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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 103, pp. 1--8.\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{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/130\hfil Asymptotic behavior of solutions]
{Asymptotic behavior of solutions to a first-order
non-homogeneous delay differential equation}

\author[Q. Zhou \hfil EJDE-2011/103\hfilneg]
{Qiyuan Zhou}

\address{Qiyuan Zhou \newline
College of Mathematics and Computer Science,
Hunan University of Arts and Science,
 Changde, Hunan 415000, China}
\email{zhouqiyuan65@yahoo.com.cn}

\thanks{Submitted May 20, 2011. Published August 10, 2011.}
\subjclass[2000]{34C12, 39A11}
\keywords{Asymptotic behavior; delay differential equation}

\begin{abstract}
 In this article, we study the asymptotic behavior of solutions
 to  the delay differential equation
 $$
 x'(t)=f(t,x(t),x(t-r(t)))\,.
 $$
 It is shown that every  solution tends to either $\infty$
 or a constant as $t\to \infty$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction}

 Consider the delay differential equation
\begin{equation}
x'(t)=f(t,x(t),x(t-r(t))), \label{e1.1}
\end{equation}
where $f\in C(\mathbb{R} \times \mathbb{R} \times \mathbb{R} )$
and $r\in C(\mathbb{R})$.
In this article, we assume the following: 
$f(t,u,v)$ is non-increasing in $u$;
the delay may be unbounded from above, but it is bounded from below
by a positive constant, $0<\tau \leq r(t)$; 
the function $\lambda(t):=t-r(t)$ is non-decreasing, and 
$\lim_{t\to \infty}\lambda(t)=\infty$.
Let
\begin{equation} \label{e1.1a}
\alpha(s)=\sup\{t:\lambda(t)=s\}.
\end{equation}
Then $\alpha(\lambda(t))\geq t$ and $\alpha(s)>s$,
and when $\lambda(t)$ is strictly increasing,
$\alpha$ is the inverse function of $\lambda$; i.e.,
$\alpha(\lambda(t))=t$.

The initial condition for \eqref{e1.1} is a
continuous function $\phi$ such that
\[
x(t)=\phi(t) \quad \text{for } t\in
E_{t_0}:=\{t_0\}\cup\{t-r(t)<t_0: t\geq t_0\}.
\]
Examples of equation \eqref{e1.1} include the following:
\begin{gather}
 x'(t)  =-x^{1/3}(t)+ x^{1/3}(t-r(t)), \label{e1.3}\\
 x'(t)  =p(t)[-x^{1/3}(t)+ x^{1/3}(t-r(t))]+q(t),  \label{e1.4}
\end{gather}
where  $r$,  $p$ and $q$  are continuous functions.
For $r(t)$ a positive constant, Bernfeld and Haddock \cite{b1}
proposed the  conjecture:
\begin{quote}
 Every solution of \eqref{e1.3} tends to a constant as
 $t\to \infty$,
\end{quote}
which was proved by Ding \cite{d1}.
Chen \cite{c2} obtained the following result:
\begin{quote}
 If $p(t)$ is bounded and  $q\in L^1[0,\infty)$, then every
 solution of \eqref{e1.4} tends to a constant as
   $t\to \infty$.
\end{quote}
Subsequently, Chen \cite{y1}  considered  \eqref{e1.1} and obtained
the following result.

\begin{theorem} \label{thmA}
Assume that $f(t,u,v)$ is strictly decreasing in $u$, and
\[
 f(t,u,v)\leq p(t)G(u,v)+q_1(t)u+q_2(t)v+q_{3}(t),
\]
where
$G(u,v)\in C(\mathbb{R}\times\mathbb{R})$ and
$p,q_i\in C(\mathbb{R})$ satisfying the following conditions:
$G(u,v)$ is strictly decreasing in $u$, and strictly increasing in $v$;
$G(u,u)\equiv 0$ for all $u\in\mathbb{R}$;
$G(u_1,u_2)>0$ for all $u_2>u_1$;
$q_i\in L^1[0,\infty)$ $(i=1, 2, 3)$, $p, q_1, q_2$;
and
\begin{equation}
\int_t ^{\alpha(t)}p(s)ds\leq M,\quad t\in\mathbb{R}\,. \label{e1.5}
\end{equation}
Then every solution of  \eqref{e1.1} is  bounded above,
and tends to either a constant or to $-\infty$,
as $t\to \infty$.
\end{theorem}

Recently, Yi \cite{y1} pointed out a mistake in \cite[Proposition 4]{c2}.
Unfortunately, the similar mistake appears in
 \cite[Lemma 1]{c1} and \cite[Lemma 2]{c2}.
Moreover,  we found that  \cite[condition (1.5)]{c2}
should be strengthened  to be
\begin{equation}
p(t)> 0, \quad
\int_t ^{\alpha(t)}p(s)ds\leq M,\quad \forall t\in\mathbb{R}\,.
 \label{e1.6}
\end{equation}
The main purpose of this paper is to show
the convergence of the solutions of \eqref{e1.1}.
Our approach is quite different from the one in \cite{d1,y1},
and our conditions are weaker than those in \cite{y1}.
Ofcourse, our proofs ovoid the mistakes in \cite{c1,c2}.

\section{Preliminary results}

We start with a well known result in differential equations.


\begin{lemma}[See \cite{t1}] \label{lem2.1}
 Let $x_0\in\mathbb{R}$, $\beta>0$,
$h\in C([x_0, x_0+\beta]\times \mathbb{R}, \mathbb{R})$,
and $h$ be a non-increasing in the second variable.
Then the initial value problem
\begin{equation} \label{e2.1}
\begin{gathered}
\frac{dy}{dx}=h(x,y)\\
y(x_0)=y_0
\end{gathered}
\end{equation}
has a unique solution  on the interval $[x_0, x_0+\beta]$.
\end{lemma}

\begin{lemma} \label{lem2.2}
Assume  $\phi:E_{t_0}\to \mathbb{R}$ is a continuous function.
Then the initial-value problem
\begin{equation}
\begin{gathered}
  x'(t)=f(t,x(t),x(t-r(t))),\quad t\geq t_0 \\
x(t)=\phi(t),\quad t\in E_{t_0}
\end{gathered} \label{e2.4}
\end{equation}
has a unique solution on $[t_0,\infty)$.
\end{lemma}

\begin{proof}  We find the solution on intervals of length
$\tau$, where $\tau$ is the lower bound for the delay $r(t)$.
For $t\in [t_0,t_0+\tau]$, let
$$
h(t,x(t)))=f(t,x(t),x(t-r(t)))=f(t,x(t),\phi(t-r(t))).
$$
Then by Lemma \ref{lem2.1}, there exists a unique solution $x(t)$
on $[t_0, t_0+\tau]$.
Recursively, we can build a unique solution of \eqref{e2.4}
for any interval $[t_0,T]$. The proof  is complete.
\end{proof}

Using the same argument as in \cite[Proposition 3]{d1}, we can
prove the following result.

\begin{lemma} \label{lem2.3}
 Suppose that $G(u,u)\equiv 0$ for all $ u\in\mathbb{R}$,
$ G(u,v)$ is non-increasing in $u$, and
non-decreasing in $v$.
Then the initial-value problem
\begin{gather}
\frac{du }{dt}=G(u,k), \label{e2.6} \\
u(t_0)=u_0,  \label{e2.7}
\end{gather}
where $k$ is constant in $\mathbb{R}$,
has a unique solution $u=u(t,k)$ on $[t_0,+\infty)$, and
the function $\phi(k)=u(t,k)$
is  continuous in $k$.
\end{lemma}

\begin{lemma} \label{lem2.4}  Suppose that
$G(u,u)\equiv 0$ for all $ u\in\mathbb{R}$,  $ G(u,v)$ is
 non-increasing in $u$, and
 non-decreasing in  $v$.
Consider the initial-value problem
\begin{gather}
\frac{du }{dt}=p(t)G(u,c+\varepsilon),   \label{e2.8}\\
u(t_0)=u_0,\quad  u_0<c, \label{e2.9}
\end{gather}
where $c$ is a nonzero constant and $\varepsilon$
is a parameter such that $0\leq\varepsilon\leq |c|/2$.
Let $u(t,t_0,\varepsilon)$ denote the solution
to \eqref{e2.8}-\eqref{e2.9}.
Assume that
\begin{itemize}
\item[(A1)] for each $\eta\neq 0$, and $t_0$ in $\mathbb{R}$,
the initial-value problem $\frac{du }{dt}= G(u, \eta )$,
$u(t_0)=\eta$ has a unique left-hand solution.
\end{itemize}
Moreover, assume that there exists a postive constant $M$
such that
\begin{equation}
p(t)>0, \quad \int_t ^{\alpha(t)}p(s)ds\leq M \quad
\text{for all } t\in\mathbb{R}\,, \label{e2.10}
\end{equation}
where $\alpha(t)$ is defined by \eqref{e1.1a}.
Then there exists a positive constant $\mu$,
independent of $t_0$ and $\varepsilon$,
such that
$$
u(t,t_0,\varepsilon)\leq c+\varepsilon-\mu, \quad\text{for }
t_0\leq t\leq\alpha(t_0)\,.
$$
\end{lemma}

\begin{proof}
 The change of variables
\begin{equation}
s=\int_{t_0}^{t}p(\xi)d\xi, \quad v(s)=u(t)  \label{e2.11}
\end{equation}
 transform \eqref{e2.8}--\eqref{e2.9} into
\begin{gather}
\frac{d v}{ds}=G(v(s),c+\varepsilon),\quad s\geq 0,  \label{e2.8p}\\
v(0)=u_0, \quad u_0<c . \label{e2.9p}
\end{gather}
From Lemma \ref{lem2.3}, there exists a unique solution $v(s,\varepsilon)$
defined on $[0,\infty)$.
Since $ c+\varepsilon$ is a continuous function in $\varepsilon$,
 it follows that
$$
\psi(\varepsilon)=(c+\varepsilon)-\bar{u}(M,\varepsilon)
 $$
is also a continuous function in $\varepsilon$.
Note that because $G(u,u)=0$, the constant  $c+\varepsilon$
is also solution to \eqref{e2.8p}.
From the uniqueness of the solutions to
\eqref{e2.8p}--\eqref{e2.9p}, and $u_0<c<c+\varepsilon$,
 we obtain
\begin{equation}
v(s,\varepsilon)<c+\varepsilon\quad \text{for all }
s\in[0,+\infty).\label{e2.12}
\end{equation}
This implies $\psi(\varepsilon)>0$.
Let
\begin{equation}
\mu=\min_{0\leq\varepsilon\leq|c|/2}
\psi(\varepsilon)>0.\label{e2.13}
\end{equation}
It follows from \eqref{e2.12},  \eqref{e2.8p},
$G(u,u)=0$, and $G(u,v)$ being non-increasing in $u$,
that $\frac{\partial v}{\partial s}v(s,\varepsilon) \geq 0$,
and
$$
v(s,\varepsilon)\leq v(M,\varepsilon)\quad
\text{for all }s\in[0,M],
$$
which implies
\begin{equation}
v(s,\varepsilon)\leq c+\varepsilon-\psi(\varepsilon)
\leq c+\varepsilon-\mu\quad
\text{for all }s\in[0,M].\label{e2.14}
\end{equation}
By the relationship between  the initial value problems
\eqref{e2.8}-\eqref{e2.9} and  \eqref{e2.8p}--\eqref{e2.9p},
it follows from  \eqref{e2.10} that
\begin{equation}
u(t,t_0,\varepsilon)\equiv v(s(t),\varepsilon) \quad
\text{for } t_0\leq t\leq\alpha(t_0).\label{e2.15}
\end{equation}
Again from \eqref{e2.10} and \eqref{e1.1a},  we obtain
\begin{equation}
s(\alpha(t_0))\leq M.\label{e2.16}
\end{equation}
By \eqref{e2.14}, \eqref{e2.15} and \eqref{e2.16},  we have
$$
u(t,t_0,\varepsilon)\leq c+\varepsilon-\mu\quad
\text{for } t_0\leq t\leq\alpha(t_0).
$$
Since $M$ is independent of $t_0$, and $\mu$
is  independent of $t_0$ and $\varepsilon$,
the proof is complete.
\end{proof}

By a similar argument, we can prove the following result.


\begin{lemma} \label{lem2.5}
 Suppose that $G(u,u)\equiv 0$ for all $u\in\mathbb{R}$,
$G(u,v)$ is  non-increasing in $u$, and
 non-decreasing in $v$. Consider the initial-value
problem
\begin{gather}
\frac{du }{dt}=p(t)G(u,c-\varepsilon),   \label{e2.17}\\
u(t_0)=u_0,\quad u_0>c,  \label{e2.18}
\end{gather}
where $c$ is a nonzero constant and $\varepsilon$
is a parameter such that $0\leq\varepsilon\leq |c|/2$.
Denote by $u=u(t,t_0,\varepsilon)$
be the solution of the initial value problem.
If (A1) and \eqref{e2.10} hold,
then there exists a positive constant $\nu$ independent
of $t_0$ and $\varepsilon$
such that
$$
u(t,t_0,\varepsilon)\geq (c-\varepsilon)+\nu\quad
\text{for } t_0\leq t\leq\alpha(t_0).
$$
\end{lemma}


\begin{remark} \label{rmk2.2} \rm
If (A1) holds for all $\eta\in\mathbb{R}$ and $\varepsilon\in [0, 1]$,
using the method  in the proof of Lemma \ref{lem2.4},
we can  show that the conclusions in Lemmas \ref{lem2.4} and
\ref{lem2.5} hold
for any $c\in\mathbb{R}$.
\end{remark}

\section{Main results}

\begin{theorem} \label{thm3.1}
 Assume that $f(t,u,v)$ non-increasing in $u$, and
\begin{equation} \label{e31i}
 f(t,u,v)\leq p(t)G(u,v)+q_1(t)u+q_2(t)v+q_{3}(t).
\end{equation}
where $G(u,v)\in C(\mathbb{R}\times\mathbb{R})$ and
$p,q_i\in C(\mathbb{R})$ satisfying the following conditions:
$G(u,v)$ is non-increasing in $u$, and non-decreasing in $v$;
$G(u,u)\equiv 0$ for all $u\in\mathbb{R}$;
$q_i\in L^1[0,\infty)$ $(i=1, 2, 3)$;
and $p, q_1, q_2$ are non-negative;
and {\rm (A1)} and \eqref{e2.10} hold.
Then every solution of \eqref{e1.1} is
bounded above. Furthermore, if $\limsup_{t\to \infty} x(t)\neq 0$,
then $x(t)$ tends to either a constant or
to $-\infty$ as $t\to \infty$.
\end{theorem}

\begin{proof}
 We first prove that every solution of \eqref{e1.1} is
bounded above. Let
$$
y_1(t)=\max\{\max_{t_0-r(t_0)\leq s\leq t}x(s),\,1\}, \quad
S_1=\{t\geq t_0: y_1(t)=x(t)\}.
$$
Let $D^{+}$ denote the upper right derivative.
Then $D^{+}y_1(t)=0$ for $t\in[t_0,+\infty)\backslash S_1$,
and $D^{+}y_1(t)\leq\max\{x'(t),0\}$  a.e. on $S_1$.
From \eqref{e1.1} and  \eqref{e31i},
\begin{equation} \label{e3.1}
\begin{split}
x'(t)
&\leq p(t)G(x(t),x(t-r(t)))+q_1(t)x(t)+q_2(t)x(t-r(t))+q_{3}(t)\\
&\leq p(t)G(x(t),y_1(t))+q_1(t)y_1(t)+q_2(t)y_1(t)+q_{3}(t)\\
&\leq p(t)G(x(t),y_1(t))+q_1(t)y_1(t)+q_2(t)y_1(t)+|q_{3}(t)|
\quad \forall t\geq t_0.
\end{split}
\end{equation}
Since $G(u,u)\equiv 0$ for all $u\in\mathbb{R}$,
and $ D^{+}y_1(t)\leq\max\{x'(t),0\}$ a.e.  on $[t_0,+\infty)$,
we obtain
$$
D^{+}y_1(t)\leq q_1(t)y_1(t)+q_2(t)y_1(t)+|q_{3}(t)|\quad
\text{a.e.  on } [t_0,+\infty).
$$
From $y_1(t)\geq1$, we have
$$
\frac{D^{+}y_1(t)}{y_1(t)}\leq q_1(t)+q_2(t)+|q_{3}(t)|\quad
\text{a.e.  on }[t_0,+\infty).
$$
Again from the monotonicity of $y_1(t)$, we obtain that $y_1(t)$
is differentiable almost everywhere on $[t_0,\infty)$.
Thus
$$
\ln\big(\frac{y_1(t)}{y_1(t_0)}\big)
\leq \int_{t_0}^{+\infty}q_1(t)dt
     +\int_{t_0}^{+\infty}q_2(t)dt+\int_{t_0}^{+\infty}|q_{3}(t)|dt
    <+\infty\quad \forall t\geq t_0,
$$
which implies $y_1(t)$ is  bounded above; thus  $x(t)$
is also bounded above.
Set $A=\limsup_{t\to \infty}x(t) <\infty$.
If $\limsup_{t\to \infty}x(t)=-\infty$,
then $\lim_{t\to \infty}x(t)=-\infty$,
which implies that Theorem \ref{thm3.1} holds.

Next we  assume that $A$ is a nonzero real number
and show that $\lim_{t\to \infty}x(t)=A$.
By contradiction, assume that $\lim_{t\to \infty}x(t)$
does not exist.
For each  $\mu_1\in [0, |A|/2]$, there exists
let $t^*>t_0$  large enough such that
\begin{gather}
x(t)\leq A+\mu_1, \quad x(t-r(t))\leq A+\mu_1\quad \forall t\geq t^*,
 \label{e3.2}\\
\int_{t^*}^{+\infty} [(q_1(t)+q_2(t))A+\mu_1+|q_{3}(t)|]dt\leq\mu_1.
\label{e3.3}
\end{gather}
For $t\geq t^*$, let
\begin{equation}
n_1(t)=x(t)-\int_{t^*}^{t}[(q_1(s)+q_2(s))A+\mu_1+|q_{3}(s)|]ds\,.
 \label{e3.4}
\end{equation}
Obviously, $n_1(t)$ is bounded above, and
$\lim_{t\to \infty}n_1(t)$ does not exist.
Let $B=\limsup_{t\to \infty}n_1(t)$ and
$b=\limsup_{t\to \infty}n_1(t)$; Thus $b<B\leq A$.
For $b<H<B$, there exists a sequence $\{t_m\}_{m=1}^\infty$
satisfying $n_1(t_{m})=H$, $t_m>t^*$ and $t_m\to \infty$ as
$m\to \infty$.
It follows from \eqref{e31i} and \eqref{e3.2}
that
$$
x'(t)\leq p(t)G(x(t),A+\mu_1)+(q_1(t)+q_2(t))A
+\mu_1+|q_{3}(t)|\quad \text{for all } t\geq t^*.
$$
 From \eqref{e3.4}, we obtain $x(t)\geq n_1(t)$
 and
\begin{equation}
n_1'(t)\leq p(t)G(n_1(t),A+\mu_1)\quad
\text{for all } t\geq t^*.\label{e3.5}
\end{equation}
For each $m$, we consider the initial-value problem
\begin{equation}
\begin{gathered}
u'(t)=p(t)G(u(t),A+\mu_1) \\
u(t_m)=H, \quad H<A\,.
\end{gathered} \label{e3.6}
\end{equation}
By Lemma \ref{lem2.4}, this problem has a  unique solution $u=u(t)$ on
$[t_m,+\infty)$, and there exists a $\mu>0$
independent of $t_m$ and of $\mu_1$,
such that
$$
u(t)\leq A+\mu_1-\mu \quad \text{for } t_m\leq t\leq\alpha(t_m).
$$
Then, by the comparison theorem and \eqref{e3.5}, we  obtain
$$
n_1(t)\leq u(t)\leq A+\mu_1-\mu\quad \text{for }
 t_m\leq t\leq\alpha(t_m),
$$
thus $ x(t)\leq A+2\mu_1-\mu$ for $t_m\leq t\leq\alpha(t_m)$.
Choosing $\mu_1\in (0,\mu/4]$, we have
\begin{equation}
x(t)\leq A-\frac{\mu}{2}\quad \text{for }
t_m\leq t\leq\alpha(t_m),\; m=1,2,\dots. \label{e3.7}
\end{equation}

    On the other hand, define
$$
y_2(t)=\max_{\lambda(t)\leq s\leq t}x(s),\quad
S_2=\{t:t\in[t^*,\infty),y_2(t)=x(t)\}.
$$
Then $D^{+}y_2(t)\leq0$ for all $t\in[t^*,\infty)\backslash S_2$,
and $D^{+}y_2(t)\leq\max\{x'(t),0\}$ for all $t\in S_2$.
Hence
\begin{equation}
D^{+}y_2(t)\leq (q_1(t)+q_2(t))(A+\mu_1)+|q_{3}(t)|\quad \forall
t\geq t^*. \label{e3.8}
\end{equation}
For $t\geq t^*$, denote
$$
n_2(t)=y_2(t)-\int_{t^*}^{t}[(q_1(s)+q_2(s))(A+\mu_1)+|q_{3}(s)|]ds.
$$
From \eqref{e3.8}, we obtain
$D^{+}n_2(t)\leq0$  for all $t\geq t^*$;
therefore, $n_2(t)$ is non-increasing.
Since $\limsup_{t\to \infty}x(t)>-\infty$, 
$$
\lim_{t\to \infty}y_2(t)=\lim_{t\to \infty}n_2(t)
+ \lim_{t\to \infty}\int_{t^*}^{t}
[(q_1(s)+q_2(s))(A+\mu_1)+|q_{3}(s)|]ds
$$
exists as real number. From the definition of $y_2$ and
the the fact that 
$\{s:\lambda(t)\leq s\leq t, t\geq t_0\}\supset [t_0,\infty)$,
it follows that
\begin{equation}
\lim_{t\to \infty}y_2(t)
=\lim_{t\to \infty}\max_{\lambda(t)\leq s\leq t}x(s)
=\limsup_{t\to \infty} x(t)=A. \label{e3.9}
\end{equation}
Since $\lambda(t)\to+\infty$ as $t\to+\infty$, for each
$t_m$ there exists $t_m'$ such that
$t_m=\lambda(t_m')$.
Then $\alpha(t_m)=\alpha(\lambda(t_m'))\geq t_m'$ and
$t_m'\geq t_m\geq t^*$.
By \eqref{e3.7},
\begin{equation}
y_2(t_m')\leq A-\frac{\mu}{2}\quad \text{for  }
 m=1,2,\dots.\label{e3.10}
\end{equation}
However, \eqref{e3.9} implies
$\lim_{m\to+\infty}y_2(t_m')=A$
which contradicts \eqref{e3.10}.
Hence $\lim_{t\to \infty}x(t)$ exists, and
$\lim_{t\to \infty}x(t)=A$. This completes the proof.
\end{proof}

In a similar fashion, by using Lemma \ref{lem2.5}, we can show the following
result.

\begin{theorem} \label{thm3.2}
Assume $f(t,u,v)$ non-increasing in  $u$, and
\begin{equation} \label{e32i}
 f(t,u,v)\geq p(t)G(u,v)+q_1(t)u+q_2(t)v+q_{3}(t),
\end{equation}
where $G(u,v)\in C(\mathbb{R}\times\mathbb{R})$ and
$p,q_i\in C(\mathbb{R})$ satisfying the following conditions:
$G(u,v)$  non-increasing in $u$, and non-decreasing in $v$;
$G(u,u)\equiv 0$ for all $u\in\mathbb{R}$;
$q_i\in L^1[0,\infty)$ $(i=1, 2, 3)$, $p, q_1, q_2$ are
nonnegative;
and {\rm (A1)} and \eqref{e2.10} hold.
Then every solution of \eqref{e1.1} is
bounded below. Furthermore, if $\limsup_{t\to \infty} x(t)\neq 0$,
then $x(t)$ tends to either a constant or to $\infty$ as $t\to \infty$.
\end{theorem}

\begin{theorem} \label{thm3.3}
Consider the differential equation
\begin{equation}
 x'(t)=p(t)G(x(t),x(t-r(t)))+q_1(t)x(t )+q_2(t)x(t-r(t))+q_{3}(t),
\label{e3.12}
\end{equation}
where $G(u,v)\in C(\mathbb{R}\times\mathbb{R})$ and
$p,q_i\in C(\mathbb{R})$ satisfying the following conditions:
$G(u,v)$ is non-increasing in $u$, and non-decreasing in $v$;
$G(u,u)\equiv 0$ for all $u\in\mathbb{R}$;
$q_i\in L^1[0,\infty)$ $(i=1, 2, 3)$, $p, q_1, q_2$ are
nonnegative;
and  {\rm (A1)} and \eqref{e2.10} hold.
Then every solution of \eqref{e3.12} tends to  a constant as
$t\to\infty$.
\end{theorem}

The proof of the above theorem follows immediately from
Theorems \ref{thm3.1} and \ref{thm3.2}.


\begin{remark} \label{rmk3.1} \rm
 Let   $G(u,v)=-u^{\theta}+v^{\theta}$, where $\theta$
is the ratio of two odd positive integers.
Then $G(u,v)$ is strictly decreasing in  $u$,  and is
strictly increasing in $v$.
Moreover, $G(u, \eta)$ is continuously differentiable when
$u \neq 0$.
 Applying Cauchy's uniqueness and existence theorem, we
conclude that   assumption (A1) holds.
Therefore,  Theorem \ref{thm3.3}  confirms the
Bernfeld-Haddock conjecture.
\end{remark}

From Remark \ref{rmk2.2}, and using a similar argument as in the
proof of Theorem \ref{thm3.1}, we can also show the following result,
under the assumption
\begin{itemize}
\item[(A1')] For each $\eta$ and $t_0$ in $\mathbb{R}$, the
initial-value problem
$\frac{du }{dt}= G(u,\eta)$, $u(t_0)=\eta$
has a unique left-hand solution.
\end{itemize}

\begin{theorem} \label{thm3.4}
Assume {\rm (A1')}. Under the hypotheses of Theorem \ref{thm3.1},
 every solution of \eqref{e1.1} is bounded above,
and tends to either a constant or to $-\infty$,  as $t\to \infty$.
\end{theorem}

\begin{theorem} \label{thm3.5}
Assume {\rm (A1')}. Under the hypotheses of Theorem \ref{thm3.2},
every solution of \eqref{e1.1} is bounded below, and tends
to either a constant or to $+\infty$, as $t\to \infty$.
\end{theorem}

The proofs of the two theorems above are similar to the
proof of Theorem \ref{thm3.1}: Replace
$\mu_1\in [0,|A|/2]$ with $\mu_1\in [0,1]$, and then
use Remark \ref{rmk2.2}.

\begin{remark} \label{rmk3.3} \rm
Note that the results in \cite{c1,c2} can be obtained
only by assuming condition (A1'),  and the strengthened
condition \eqref{e2.10}.
Since the function $G(u,v)$ in this article satisfies weaker
conditions than those in \cite{c1,c2}, their results there are special
cases in this article.
When $r(t)$ is constant and $p(t)$ is a bounded and positive
function, \eqref{e2.10} holds naturally.
Hence, our results  include those in \cite{d1,y1},
and naturally extend the Bernfeld-Haddock conjecture.
\end{remark}

\subsection*{Acknowledgments}
The author would like to thank the anonymous referees for their
helpful comments and suggestions. 
This work was supported by the
Scientific Research Fund of Hunan Provincial Natural Science
Foundation of PR China (Grants No. 11JJ6006,  No. 10JJ6011), and
the Natural Scientific Research Fund of Hunan Provincial Education
Department of PR China (Grants No. 10C1009, No. 09B072).

\begin{thebibliography}{0}

\bibitem{b1} S. R. Bernfeld and J. R. Haddock;
\emph{A variation of Razumikhims method
for retarded functional equstions}, Nonlinear Systems and
Applications, An International Conference, Acadimic Press New
York, 1977, 561-566.

\bibitem{c1} B. S. Chen;
\emph{Asymptotic behavior of a class of nonautonomous
  retarded differential equations} (in Chinese),
Chinese Science Bulletin, 1988, 6: 413-415.

\bibitem{c2}  B. S. Chen;
\emph{Asymptotic behavior of solutions of some infinite
  retarded differential equations} (in Chinese),  Acta Math. Sinica,
  1990, 3: 353-358.

\bibitem{c3} E. A. Coddington and N. Levinson;
\emph{Theory of Ordinary Differential Equations},
New York: McGraw-Hill, 1955, 53-54.

\bibitem{d1} T. Ding;
\emph{Asymptotic behavior of  solutions of some retarded differential
   equations}. Science in China (Series A), 1982,
Vol. XXV No. 8: 363-370.

\bibitem{t1}  Canqin Tang, Bingwen Liu;
\emph{Existence and uniqueness of solutions
for a kind of ordinary differential equations} (in Chinese), College
Mathematics, 27(2)  (2011): 123-124.

\bibitem{y1} T. S. Yi, L. H. Huang;
\emph{A generalization of the  Bernfeld-Haddock
conjecture and its proof} (in Chinese), Acta Math. Sinica, 2007,
50(2): 261-270.


\end{thebibliography}

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