{s}v(u)\,du}w(s)\,ds\leq \frac{\delta_0}{4J}$ 
for all $ n_{k}>n_{k_{1}} $, where $ \delta_0>0 $ satisfies 
$ 2\delta_0Je^{M}+\alpha<1 $.

Now, we consider the solution $ z(t)=z(t,t_{n_{k_{1}}}, \phi) $ of 
\eqref{transformed} with $ \phi(t_{n_{k_{1}}})=\delta_0$
and $ \phi(s)\leq \delta_0 $ for $ s\leq t_{n_{k_{1}}} $, and we may
 choose $ \phi $ such that $ |z(t)|\leq 1 $ for $ t\geq t_{n_{k_{1}}} $ and
\begin{equation}\label{8}
\begin{aligned}
&\phi(t_{n_{k_{1}}})-\int_{t_{n_{k_{1}}}-r(t_{n_{k_{1}}})}^{t_{n_{k_{1}}}}
\Big(v(s)-a(s)-\frac{p'(s)}{p(s)}\Big)\phi(s)\,ds \\
&-\frac{p^{2}(t_{n_{k_{1}}}-r(t_{n_{k_{1}}}))}{2p(t_{n_{k_{1}}})}
\frac{c(t_{n_{k_{1}}})}{1-r'(t_{n_{k_{1}}})}\phi^{2}
(t_{n_{k_{1}}}-r(t_{n_{k_{1}}}))\geq\frac{1}{2}\delta_0.
\end{aligned}
\end{equation}
So, it follows from \eqref{8} with $ z(t)=(Pz)(t) $ that for $ k\geq k_{1} $,
\begin{equation}\label{9}
\begin{aligned}
&\Big|z(t_{n_{k}})-\frac{p^{2}(t_{n_{k}}-r(t_{n_{k}}))}{2p(t_{n_{k}})}
 \frac{c(t_{n_{k}})}{1-r'(t_{n_{k}})}z^{2}(t-r(t_{n_{k}}))\\
&\quad -\int_{t_{n_{k}}-r(t_{n_{k}})}^{t_{n_{k}}}
 \Big[v(s)-a(s)-\frac{p'(s)}{p(s)}\Big]z(s)\,ds\Big| \\
&\geq \frac{1}{2}\delta_0e^{-\int_{t_{n_{k_{1}}}}^{t_{n_{k}}}v(u)\,du}
 -\int_{t_{n_{k_{1}}}}^{t_{n_{k}}}
  e^{-\int_{s}^{t_{n_{k}}}v(u)\,du}w(s)\,ds \\
&=e^{-\int_{t_{n_{k_{1}}}}^{t_{n_{k}}}v(u)\,du} 
 \Big[\frac{1}{2}\delta_0-e^{-\int_0^{t_{n_{k_{1}}}}v(u)\,du}
\int_{t_{n_{k_{1}}}}^{t_{n_{k}}}e^{\int_0^{s}v(u)\,du} w(s)\,ds\Big] \\
&\geq e^{-\int_{t_{n_{k_{1}}}}^{t_{n_{k}}}v(u)\,du}
 \Big[\frac{1}{2}\delta_0-J
\int_{t_{n_{k_{1}}}}^{t_{n_{k}}}e^{\int_0^{s}v(u)\,du}w(s)\,ds\Big] \\
&\geq\frac{1}{4}\delta_0e^{-\int_{t_{n_{k_{1}}}}^{t_{n_{k}}}v(u)\,du}
 \geq \frac{1}{4}\delta_0e^{-2M}>0.
\end{aligned}
\end{equation}
On the other hand, suppose that the solution of \eqref{transformed} $ 
z(t)=z(t,t_{n_{k_{1}}}, \phi)\to 0 $ as $ t\to \infty $. 
Since $ t_{n_{k}}-r(t_{n_{k}})\to \infty $ as $ k\to \infty $, 
and (ii) holds, we have
\begin{align*}
&z(t_{n_{k}})-\frac{p^{2}(t_{n_{k}}-r(t_{n_{k}}))}{2p(t_{n_{k}})}
\frac{c(t_{n_{k}})}{1-r'(t_{n_{k}})}z^{2}(t-r(t_{n_{k}}))\\
&-\int_{t_{n_{k}}-r(t_{n_{k}})}^{t_{n_{k}}}\Big[v(s)-a(s)
 -\frac{p'(s)}{p(s)}\Big]z(s)\,ds\to 0 \quad \text{as } k\to \infty,
\end{align*}
which contradicts \eqref{9}. Hence condition \ref{thm1iv} is necessary for the 
asymptotic stability of the zero solution of \eqref{transformed}.

Since $ p(t) $ is a positive bounded function, from the above arguments 
we obtain that \ref{thm1iv} is a necessary and sufficient condition for the 
asymptotic stability of the zero solution of \eqref{1}.
\end{proof}

When $ g(x)=x^{2} $ in \eqref{1}, we have the following result.

\begin{corollary}\label{coro1}
Suppose the following conditions are satisfied:
{\rm (i)} the delay $ r(t) $ is twice differentiable with $ r'(t)\neq 1 $, 
and $t-r(t)\to \infty$ as $t\to \infty$;

{\rm (ii)} there exists a bounded function $ p:[r_0, \infty)\to (0, \infty) $ 
with $ p(0)=1 $ such that $ p'(t) $ exists on $ [r_0, \infty) $, 
and there exists a constant $ \alpha \in (0,1) $ and an arbitrary continuous 
functions $ v: [r_0, \infty)\to \mathbb{R} $ such that
\begin{equation}\label{alpha1-1}
\begin{aligned}
&l\Big\{\big|\frac{c(t)p^{2}(t-r(t))}{p(t)(1-r'(t))}\big|
 +\int_0^{t}|\overline{k}(s)-2\overline{b}(s)|e^{-\int_{s}^{t}v(u)\,du}\,ds\Big\}\\
&+\int_{t-r(t)}^{t}\big|v(s)-a(s)-\frac{p'(s)}{p(s)}\big|\,ds\\
&+\int_0^{t}e^{-\int_{s}^{t}v(u)\,du}|v(s)|
 \int_{s-r(s)}^{s}\big|v(u)-a(u)-\frac{p'(u)}{p(u)}\big|\,du\,ds\\
&+\int_0^{t}e^{-\int_{s}^{t}v(u)\,du}
\big|v(s-r(s))-a(s-r(s))-\frac{p'(s-r(s))}{p(s-r(s))}\big||1-r'(s)|\,ds \\
&\leq \alpha,
\end{aligned}
\end{equation}
where $ \overline{k}(s) $ is defined as in {\rm{\eqref{overline}}},
\begin{equation}\label{chapter3-6-b1}
\overline{b}(s)=\frac{b(s)p^{2}(s-r(s))+c(s)p(s-r(s))p'(s-r(s))}{p(s)},
\end{equation}
and $ l>0 $ is defined as in {\rm{\eqref{two constants}}};
\item[$\rm{(iii)}$] and such that
\begin{align*}
\liminf_{t\to \infty}\int_0^{t}v(s)\,ds>-\infty.
\end{align*}
Then the zero solution $ x(t, \phi) $ of {\rm{\eqref{1}}} is asymptotically
 stable if and only if
\begin{equation} \label{coro1iv}
\int_0^{t}v(s)\,ds\to \infty \quad\text{as }  t\to \infty.
\end{equation}
\end{corollary}

\section{Proof of Theorem \ref{thm2}}\label{sec3}

 We start with some preparations. Define
\begin{align*}
S_{\phi}^{l}&= \Big\{ \varphi\in C([r_0, \infty), \mathbb{R}):
 \|\varphi\|=\sup_{t \geq r_0}|\varphi(t)|\leq l, \; \varphi(t)=\phi(t)\\ 
&\quad \text{for } t\in[r_0, 0], \; \varphi(t)\to 0  \text{ as } 
 t\to \infty \Big\}.
\end{align*}
Then $ S_{\phi}^{l} $ is complete metric space with metric 
$\rho(x,y)=\sup_{t\geq r_0}\{|x(t)-y(t)|\}$.

Let $ z(t)=\phi(t) $ on $[r_0, 0] $, and let $x(t)=p(t)z(t)$, for $ t\geq 0$, 
from \eqref{2}, we obtain
\begin{equation}\label{transformed1}
\begin{aligned}
z'(t)&= -\big(a(t)+\frac{p'(t)}{p(t)}\big)z(t)
 +\frac{c(t)p(t-r(t))p'(t-r(t))}{p(t)}z^{2}(t-r(t))\\
&\quad +\frac{c(t)p^{2}(t-r(t))}{p(t)}z(t-r(t))z'(t-r(t)) \\
&\quad +\int_{t-r(t)}^{t}\frac{p^{\gamma}(s)K(t,s)}{p(t)}g(z(s))\,ds.
\end{aligned}
\end{equation}
Since $ p(t) $ is bounded, we only need to prove that the zero solution 
of \eqref{transformed1} is asymptotically stable.

If we multiply both sides of \eqref{transformed1} by $ e^{\int_0^{t}v(s)\,ds} $, 
integrate from $ 0 $ to $ t $, and perform an integration by parts, we obtain
\begin{align*}
z(t)
&= \Big\{\phi(0)-\int_{-r(0)}^{0}\Big(v(s)-a(s)-\frac{p'(s)}{p(s)}\Big)\phi(s)\,ds\\
&\quad  -\frac{p^{2}(-r(0))}{2p(0)}\frac{c(0)}{1-r'(0)}\phi^{2}(-r(0))\Big\}
 e^{-\int_0^{t}v(s)\,ds}\\
&\quad +\frac{p^{2}(t-r(t))}{2p(t)}\frac{c(t)}{1-r'(t)}z^{2}(t-r(t))\\
&\quad +\int_{t-r(t)}^{t}\left[v(s)-a(s)-\frac{p'(s)}{p(s)}\right]z(s)\,ds\\
&\quad -\int_0^{t}e^{-\int_{s}^{t}v(u)\,du}v(s)\int_{s-r(s)}^{s}
 \Big(v(u)-a(u)-\frac{p'(u)}{p(u)}\Big)z(u)\,du\,ds\\
&\quad  +\int_0^{t}e^{-\int_{s}^{t}v(u)\,du}
 \Big(v(s-r(s))-a(s-r(s))-\frac{p'(s-r(s))}{p(s-r(s))}\Big)\\
& \quad \times (1-r'(s))z(s-r(s))\,ds\\
&\quad  -\frac{1}{2}\int_0^{t}e^{-\int_{s}^{t}v(u)\,du}
\big(\overline{k}(s)-2b_{1}(s)\big)z^{2}(s-r(s))\,ds\\
&\quad  + \int_0^{t}e^{-\int_{s}^{t}v(u)\,du}
 \int_{s-r(s)}^{s}\frac{K(s, u)p^{\gamma}(u)}{p(s)}g(z(u))\,du\,ds
:=\sum_{i=1}^{7}I_{i}(t)
\end{align*}
where $ \overline{k}(s) $ and $ b_{1}(s) $ are defined as in \eqref{overline} 
and \eqref{b} respectively.

\begin{lemma}\label{lemma1}
Let $ z\in S_{\phi}^{l} $ and define an operator by $ (Pz)(t)=\phi(t)$ for 
$ t\in [r_0, 0] $ and for $ t\geq 0$, $(Pz)(t)=\sum_{i=1}^{7}I_{i}(t)$. 
If conditions {\rm (i)-(iii)} in Theorem \ref{thm2} are satisfied, then 
there exists $ \delta>0 $ such that for any $ \phi:[r_0, 0]\to (-\delta, \delta) $,
 we have that $ P: S_{\phi}^{l}\to S_{\phi}^{l} $ and $ P $ is a contraction 
mapping with respect to the metric defined on $ S_{\phi}^{l} $.
\end{lemma}

\begin{proof}
Set $J=\sup_{t\geq 0}\{e^{-\int_0^{t}v(s)\,ds}\}$, by (iii), 
$J$ is well defined.
Suppose that (iii) holds. Using the similar arguments as as the proof of 
Theorem \ref{thm1}, we obtain that $ P\varphi\in S_{\phi}^{l} $ for 
$ \varphi\in S_{\phi}^{l} $. Now, we show that $ P $ is a contraction mapping. 
In fact, for $ \varphi, \eta \in S_{\phi}^{l} $, by using condition 
\eqref{alpha2} in Theorem \ref{thm2}, we obtain that
\begin{align*}
&\big|(P\varphi)(t)-(P\eta)(t)\big|\\
&\leq\big|\frac{p^{2}(t-r(t))}{2p(t)}\frac{c(t)}{1-r'(t)}\big|2l 
 \|\varphi-\eta \|\\
&\quad +\int_{t-r(t)}^{t}\big|v(s)-a(s)-\frac{p'(s)}{p(s)}
\big|\, \|\varphi-\eta \|\,ds\\
&\quad +\int_0^{t}e^{-\int_{s}^{t}v(u)\,du}|v(s)|
 \int_{s-r(s)}^{s}\big|v(u)-a(u)-\frac{p'(u)}{p(u)}\big|\,
 \|\varphi-\eta \|\,du\,ds\\
&\quad +\int_0^{t}e^{-\int_{s}^{t}v(u)\,du}
 \big|v(s-r(s))-a(s-r(s))-\frac{p'(s-r(s))}{p(s-r(s))}\big|\\
&\quad \times  |1-r'(s)|\|\varphi-\eta \|\,ds\\
&\quad +\frac{1}{2}\int_0^{t}|\overline{k}(s)-2b_{1}(s)| 2l \|\varphi-\eta \|\\
&\quad +L\int_0^{t}e^{-\int_{s}^{t}v(u)\,du}
 \int_{s-r(s)}^{s}\frac{|K(s, u)|p^{\gamma}(u)}{p(s)}\,du\cdot \|\varphi-\eta\|\,ds\\
&\quad  \leq \alpha \|\varphi-\eta \|.
\end{align*}
Hence, we obtain that $ P: S_{\phi}^{l}\to S_{\phi}^{l}$ is a contraction mapping.
\end{proof}
 
\begin{proof}[Proof of Theorem \ref{thm2}]
Let $ P $ be defined as in Lemma \ref{lemma1}. By the contraction mapping 
principle, $ P $ has a unique fixed point $ z $ in $ S_{\phi}^{l} $ which is
 a solution of \eqref{2} with $ z(t)=\phi(t) $ on $ [r_0, 0] $ and 
$ z(t)\to 0 $ as $ t\to \infty $.

Let $ \epsilon>0 $ be given, then we choose $ m>0 $ so that
 $ m<\min\{l, \epsilon\} $.
Replacing $ l $ with $ m $ in $ S_{\phi}^{l} $, we obtain there is a 
$ \delta>0 $ such that $ \|\phi\|<\delta $ implies that the unique solution 
of \eqref{2} with $ z(t)=\phi(t)  $ on $ [r_0, 0] $ satisfies
 $ |z(t)|\leq m<\epsilon $ for all $ t\geq r_0 $. This shows that the zero 
solution of \eqref{2} is asymptotically stable if \ref{thm2iv} holds.

Following the similar arguments as the proof of Theorem \ref{thm1},
 we obtain that \ref{thm2iv} is necessary for the asymptotic stability of the 
zero solution of \eqref{2}. The proof is complete.
\end{proof}

When $ g(x)=x^{2}$, we have the following corollary.

\begin{corollary}\label{coro2}
Suppose the following conditions are satisfied:

{\rm (i)} the delay $ r(t) $ is twice differentiable, $ r'(t)\neq 1 $, 
 $t-r(t)\to \infty$ as $t\to \infty$;

{\rm (ii)} there exists a bounded function $ p:[r_0, \infty)\to (0, \infty)  $ 
with $ p(0)=1 $ such that $ p'(t) $ exists on $ [r_0, \infty) $, and there 
exists a constant $ \alpha \in (0,1) $, a constant $ l>0 $ and a continuous 
functions $ v: [r_0, \infty)\to \mathbb{R} $ such that
\begin{equation}\label{al2}
\begin{aligned}
&l\Big\{\big|\frac{c(t)p^{2}(t-r(t))}{p(t)(1-r'(t))}\big|
 +\int_0^{t}\Big[|\overline{k}(s)-2b_{1}(s)| \\
&\quad  +2\int_{s-r(s)}^{s}\big|\frac{K(s,u)p^{2}(u)}{p(s)}\big|\,du \Big]
 e^{-\int_{s}^{t}v(u)\,du}\,ds\Big\} \\
&\quad  +\int_{t-r(t)}^{t}\big|v(s)-a(s)-\frac{p'(s)}{p(s)}\big|\,ds \\
&\quad  +\int_0^{t}e^{-\int_{s}^{t}v(u)\,du}
 \big|v(s-r(s))-a(s-r(s))-\frac{p'(s-r(s))}{p(s-r(s))}\big||1-r'(s)|\,ds \\
&\quad  +\int_0^{t}e^{-\int_{s}^{t}v(u)\,du}|v(s)|
 \int_{s-r(s)}^{s}\big|v(u)-a(u)-\frac{p'(u)}{p(u)}\big|\,du\,ds\leq\alpha,
\end{aligned}
\end{equation}
where $\overline{k}(s)$ and $ b_{1}(s) $ are defined as in \eqref{overline}
 and \eqref{b};

{\rm (iii)} and such that
\[
\liminf_{t\to \infty}\int_0^{t}v(s)\,ds>-\infty.
\]
Then the zero solution of \eqref{2} with a small initial function $ \phi $ 
is asymptotically stable if and only if
\begin{equation} \label{cor2iv}
\int_0^{t}v(s)\,ds\to \infty \quad\text{as }  t\to \infty.
\end{equation}
\end{corollary}


\section{Examples}\label{sec: AR}

\begin{example}\rm
Consider the  nonlinear neutral differential equation
\begin{equation}\label{ex1}
x'(t)-c(t)x(t-r(t))x'(t-r(t))=-a(t)x(t)+b(t)x^2(t-r(t))
\end{equation}
for $ t\geq 0 $, where $a(t)=\frac{2}{t+1}$, $c(t)=0.95$, $ r(t)=0.05t $, 
$ l=1$, $ b(t) $ satisfies
$|\overline{k}(s)-2\overline{b}(s)|\leq \frac{0.3}{s+1}$, then the zero 
solution of \eqref{ex1} is asymptotically stable.
\end{example}

\begin{proof}
We check  condition \eqref{alpha1-1} in Corollary \ref{coro1},
choosing $ v(t)=\frac{1.5}{t+1} $ and $ p(t)=\frac{1}{t+1} $, we obtain that
\begin{align*}
&l\Big\{\big|\frac{c(t)p^{2}(t-r(t))}{p(t)(1-r'(t))}\big|
 +\int_0^{t}e^{-\int_{s}^{t}v(u)\,du}|\overline{k}(s)-2\overline{b}(s)|\,ds\Big\}\\
&+\int_{t-r(t)}^{t}\left|v(s)-a(s)-\frac{p'(s)}{p(s)}\right|\,ds\\
&+\int_0^{t}e^{-\int_{s}^{t}v(u)\,du}|v(s)|\int_{s-r(s)}^{s}\left|v(u)-a(u)-\frac{p'(u)}{p(u)}\right|\,du\,ds\\
&+\int_0^{t}e^{-\int_{s}^{t}v(u)\,du}
 \Big|v(s-r(s))-a(s-r(s))-\frac{p'(s-r(s))}{p(s-r(s))}\Big||1-r'(s)|\,ds\\
&<0.36+0.026+0.026+0.33+0.2=0.941<1,
\end{align*}
and since $ \int_0^{t}v(s)\,ds=\int_0^{t}\frac{1.5}{s+1}\,ds=1.5\ln(t+1)\to \infty$ 
as $t\to \infty $, $ p(t)\leq 1$, so the conditions of Corollary \ref{coro1} 
are satisfied. Therefore, the zero solution of \eqref{ex1} is asymptotically stable.
\end{proof}

Note that  $|\frac{c(t)}{1-r'(t)}|=1 $;
 therefore the result in \cite{AR} is not applicable.

\begin{example} \rm
Consider the  nonlinear neutral Volterra integral equation
\begin{equation}\label{ex2}
x'(t)-c(t)x(t-r(t))x'(t-r(t))=-a(t)x(t)+\int_{t-r(t)}^{t}K(t,s)x^{2}(s)\,ds
\end{equation}
for $ t\geq 0$, where $a(t)=\frac{2.5}{t+0.1}$, 
$c(t)=\frac{(0.95t+0.1)^{2}}{t+0.1}$, $ r(t)=0.05t$, $ l=1$,
 $ K(t, s)=\frac{1}{t+0.1} $, then the zero solution of \eqref{ex2}
 is asymptotically stable.
\end{example}

\begin{proof}
We check the condition \eqref{al2} in Corollary \ref{coro2},
choosing $ v(t)=\frac{2}{t+0.1} $ and $ p(t)=\frac{0.1}{t+0.1} $, we obtain that
\begin{align*}
&l\left|\frac{c(t)p^{2}(t-r(t))}{p(t)(1-r'(t))}\right|\\
&+ l\int_0^{t}\Big[|\overline{k}(s)-2b_{1}(s)|
+2\int_{s-r(s)}^{s}\big|\frac{K(s,u)p^{2}(u)}{p(s)}\big|\,du \Big]
 e^{-\int_{s}^{t}v(u)\,du}\,ds\\
&+\int_{t-r(t)}^{t}\left|v(s)-a(s)-\frac{p'(s)}{p(s)}\right|\,ds\\
&+\int_0^{t}e^{-\int_{s}^{t}v(u)\,du}|v(s)|\int_{s-r(s)}^{s}
 \big|v(u)-a(u)-\frac{p'(u)}{p(u)}\big|\,du\,ds\\
&+\int_0^{t}e^{-\int_{s}^{t}v(u)\,du}\left|v(s-r(s))-a(s-r(s))
 -\frac{p'(s-r(s))}{p(s-r(s))}\right||1-r'(s)|\,ds\\
&<0.106+0.416+0.026+0.026+0.25=0.824<1,
\end{align*}
and since $ \int_0^{t}v(s)\,ds=\int_0^{t}\frac{2}{s+0.1}\,ds=2\ln(t+0.1)\to \infty$ 
as $t\to \infty $, $ p(t)\leq 1 $, so the conditions of Corollary \ref{coro2} 
are satisfied. Therefore, the zero solution of \eqref{ex2} is asymptotically stable.
\end{proof}

Note that  $|\frac{c(t)}{1-r'(t)}|=\frac{(0.95t+0.1)^{2}}{0.95(t+0.1)}\to \infty$ 
as $t\to \infty $; therefore the  result in \cite{AR} is not applicable.

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