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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 69, pp. 1--12.\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-2017/69\hfil Asymptotic behavior of solutions]
{Asymptotic behavior of solutions to a non-autonomous
system of two-dimensional differential equations}

\author[S. Xiao \hfil EJDE-2017/69\hfilneg]
{Songlin Xiao}

\address{Songlin Xiao \newline
School of Mathematics and Information Science, Guangzhou University,
Guangzhou 510006, China}
\email{xiaosonglin2017@163.com, Phone +8602039366859, Fax +8602039366859}

\thanks{Submitted February 21, 2017. Published March 10, 2017.}
\subjclass[2010]{34C12, 39A11}
\keywords{Bernfeld-Haddock conjecture;
non-autonomous  differential equation; 
\hfill\break\indent time-varying delay; asymptotic behavior}

\begin{abstract}
 This article concerns the two-dimensional Bernfeld-Haddock conjecture
 involving non-autonomous delay differential equations. Employing the
 differential inequality theory, it is shown that every  bounded
 solution  tends to a constant vector as $t\to \infty$.
 Numerical simulations are carried out to verify our theoretical findings.
\end{abstract}

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

\section{Introduction}

In  1976,  Bernfeld and Haddock \cite{b1} proposed  the following conjecture.
\begin{quote}
Every solution of the
delay differential equation
\begin{equation}
  x'(t)=-x^{1/3}(t)+x^{1/3}(t-r),\label{e1.1}
\end{equation}
  where $r>0$, tends to a constant as $t\to \infty$.
\end{quote}
To confirm the above conjecture, variants of the above equation,
which have been used as models for   some population growth and the
spread of epidemics,   have
  received considerable attention (see, for example, 
\cite{c1,c2,d1,d2,d3,x1, y1,y2,y3,y4,z1,z2}  and   the references  therein).
In particular, the
asymptotic behavior of the autonomous equations
\begin{equation}
  x'(t)=-F(x(t))+G(x(t-r )),       \label{e1.2}
\end{equation}
and
\begin{equation}
\begin{gathered}
x'_1(t)=-F_1(x_1(t))+G_1(x_2(t-r_2)),\\
x'_2(t)=-F_2(x_2(t))+G_2(x_1(t-r_1)),
\end{gathered}  \label{e1.3}
\end{equation}
 and  non-autonomous equations
\begin{equation}
      x'(t)  =p(t)[-x^{1/3}(t)+ x^{1/3}(t-r_3(t))] ,  \label{e1.4}
\end{equation}
have been  studied in \cite{d1,d2,d3,x1,y1,y2,y3,y4,z1} and 
\cite{c1,c2,z2}, respectively.
Here, $r,  r_1$ and $r_2$ are positive constants, $F $, $G $, 
$F_i$, $G_i \in C(\mathbb{R}, \mathbb{R})$,  $F $  and $F_i$
are nondecreasing on $\mathbb{R} $,
$p, r_3\in C(\mathbb{R}, (0, +\infty) )$, $ r_3(t) >0$, $p(t)>0$, $i=1,2$.

Furthermore, it was shown in the above   mentioned references that each bounded
solution of above equations   tends to a constant solution as $t\to\infty$.
We also found that the main methods mentioned above include two kinds,
one is the analysis method   of the monotone dynamical system 
\cite{x1,y1,y2,y3,y4,z1},
the other is the differential inequality analysis   technique 
\cite{c1,c2,d1,d2,d3,z1}.
 As pointed out in \cite{y4}, there were some errors  in several existing works
in \cite{c1,c2,d1,d2,d3,z1}, and the uniqueness of the left-hand solution of
the following differential equation
\begin{equation}
\begin{gathered}
x'(t)=-F(x(t))+F(c),\\
x(t_0)=x_0 \quad  \text{for } t_0,x_0\in \mathbb{R} ,
\end{gathered} \label{e1.5}
\end{equation}
played  a crucial role  in the discussion of above  references.   
Consequently, to improve the  proof in \cite{c1,c2,d1,d2,d3}, Ding
adopted  the following  additional  assumption:
\begin{itemize}
\item [(A1)]  If $c\neq 0$ then the solution
to \eqref{e1.5} on the interval $(t_0-\delta, t_0]$ is unique,
where $\delta$ is a positive constant
 (this soluution is called left-hand solution in \cite{y1})
\end{itemize}
This assumption is also included in \cite[Appendix]{y4}.

 On the other hand,  delays in population and
ecology models are usually time-varying and  usually can be
generalized as the   non-autonomous functional differential
equation. Thus, we can generalize the equation \eqref{e1.3} in
two-dimensional Bernfeld-Haddock conjecture to the following
non-autonomous delay differential equations:
\begin{equation}
\begin{gathered}
x'_1(t)=\gamma_1(t)[-F_1(x_1(t))+G_1(x_2(t-\tau_2(t)))],\\
x'_2(t)=\gamma_2(t)[-F_2(x_2(t))+G_2(x_1(t-\tau_1(t)))],
\end{gathered} \label{e1.6}
\end{equation}
and $F_i, G_i\in C(\mathbb{R}, \mathbb{R}),\gamma_i, \tau_i
\in  C(\mathbb{R}, (0,  +\infty) )$, $i=1,2$. Moreover, it is assumed
that $F_i$ is strictly increasing on $\mathbb{R }$,
$F_i$ is continuous differentiable on $\mathbb{R}\setminus \{0\}$,  and
\begin{equation}
F_i(0)=0, \quad F_i'(x)>0  \text{for all }  x\in \mathbb{R}\setminus \{0\},\;
 i=1,2. \label{e1.7}
\end{equation}
for $i=1,2$. It is worth
noting that system \eqref{e1.6} includes equation \eqref{e1.2} as a special case.
In fact, if $\tau_1(t)=\tau_2(t)=r $  and consider the synchronized
solutions of \eqref{e1.6} with $x_1(t)=x_2(t)=\varphi_0(t)$ for
$t\in [-r,0]$, then system \eqref{e1.6} reduces to equation \eqref{e1.2}.
Obviously, \eqref{e1.1}, \eqref{e1.3} and \eqref{e1.4} are the special
 cases of \eqref{e1.6}. It is well
known that a non-autonomous delay differential equation generally
does not generate a semiflow and hence methods for differential
equations with constant delays  \cite{x1,y1,y2,y3,y4,z1}
are not suitable for \eqref{e1.6}.
Moreover, the irregularity of the set of equilibria seems to cause
some difficulties in the study of system \eqref{e1.6} now. Hence, to the
best of our knowledge, there is no result on the asymptotic behavior
of solutions  of non-autonomous delay differential equations \eqref{e1.6}
before.

Motivated by the above discussions, we aim  to employ a novel
argument to  prove   that   every   solution of \eqref{e1.6} tends to
a constant vector as $t\to+\infty$.

Throughout this article, for a bounded and continuous function $g$
defined on $\mathbb{R}$, we denote
\[
g^{+}=\sup_{t\in \mathbb{R}}g(t)\quad \text{and} \quad
g^{-}=\inf_{t\in \mathbb{R}}g(t).
\]
It will be always assumed that
$$
r=\max\{\tau_1^{+}, \tau_2^{+}\}\geq \tau^{*}=\min\{\tau_1^{-}, \tau_2^{-}\}>0, \quad
0<\gamma^{-}_i\leq \gamma^{+}_i<+\infty, \quad
i\in J=\{1,2\}.
$$
We will  denote
 $C=C([-\tau_1^{+},0],\mathbb{R} )\times C([-\tau_2^{+},0],\mathbb{R} )$
as the Banach space equipped  with a supremum norm.  We define the
initial condition
\begin{equation}
x_i(t_0+\theta)=\varphi_i(\theta), \quad
\theta\in [-\tau_i^{+},0],  \quad t_0 \in \mathbb{ R}, \quad
\varphi=(\varphi_1,\varphi_2)\in C  , \quad i\in J. \label{e1.8}
\end{equation}
We write $ x(t; t_0, \varphi)=(x_1(t; t_0, \varphi)$,
$x_2(t; t_0, \varphi)) $ to denote the solution of the initial value problem
\eqref{e1.6} and \eqref{e1.8}. Also, let $[t_0,\eta(\varphi))$ be the maximal
right-interval of existence of $x(t; t_0, \varphi)$.

The remaining  of this paper is organized as follows. In Section 2,
we recall some relevant results, and give a  detailed proof on
the     boundedness and global existence of every solution
  for \eqref{e1.6} with the initial
 condition   $\eqref{e1.8}$. Based on the preparation in Section 2,
  we state and prove our main result  in Section 3.
In Section 4, we give some examples to
illustrate the effectiveness of the obtained results by numerical
simulations.

\section{Preliminary results}

Assume  that $F:   \mathbb{R}  \to  \mathbb{R}  $ is   continuous
and strictly increasing, and
\begin{equation}
\begin{gathered}
F(0)=0, \quad F(x) \text{ is continuous differentiable on }
 \mathbb{R}\setminus  \{0\}, \\
 F'(x)>0  \text{ for all } x\in \mathbb{R}\setminus  \{0\}.
\end{gathered} \label{e2.1}
\end{equation}
Then, $F$ satisfies {\rm (A1)}.  From
\cite[Lemma 2.1, Propositions 4* and 5*]{l1}, we have the following results.

\begin{proposition} \label{prop2.1}
Consider the differential equation
\begin{equation}
    u'=-F(u)+F(c+\varepsilon),       \label{e2.2}
\end{equation}
where $c\neq 0$ is a given constant, $\varepsilon$  is a parameter satisfying
$0\le\varepsilon\le  |c|/2$, and the initial condition is
\begin{equation}
    u(t_0)=u_0 \quad (u_0<c).     \label{e2.3}
\end{equation}
Let $u=u(t;t_0,u_0)$ be the solution of the initial value problem
\eqref{e2.2} and \eqref{e2.3}, and $\alpha>0$ be a given constant.
Then there exists a   positive real number $\mu$ independent of $t_0$ and
$\varepsilon$ such that
$$
  (c+\varepsilon)-u(t;t_0,u_0)\ge\mu>0
  \quad \text{for } t\in [t_0,t_0+\alpha].
$$
\end{proposition}

\begin{proposition} \label{prop2.2}
   Consider the differential equation
\begin{equation}
    u'=-F(u)+F(c-\varepsilon),       \label{e2.4}
\end{equation}
where $c\neq 0$ is a given constant, $\varepsilon$   is a parameter
satisfying  $0\le\varepsilon\le   \frac{|c|}{2}$. Moreover, assume
the initial   condition
\begin{equation}
    u(t_0)=u_0 \quad  (u_0>c).     \label{e2.5}
\end{equation}
  Let $u=u(t;t_0,u_0)$ be the solution of the initial value problem
\eqref{e2.4} and \eqref{e2.5}, and $\alpha>0$ be a given constant.
Then there exists a   positive real number $\nu$ independent of $t_0$ and
$\varepsilon$ such that
\[
  u(t;t_0,u_0)-(c-\varepsilon)\ge \nu>0 \quad
 \text{for } t\in [t_0,t_0+\alpha].
\]
\end{proposition}

One can easily see that $F (x)=x^{1/3}$ satisfies \eqref{e2.1} and hence
 Propositions  \ref{prop2.1} and  \ref{prop2.2} hold in this case.


\begin{lemma}[see  \cite{l2,z2}] \label{lem2.1}
 Let $t_0\in \mathbb{R}$,
$\beta>0$, $\bar{h}\in C([t_0, t_0+\beta]\times  \mathbb{R}, \mathbb{R})$,
and $\bar{h}$ is non-increasing with respect to the
second variable. Then the initial value problem
\begin{gather*}
\frac{dx}{dt}=\bar{h}(t,x)\\
x(t_0)=x_0
\end{gather*}
has a unique solution $x=x(t)$ on  $[t_0, t_0+\beta]$.
\end{lemma}

\begin{lemma} \label{lem2.2}
 Let $\varphi \in C$. Then $x (t;t_0,\varphi)$ exists and is unique on
$[t_0, \infty)$.
\end{lemma}

\begin{proof}
Let $x (t)=x (t; t_0,\varphi)$.  We will
show that $x (t)$ exists and is unique on $[t_0,t_0+\tau^{*}]$.
To see this, let
\begin{gather*}
d_1(t)=G_1(x_2(t-\tau_2(t) ))=G_1(\varphi_2(t-\tau_2(t)-t_0 )), \\
d_2(t)=G_2(x_1(t-\tau_1(t) ))=G_2(\varphi_1(t-\tau_1(t)-t_0 ))
\end{gather*}
for any $t\in [t_0,t_0+\tau^{*}]$. Consider the solution
$x_i(t)$ of the  initial value problem
\begin{gather*}
x'_i(t) = \gamma_i(t)[-F_i(x_i(t))+d_i(t)],\\
x_i(t_0 ) = \varphi_i(0)
\end{gather*}
where $i\in J$. By Lemma \ref{lem2.1},  $x_i(t)$ exists and is unique on
$[t_0,t_0+\tau^{*}]$, $i\in J$. Hence, $x (t )$ exists and is
unique on $[t_0,t_0+\tau^{*}]$.  It follows from induction that
$x (t)$ exists and is unique on $[t_0,+\infty)$. The proof is complete.
\end{proof}

\begin{lemma} \label{lem2.3}
 Let $\varphi \in C$, and
$F_i(u)=G_i(u)$ for all $ u\in \mathbb{R}, i\in J$.
Then $x (t; t_0,\varphi)$ exists and is unique on $[t_0, +\infty)$.
Moreover, $x (t; t_0,\varphi)$ is bounded on $[t_0, +\infty)$.
\end{lemma}

\begin{proof}
 By Lemma \ref{lem2.2},
$x(t)=x (t; t_0,\varphi)$ exists and is unique on   $[t_0, +\infty)$.
Furthermore,  we claim that
$$
\alpha<x_i(t;t_0,\varphi)<\beta \ \text{ for   all } \ t\in [t_0,
+\infty), i\in J,
$$
where $\alpha$ and $\beta$ are two constants such that
$\alpha<\varphi_i(s)<\beta$ for all $s\in [-\tau_i^{+},  0]$,
$i\in J$. Suppose that the claim is not true. Then one of the
following two cases must occur:

\noindent\textbf{Case I.}  There exist $i^{*}\in J$  and  $\theta_1>t_0$ such
that
\begin{equation}
x_{i^{*}}(\theta_1;t_0,\varphi)=\beta   \text{ and  } x_{j}(t;t_0,\varphi)<\beta
\text{  for   all } t \in[t_0-\tau_{j}^{+},\theta_1),\; j\in J.  \label{e2.6}
\end{equation}

\noindent\textbf{Case II.}
There exist   $i^{*}\in J$  and  $\theta_2>t_0$ such     that
\begin{equation}
x_{i^{*}}(\theta_2;t_0,\varphi)=\alpha    \text{ and  }
\alpha< x_{j}(t;t_0,\varphi)  \text{ for all  } t \in[t_0-\tau_{j}^{+}, \theta_2),
\; j\in J. \label{e2.7}
\end{equation}
When Case I holds,  in view of \eqref{e1.6} and \eqref{e2.6},  we have
\begin{align*}
0&\leq x_{i^{*}} '(\theta_1)\\
&=\gamma_{i^{*}}(\theta_1)[-F_{i^{*}}(x_{i^{*}}
 (\theta_1))+F_{i^{*}}(x  _{\bar{i}^{*}}(\theta_1-\tau _{\bar{i}^{*}}(\theta_1))]
 \\
& <\gamma_{i^{*}}(\theta_1)[-F_{i^{*}}(\beta)+F_{i^{*}}(\beta)]\\
&=0, \quad \bar{i}^{*}\in J\setminus \{i^{*}\},
\end{align*}
which is a  contradiction.

When Case II holds, similarly we  have
\begin{align*}
0&\geq x_{i^{*}}'(\theta_2)\\
&=\gamma_{i^{*}}(\theta_2)[-F_{i^{*}}(x_{i^{*}}
 (\theta_2))+F_{i^{*}}(x  _{\bar{i}^{*}}(\theta_2-\tau _{\bar{i}^{*}}(\theta_2))]\\
& > \gamma_{i^{*}}(\theta_2)[-F_{i^{*}}(\alpha)+F_{i^{*}}(\alpha)]\\
&=0, \quad \bar{i}^{*}\in J\setminus \{i^{*}\},
\end{align*}
which is also a contradiction. Thus we have proved the claim and completed
the proof.
\end{proof}

\section{Main result}

The purpose of this section is to show  that every bounded solution
of \eqref{e1.6} tends to a constant  as $t\to +\infty$, which is
our main result in this paper.

\begin{theorem} \label{thm3.1}
 Assume either  $G_i\ge F_i$ or $G_i\le F_i$, $i\in J$,
 Then every bounded solution  of the
initial value problem \eqref{e1.6} and \eqref{e1.8} tends to a constant
vector as $t\to +\infty$.
\end{theorem}

\begin{proof}
Note that Theorem \ref{thm3.1} is equivalent to the statement:
If either  $G_i\ge F_i (i\in J)$ or $G_i\le F_i  (i\in J)$
holds and $\varphi\in C$  such that $x_i(t; t_0,\varphi)$
   is bounded for all $t\in \mathbb{R}$ and $i\in J$, then
$$
 l_i =\liminf_{t\to +\infty} x_i(t; t_0,\varphi)
=\limsup_{t\to +\infty} x_i(t; t_0,\varphi)=L_i, i\in J.
$$
We only consider the case where $G_i\le F_i (i\in J)$ since  the
case where $G_i\ge F_i (i\in J)$ can proved similarly.
Let
\begin{gather*}
x_i(t)= x_i(t;t_0,\varphi),\quad  \text{for  all }  t \ge t_0 , i\in J, \\
y_i(t)=\max_{t-r \leq s\leq t}x_i(s),  \quad
u_i(t)=\min_{t-r\leq s\leq t}x_i(s) \quad  \text{for  all }  t \ge t_0+r,\; i\in J, \\
y(t)=\max\{y_1(t), y_2(t)\}, u(t)=\min\{u_1(t), u_2(t)\}, \\
S=\{ t| t\in [t_0+r,  +\infty), y(t)= x_i(t ) \quad \text{for some } i\in J \}.
\end{gather*}

Firstly, we show $D^+y(t)\le 0$ for all $t\ge t_0+r$. We distinguish
two cases to finish the proof.
\smallskip

\noindent\textbf{Case 1.} $t\in [t_0+r,  +\infty)\setminus S$. Then there
exist $i_0\in J $ and $t^{*}\in [t-r, t)$ such that
$$
y(t)=y_{i_0}(t)=\max_{t-r\leq s\leq t}x_{i_0}(s)
=x_{i_0}(t^{*})>\max\{x_1(t), x_2(t)\}.
$$
From the continuity of $x_i(\cdot)$ at $t$, we can choose a positive
constant $\delta<r$ such that
$$
    x_i(s)<x_{i_0}(t^{*}) \quad \text{ for all  } s\in [t,
    t+\delta],\;  i\in J,
$$
which yields
$$
x_i(s)\leq x_{i_0}(t^{*})=\max_{t-r\leq
s\leq t}x_{i_0}(s)=y_{i_0}(t)=y (t) \quad \text{for all  } s\in
[t-r, t+\delta], \; i\in J.
$$
It follows that
\begin{align*}
y(t+h)&= \max\big\{\max_{t+h-r\leq
s\leq t+h}x_1(s),  \max_{t+h-r\leq s\leq t+h}x_2(s)\big\}\\
&\leq \max\big\{\max_{t -r\leq s\leq t+\delta}x_1(s),
 \max_{t -r\leq s\leq  t+\delta}x_2(s)\big\}\\
&\leq   \max_{t-r\leq s\leq t}x_{i_0}(s)=y_{i_0}(t)=y (t)  \text{ for all  }
 h\in (0, \delta),
\end{align*}
and hence
$$
D^{+}y(t)=\limsup_{h\to 0^{+}}\frac{y(t+h)-y(t)}{h}
 \leq\limsup_{h\to 0^{+}}\frac{y(t )-y(t)}{h}=0.
$$


\noindent\textbf{Case 2.}
$t\in   S$.  Then there exists $i_0\in J $ such that
$$
y(t)=y_{i_0}(t)=x_{i_0}(t)=\max_{t-r\leq s\leq t}x_{i_0}(s).
$$
Then  \eqref{e1.6} implies
\begin{align*}
0&\leq  x_{i_0}'(t)\\
&= \gamma_{i_0 }(t)[-F_{i_0 }(x_{i_0}(t))
 +G_{i_0 }(x_{\bar{i}_0 }(t-\tau_{\bar{i}_0 }(t)))]\\
&\leq \gamma_{i_0 }(t)[-F_{i_0 }(x_{i_0 }(t))
 +F_{i_0}(x_{\bar{i}_0 }(t-\tau_{\bar{i}_0 }(t)))]\\
&\leq \gamma_{i_0 }(t)[-F_{i_0 }(x_{i_0 }(t))+F_{i_0 }(x_{i_0}(t))]\\
&= 0 , \quad \text{where } \bar{i}_0\in J\setminus\{i_0\},
\end{align*}
which gives $ x_{i_0}'(t)=0$. Let $\rho=\frac{1}{2} \tau ^{*}$.
 Obviously, $\rho>0$. First we assume that $y(s)= x_{i_0}(s)$ for all
$s\in(t, t+\rho]$. Then we have
\begin{align*}
D^{+}y(t)&= \limsup_{h\to 0^{+}}\frac{y(t+h)-y(t)}{h}\\
&= \limsup_{h\to 0^{+}}\frac{y(t+h)-x_{i_0}(t)}{h}\\
&=  \limsup_{h\to 0^{+}}\frac{x_{i_0}(t+h)-x_{i_0}(t)}{h}\\
&= x_{i_0}^{'}(t)\\
&= 0, \quad \text{where } 0<h<\rho.
\end{align*}
Now assume that there exists $s_1\in (t, t+\rho]$ such that
$y(s_1)> x_{i_0 }(s_1)$. Consequently, one can show that
either
\begin{equation}
y(s_1)=y_{i_0 }(s_1)=\max_{s_1-r\leq s\leq s_1}x_{i_0}(s) \label{e3.1}
\end{equation}
or
\begin{equation}
y(s_1)=y_{\bar{i}_0 }(s_1)=\max_{s_1-r\leq s\leq s_1}x_{\bar{i}_0}(s)>y_{i_0 }(s_1),
  \quad \text{where }  \bar{i}_0\in J\setminus\{i_0\}, \label{e3.2}
\end{equation}
holds.

If \eqref{e3.1} holds,  we can choose a constant
$\tilde{t}\in [s_1-r,s_1)$ such that
$$
y(s_1)=x_{i_0 }(\tilde{t})=\max_{s_1-r\leq s\leq s_1}x_{i_0}(s).
$$
This, together with the fact that $t-r<s_1-r\leq t+\rho-r<t<s_1$, implies
 $$
x_{i_0}(\tilde{t})
\geq x_{i_0}(t )=y(t)=y_{i_0}(t)=\max_{t-r\leq s\leq
t}x_{i_0}(s).
$$
We claim that
\begin{equation}
x_{i_0}(\tilde{t}) = x_{i_0}(t )=y(t)=y_{i_0}(t).\label{e3.3}
\end{equation}
Otherwise, $x_{i_0}(\tilde{t})> x_{i_0}(t )$. Then $t<\tilde{t}<s_1$ and
\begin{align*}
0&\leq x_{i_0}'(\tilde{t})
=\gamma_{i_0}(\tilde{t})[-F_{i_0}(x_{i_0} (\tilde{t}))
 +G_{i_0}(x _{\bar{i}_0}(\tilde{t}-\tau_{\bar{i}_0}(\tilde{t})))] \\
&\leq \gamma_{i_0}(\tilde{t})[-F_{i_0}(x_{i_0} (\tilde{t}))+F_{ i_0}(x
_{\bar{i}_0}(\tilde{t}-\tau_{\bar{i}_0}(\tilde{t})))]\,.
\end{align*}
It follows that
$$
F_{i_0}(x_{i_0} (\tilde{t}))\leq F_{ i_0}(x
_{\bar{i}_0}(\tilde{t}-\tau_{\bar{i}_0}(\tilde{t})))
$$
and
\begin{equation}
 x _{\bar{i}_0}(\tilde{t}-\tau_{\bar{i}_0}(\tilde{t})) \geq x_{i_0}
(\tilde{t})>x_{i_0}(t ). \label{e3.4}
\end{equation}
Noting that $t-r\leq t-\tau_{\bar{i}_0}(\tilde{t})
<\tilde{t}-\tau_{\bar{i}_0}(\tilde{t})<\tilde{t}-\rho<t< s_1$,
we have
$$
x _{i_0}(\tilde{t})\leq x_{\bar{i}_0}
(\tilde{t}-\tau_{\bar{i}_0}(\tilde{t}))\leq \max_{t-r\leq
s\leq t}x_{\bar{i}_0}(s) \leq y(t)=x_{i_0}(t ),
$$
which contradicts with \eqref{e3.4}. Thus we have proved the claim.
It follows that
$$
\max_{t-r\leq s\leq s_1}x_{i_0}(s) =x_{i_0}(t),
$$
which, together the fact that
$$
t-r<s_1-r\leq t+\rho-r<t<s_1, \quad
 y_{\bar{i}_0}(t)\leq y_{i_0}(t), \quad
 y_{\bar{i}_0}(s_1)\leq y_{i_0}(s_1),
$$
yields
\[
\max_{t-r\leq s\leq s_1}x_{\bar{i}_0}(s)
\leq \max_{t-r\leq s\leq s_1}x_{i_0}(s)
 =x_{i_0}(t)=y(t), \quad y(t+h)=x_{i_0}(t )
\]
for all $0<h<s_1-t$,
and hence
\begin{align*} D^{+}y(t)
&= \limsup_{h\to 0^{+}}\frac{y(t+h)-y(t)}{h}\\
&= \limsup_{h\to 0^{+}}\frac{y(t+h)-x_{i_0}(t)}{h}\\
&=  \limsup_{h\to 0^{+}}\frac{x_{i_0}(t )-x_{i_0}(t)}{h} = 0.
\end{align*}
If \eqref{e3.2} holds,  we can choose a constant
$\bar{t}\in [s_1-r, s_1]$ such that
\begin{equation}
y(s_1)=x_{\bar{i}_0 }(\bar{t})
=\max_{s_1-r\leq s\leq s_1}x_{\bar{i}_0}(s)>y_{i_0 }(s_1)\geq x_{i_0 }(t).
\label{e3.5}
\end{equation}
Clearly,  $t<\bar{t}\leq s_1$ and
\begin{align*}
0& \leq x_{\bar{i}_0}'(\bar{t})
 =\gamma_{\bar{i}_0}(\bar{t})[-F_{\bar{i}_0}(x_{\bar{i}_0}
(\bar{t}))+G_{\bar{i}_0}(x _{i_0}(\bar{t}-\tau_{i_0}(\bar{t})))] \\
&\leq \gamma_{\bar{i}_0}(\bar{t})[-F_{\bar{i}_0}(x_{\bar{i}_0} (\bar{t}))
 +F_{\bar{i}_0}(x _{i_0}(\bar{t}-\tau_{i_0}(\bar{t})))] ,
\end{align*}
which follows that
$$
F_{\bar{i}_0}(x_{\bar{i}_0} (\bar{t}))
\leq F_{\bar{i}_0}(x _{i_0}(\bar{t}-\tau_{i_0}(\bar{t})))]
$$
and
\begin{equation}
x _{i_0}(\bar{t}-\tau_{i_0}(\bar{t})) \geq x_{\bar{i}_0}
(\bar{t})>x_{i_0}(t ). \label{e3.6}
\end{equation}
Noting that $t-r\leq t-\tau_{i_0}(\bar{t}) <\bar{t}-\tau_{i_0}(\bar{t})
<\bar{t}-\rho<t< s_1$, we have
$$
x_{\bar{i}_0} (\bar{t})\leq x _{i_0}(\bar{t}-\tau_{i_0}(\bar{t}))
\leq \max_{t-r\leq s\leq t}x_{i_0}(s) \leq y(t)=x_{i_0}(t ),
$$
which contradicts with \eqref{e3.6}. Thus,  \eqref{e3.2} does not hold.  It
proves that  $D^+y(t)\le 0$ for all $t\ge t_0+r$.

 Secondly, using similar arguments as those  in the proof of
$D^{+}y(t)\leq 0$, we can obtain
$$
D^{-}u(t)\geq 0 \quad  \text{for  all } t \ge t_0+r.
$$

From the above results, we see that $y$ is non-increasing and $u$ is
non-decreasing on $[t_0+r, +\infty)$. In view of the boundedness
of $x$, we obtain
\begin{gather*}
\lim _{t\to  +\infty}y(t)=A  \geq \lim _{t\to +\infty}u(t)=B,\\
A\geq L_i\geq l_i\geq B,\quad i\in J.
\end{gather*}
It suffices to show that $L_i= l_i, i\in J$.
 Suppose that, on
the contrary, either  $L_1> l_1 $ or $L_2> l_2 $ holds. We
next consider that $L_1> l_1 $. (The situation is analogous for
$L_2> l_2 $.)  Then, it is easily to see that $ B<A$, and $A$
and $B$ are not zero simultaneously. Without loss of generality, we
assume that $A\neq 0$ since the proof for the case of $B\neq0$ is
quite similar. For $\bar{H}\in (l_1, L_1)\subset (B,A)$, we can
choose $t_0^{*}>t_0+r $ and
$\{\tau_m\}_{m=1}^{+\infty} \subset [t_0^{*}+r, +\infty)$ such that
$$
x_1(\tau_m)=\bar{H}, \quad
\lim _{m\to +\infty}\tau_m=+\infty,  \quad
 x_i(t)\leq A+\frac{|A|}{2} \quad  \forall t\in [t_0^{*}, +\infty),\;
i=1,2.
$$
Then, for an arbitrary positive integer $ m $, it follows from
the monotonicity and definition of $y(t)$ that
\[
F_1(A )\leq F_1(y ( \tau _m))=F_1 (A+\varepsilon_m) ,\quad
0\leq \varepsilon_m\leq \frac{|A|}{2}, \quad
\varepsilon_m =y ( \tau _m)-A\to  0
\]
as $m\to +\infty)$.
 In the light of the fact that $\gamma^{+}\geq \gamma^{-}>0$ and
$$
y ( \tau _m)\geq y ( t)\geq x_i( t)  \quad \text{for  all }
 t\in [\tau _m, \tau _m+3r],\; i\in J,
$$
we obtain
$$
-F_1(x _1(t))+F_1(y ( \tau_m)) \geq 0 \quad \text{for  all }
 t\in [\tau _m, \tau _m+3r],
$$
and
\begin{equation}
\begin{aligned}
 x_1'(t) &= \gamma_1(t)[-F_1(x_1(t))+G_1(x_2(t-\tau_2(t)))]\\
&\leq \gamma_1(t)[-F_1(x _1(t))+F_1(x_2(t-\tau_2(t)))]\\
&\leq \gamma_1(t)[-F_1(x _1(t))+F_1(y ( \tau_m))]\\
&\leq  \gamma_1^{+}[-F_1(x_1(t))+F_1(A+\varepsilon_m)] \quad
 \text{for all } t\in [\tau_m, \tau _m+3r].
\end{aligned} \label{e3.7}
\end{equation}
Denote $v(t)=v(t;\tau _m, \varepsilon_m) $ the
solutions of the initial-value problem
\begin{equation}
v '(t)= \gamma^{+}[-F_1(v(t))+F_1(A+\varepsilon_m)], \quad
   v(\tau_m)=\bar{H}. \label{e3.8}
\end{equation}
 Note that $\bar{H}<A$.  Proposition \ref{prop2.1}   implies that
$$
A+\varepsilon_m-v(t;\tau _m, \varepsilon_m)\geq \mu>0,  \, t\in [\tau _m, \tau _m+3r],
$$
where the positive constant
$\mu$ is independent of $\tau _m$ and $\varepsilon_m$.
Furthermore, from \eqref{e3.7} and \eqref{e3.8}, we have
\begin{gather}
x_1(t)\leq v(t)<A+\varepsilon_m-\mu, \quad t\in [\tau _m, \tau _m+3r],\label{e3.9}\\
y_1(s)=\max_{s-r\leq t\leq s}x_1(t)<A+\varepsilon_m-\mu, \quad
 s\in [\tau _m+r, \tau _m+3r], \nonumber \\
y_1(\tau _m+2r)\leq y_1(\tau _m+r) <A+\varepsilon_m-\mu. \label{e3.10}
\end{gather}
For $s\in [\tau _m+2r, \tau _m+3r]$, from the fact that
$$
y_2(s)=\max_{s-r\leq t\leq s}x_2(t),
$$
it follows that there exists
$t^{*}\in  [s-r, s]\subseteq [\tau _m+r, \tau _m+3r]$ such that
$$
y_2(s)=x_2(t^{*})=\max_{s-r\leq t\leq s}x_2(t)
$$
and
\begin{align*}
0&\leq x_2'(t^{*})\\ &= \gamma_2(t^{*})[-F_2(x_2(t^{*}))
 +G_2(x_1(t^{*}-\tau_1(t^{*})))]\\
&\leq \gamma_2(t^{*})[-F_2(x_2(t^{*}))+F_2(x_1(t^{*}-\tau_1(t^{*})))],
\end{align*}
which implies that
$$
y_2(s)=\max_{s-r\leq t\leq s}x_2(t)=x_2(t^{*})
\leq x_1(t^{*}-\tau_1(t^{*})) <A+\varepsilon_m-\mu,
$$
and
\begin{equation}
y_2(\tau _m+ 2r) <A+\varepsilon_m-\mu   . \label{e3.11}
\end{equation}
From \eqref{e3.10} and \eqref{e3.11}, we have
$$
y (\tau _m+ 2r)=\max\{y_1(\tau _m+ 2r), y_2(\tau _m+ 2r)\}
<A+\varepsilon_m-\mu  ,
$$
which contradicts that
$\lim _{m\to +\infty}y(\tau _m+ r)=\lim _{t\to
+\infty}y(t)=A$. Hence, $L_1= l_1 $. This completes the proof.
\end{proof}

 From Lemma \ref{lem2.3},  we have the following results for
equation \eqref{e1.6}.

\begin{corollary} \label{coro3.1}
  Let  $F_i=G_i$  $(i\in J)$.
  Then every  solution  of the initial value problem \eqref{e1.6} and 
\eqref{e1.8} tends to a constant vector as
$t\to +\infty$.
\end{corollary}

\begin{remark} \label{rmk3.1} \rm
It is worth noting that system \eqref{e1.6}
includes the  scalar equation
\begin{equation}
  x'(t)= \gamma(t)[-F(x(t))+F(x(t-\tau (t) ))],       \label{e3.12}
\end{equation}
 as a special case. In fact, if $F_i=G_i= F$,
 $\gamma_i= \gamma$, $\tau_i= \tau$ and consider the synchronized solutions of
\eqref{e1.6} with $x_1(t)= x_2(t)=\varphi(t)$ for
$t\in [- \tau ^{+} ,0]$, then system \eqref{e1.6} reduces to scalar
equation \eqref{e3.12}. This
implies that Bernfeld and Haddock conjecture is only a special case
of Corollary \ref{coro3.1} with $F(x)=x^{1/3}$ and $\gamma(t)\equiv 1$.
 Moreover, the main results  in the most recently papers   \cite{l1,l2}
are also a special case of Corollary \ref{coro3.1}. In particular, we obtain
from  Corollary \ref{coro3.1} that every solution of the following equation
$$
x'(t)=  \gamma(t)[x^{\frac{n }{m}}(t)-x^{\frac{n }{m}}(t- \tau(t))],
\gamma(t)> 0, \quad \frac{n }{m}\in (0,1),
$$
tends to a constant   as $t\to +\infty$. Here,  $\tau(t) $
and $ \gamma(t)$ are  continuous functions and  are bounded
above and below by positive constants, and
$x_{t_0} =\varphi\in C([-\tau^{+},0],\mathbb{R} )$. This  answers the second open
problem proposed in  \cite{l1}.
\end{remark}


\section{Numerical simulations}

Consider the following functional differential equations  with  time-varying delays,
\begin{gather}
   x' (t) =   - x^{1/3}  (t) +  x^{1/3}  (t-(1+|\cos t|)) ,  \quad
  x_{t_0} =\varphi\in C([-2,0],\mathbb{R} ),
  \label{e4.1} \\
 x' (t) =  (1+\cos ^{2 } t)[- x^{1/3}  (t) +  x^{1/3}  (t-(1+|\cos t|))] , \quad
           x_{t_0} =\varphi\in C([-2,0],\mathbb{R} )      \label{e4.2} \\
\begin{gathered}
x'_1(t)= (1+\cos ^{4} t)[ -x_1^{3/5}(t) + x_2^{3/5}(t -(1+|\sin t|))] ,\\
x'_2(t)= (1+3\cos ^{2} t)[ -x_2^{3/5}(t) +x_1^{3/5}(t -(1+ |\cos t|))],\\
x_{t_0} = \varphi\in C([-2,0],\mathbb{R} )\times C([-2,0],\mathbb{R} ).
\end{gathered}   \label{e4.3}
\end{gather}
It follows from Corollary \ref{coro3.1}  that for   every solution of \eqref{e4.1},
\eqref{e4.2} and \eqref{e4.3} tends to a constant  solution as 
$t\to +\infty$. Figures \ref{fig1}--\ref{fig3} support this result with the numerical
solutions of the above equations with different initial values.

\begin{figure}[ht]
\begin{center}
  \includegraphics[width=0.7\textwidth]{fig1} 
\end{center}
  \caption{Numerical solutions  of \eqref{e4.1} for initial values 
$\varphi(s)= 1+\sin s$, $2+\sin s$, $6\sin s$, $s\in[-2,0]$.}
\label{fig1}
\end{figure}

\begin{figure}[ht]
\begin{center}
   \includegraphics[width=0.7\textwidth]{fig2}
\end{center}
  \caption{Numerical solutions of  \eqref{e4.2} for initial values 
$\varphi(s)= 1+3\sin s$, $2\sin s$, $2+5\sin s$, $s\in[-2,0]$.}
\label{fig2}
\end{figure}

\begin{figure}[ht]
\begin{center}
  \includegraphics[width=0.9\textwidth]{fig3}
\end{center}
\caption{Numerical solutions of  \eqref{e4.3} for initial values 
$\varphi(s)= (-3\sin s, -3\sin s)$, $(-2\sin s, -2\sin s)$, $(-6\sin s,-6\sin s)$, 
$s\in[-2,0]$.}
\label{fig3}
\end{figure}


 Since two-dimensional Bernfeld-Haddock conjecture
involving non-autonomous delay differential equations has not been
touched in  \cite{l1,l2,z2}, one can find  that all results in the above
references  cannot be applied to  \eqref{e4.3}. Moreover, the scalar
equation in Bernfeld-Haddock conjecture has been included in
two-dimensional non-autonomous delay differential equation \eqref{e1.6},
and the conclusions related to Bernfeld-Haddock conjecture in the references
above can be summed up as a special case of the results of this
paper. This implies that our results  extend previously known results. 

\subsection*{Acknowledgements}
The author  would like to express his sincere appreciation to the
reviewers for their helpful comments in improving the presentation
and quality of this paper.

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\end{document}