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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 55, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/55\hfil Travelling wave solutions]
{Travelling wave solutions in delayed
cellular neural networks with nonlinear output}
\author[X. Liu, P. Weng, Z. Xu\hfil EJDE-2006/55\hfilneg]
{Xiuxiang Liu, Peixuan Weng, Zhiting Xu}  % in alphabetical order

\address{Xiuxiang Liu \newline
School of Mathematical Sciences,
South China Normal University,
 Guangzhou, 510631, China}
\email{liuxx@scnu.edu.cn}

\address{Peixuan Weng \newline
School of Mathematical Sciences,
South China Normal University,
 Guangzhou, 510631, China}
\email{wengpx@scnu.edu.cn}

\address{Zhiting Xu \newline
School of Mathematical Sciences,
South China Normal University,
 Guangzhou, 510631, China}
\email{xztxhyyj@pub.guangzhou.gd.cn}

\date{}
\thanks{Submitted February 15, 2006. Published April 28, 2006.}
\thanks{Supported by grants 04010364 and  10571064
 from the Natural Science Foundation of the \hfill\break\indent
 Guangdong Province, and of China}
\subjclass[2000]{34B15, 34K10}
\keywords{Cellular neural networks; lattice dynamical system;
\hfill\break\indent
 travelling wave solutions; monotone iteration}

\begin{abstract}
 This paper concerns  the existence of
 travelling wave solutions of delayed cellular neural
 networks distributed in a 1-dimensional lattice
 with nonlinear output. Under  appropriate assumptions,
 we prove the existence of travelling waves and extend
 some known results.
\end{abstract}


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

\section{Introduction}

Cellular neural networks (CNN) system were first proposed by Chua
and Yang \cite{c1,c2} as an achievable alternative to fully-connected
neural networks in electric circuit systems, so it is also called
CY-CNN system. The structure of CNN is similar to the cellular
automata that any cell in a CNN is connected only to its
neighboring cells. In the recent years, the CNN approach has been
applied to a broad scope of problems arising from, for example,
image and video signal processing, robotic and biological visions
etc. We refer the readers to \cite{h1,h2,h3} for some practical
applications. An 1-dimensional CNN without inputs is given by
\begin{equation}
\frac{{\rm d} x_i(t)}{{\rm d} t}=-x_i(t)+z+\alpha f(x_i(t))+\beta
f(x_{i+1}(t)), \label{e1.1}
\end{equation}
for $i$ in a 1-dimensional lattice $\mathbb{Z}$,  where the positive
coefficients $\alpha, \beta$ of the signal output function
$f$ constitute the so-called space-invariant template that
measures the synaptic weights of self-feedback and neighborhood
interaction. The quantity $z$ is called a threshold or bias term
and is related to the independent voltage sources in electric
circuits. Due to the finite switching speed and finite velocity of
signal transmission, the distributed delays may exist for CNN
systems. One of the models with delay is proposed in \cite{l1} with
$z=0$ by
\begin{equation}
\frac{{\rm d} x_i(t)}{{\rm d} t}=-x_i(t)+\alpha f(x_i(t))+\beta \int_0^\tau K(u)
f(x_{i+1}(t-u)){\rm d} u, \label{e1.2}
\end{equation}
where $\tau> 0 $ is a constant, $K:[0, \tau]\to [0,+\infty)$
is a piece-wise continuous function satisfying
\[
\int_0^\tau K(u){\rm d} u=1.
\]
It is assumed that the self-feedback interaction is instantaneous
and there exists delay in neighborhood interaction. A typical
output function $f$ in \eqref{e1.1} or \eqref{e1.2} is defined by
\begin{equation}
f(x)=\begin{cases}
1 & \mbox {if }  x\geq 1,\\
x& \mbox {if } |x|\leq 1,\\
-1 & \mbox {if }   x\leq -1;
\end{cases}
\label{e1.3}
\end{equation}
 see for example \cite{c1,c2,h1,h2,h3,l1}.

A travelling wave solution of \eqref{e1.2} is defined  as a solution
of \eqref{e1.2} with
\begin{equation}
x_i(t)=\phi(i-ct):=\phi(s), \quad \mbox{for all $i\in \mathbb{Z}$ and $t\in \mathbb{R}$},
\label{e1.4}
\end{equation}
where $c\in\mathbb{R}$ is the wave speed. The profile equation
for $\phi(s)$ can be written as
\begin{equation}
-c\phi'(s)=-\phi(s)+\alpha f(\phi(s))+\beta \int_0^\tau K(u)f(\phi(s+1+u)){\rm d} u.
\label{e1.5}
\end{equation}
Assuming that the synaptic connection is sufficiently large so that
\begin{equation}
\alpha+\beta>1, \label{e1.6}
\end{equation}
then there are three equilibria of \eqref{e1.5}:
\begin{equation}
x^-=-(\alpha+\beta),\quad x^0=0,\quad x^+=\alpha+\beta.\label{e1.7}
\end{equation}
Recently, Weng and Wu \cite{w1} studied the deformation and existence of travelling
wave solutions for \eqref{e1.2} with \eqref{e1.3}, which satisfy different types of
asymptotic boundary conditions. For instance one type of conditions are
\begin{equation}
\lim_{s\to -\infty}\phi(s)=x^0,\quad
\lim_{s\to\infty}\phi(s)=x^+. \label{e1.8}
\end{equation}
 Ling \cite{l1} discussed the deformation and existence of travelling
wave solutions of
\eqref{e1.2} with  a nonlinear output function $f$ on $[-1, 1]$, that is
\begin{equation}
f(x)=\begin{cases}
1 & \mbox {if }  x\geq 1,\\
\sin(\frac{\pi}{2}x)& \mbox {if }  |x|\leq 1,\\
-1 & \mbox {if }  x\leq -1.
\end{cases} \label{e1.9}
\end{equation}

This paper is a continuation of the work in \cite{l1,w1}.
We consider the existence of travelling wave
solutions of \eqref{e1.2} with a more general output
function $f$ under the following hypotheses:
\begin{itemize}
\item[(H1)]
 $f$ is a continuous odd function on $(-\infty, +\infty)$ and
satisfies:
\begin{enumerate}
\item $f(x)=1$ for $x\geq 1$;
\item $f(0)=0$ and $f$ is differentiable at $x=0$, $\mu=f'(0)\geq 1$;
\item $f$ is non-decreasing and $f(x)\leq f'(0)x$ on  $[0, 1]$.
\end{enumerate}
\end{itemize}
 Under the above hypotheses and \eqref{e1.6}, we see that
$x^-$, $x^0$ and $x^+$ are still equilibria of \eqref{e1.5}, furthermore,
\begin{equation}
f'(0)x-f(x)=o(x)\quad \mbox{as }x\to 0. \label{e1.10}
\end{equation}
 In fact, noting that $f(0)=0$, one has
$$
\lim_{x\to 0}\frac{f'(0)x-f(x)}{x} =f'(0)-\lim_{x\to
0}\frac{f(x)-f(0)}{x}=0,
$$
hence $f'(0)x-f(x)=o(x)$.

\section{Existence of Monotone Travelling Waves}

In this section, we study the existence of travelling wave solutions
of \eqref{e1.2} under the
assumption (H1). First of all, we define the characteristic function
of \eqref{e1.5} at $x^0=0$ by
\begin{equation}
\Delta(\lambda, c, x^0)=-c\lambda+1-\mu\alpha -\mu\beta\int_0^\tau
K(u){\rm e}^{\lambda(1+cu)}{\rm d} u. \label{e2.1}
\end{equation}
The characteristic function \eqref{e2.1} plays crucial roles in our study.
The following lemma is needed.

\begin{lemma} \label{lem2.1}
Assume that $\alpha\geq 1$. There exist exactly a pair of
numbers $(c_\ast, \lambda_\ast)$ with $c_\ast<0$,
$\lambda_\ast =\lambda(c_\ast)>0$ such that
\begin{itemize}
\item[(i)] $\Delta(\lambda_\ast, c_\ast, x^0)=0$,
$\frac{\partial}{\partial\lambda} \Delta(\lambda_\ast, c_\ast, x^0)=0$;
 \item[(ii)]  for $c_\ast<c\leq 0$, $\Delta(\lambda,c, x^0)<0$ for
any $\lambda\in\mathbb{R}$;
 \item[(iii)]  for any $c<c_\ast$, there exists
$\lambda_1>0$, $\varepsilon_1>0$ such that
$\Delta(\lambda_1, c, x^0)=0$, and for any small
$\varepsilon\in (0, \varepsilon_1)$ one has
$\Delta(\lambda_1 +\varepsilon, c, x^0)>0$.
\end{itemize}
\end{lemma}

\begin{proof} A simple calculation leads to
\begin{gather*}
\frac{\partial}{\partial c}\Delta(\lambda, c, x^0)
=-\lambda\Big(1+\mu\beta\int_0^\tau
K(u)u{\rm e}^{\lambda(1+cu)} {\rm d} u\Big),\\
\frac{\partial}{\partial\lambda}\Delta(\lambda, c, x^0)=-c-\mu\beta\int_0^\tau
K(u)(1+cu){\rm e}^{\lambda(1+cu)}{\rm d} u,\\
\frac{\partial^2}{\partial\lambda^2} \Delta(\lambda, c, x^0)=-\mu\beta\int_0^\tau
K(u)(1+cu)^2{\rm e}^{\lambda(1+cu)}{\rm d} u<0\,.
\end{gather*}
Note that,
\begin{gather*}
 \Delta(0, c, x^0)=1-\mu(\alpha+\beta)<0\quad \mbox{for any }c\in\mathbb{R},\\
\lim_{c\to-\infty}\Delta(\lambda, c, x^0) = +\infty \quad
\mbox{for any }\lambda>0,\\
 \Delta(\lambda, 0,
x^0)=1-\mu\alpha-\mu\beta {\rm e}^{\lambda}<0\quad \mbox{for any }\lambda\in\mathbb{R},
\end{gather*}
 and that $\Delta(\lambda,
c, x^0)$ is a concave function of $\lambda\in \mathbb{R}$ for any given $c\in \mathbb{R}$.
Therefore, there exist exactly a pair of numbers
$(c_\ast, \lambda_\ast)$ with $c_\ast<0, \lambda_\ast= \lambda(c_\ast)>0$
 satisfying (i) and (iii).


Note that $ \Delta(\lambda, 0, x^0)<0$ for any $\lambda\in\mathbb{R}$.
Furthermore, for any given $\lambda<0$, we have
$\frac{\partial\Delta}{\partial c}>0$.
Therefore,  $\Delta(\lambda, c, x^0)=0$ has no real roots for any
$c\in (c_\ast,0]$. This completes the proof.
\end{proof}

We now consider the existence of monotone travelling waves
of \eqref{e1.2} for $c<c_*$. Our approach is based on monotone
iteration, coupled with the concept of upper and lower
solutions introduced below.

\begin{definition} \label{def2.1} \rm
 A function $V:\mathbb{R}\to\mathbb{R}$ is called an upper solution of
\eqref{e1.5} if it is differentiable almost everywhere (a.e.)
and satisfies the inequality
$$
-cV'(s)\geq -V(s)+\alpha f(V(s))+\beta\int_0^\tau K(u) f(V(s+1+cu)) {\rm d} u.
$$
 Similarly, a function $ v:\mathbb{R}\to\mathbb{R}$ is called a lower solution
of \eqref{e1.5} if it is differentiable almost everywhere and
satisfies the inequality
\[
-cv'(s)\leq -v(s)+\alpha f(v(s))+\beta\int_0^\tau K(u) f(v(s+1+cu)){\rm d} u.
\]
For any $c<c_\ast$, we define following  two functions:
\begin{gather*}
V(s)=\begin{cases}
x^+&s\geq 0,\\
x^+{\rm e}^{\lambda_1 s}&s\leq 0,
\end{cases}  \\ %{e2.2}
v(s)=\begin{cases}
0&s\geq 0,\\
\eta  (1-{\rm e}^{\varepsilon s}) {\rm e}^{\lambda_1 s}&s\leq 0,
\end{cases}%{e2.3}
\end{gather*}
where $x^+$ is defined in \eqref{e1.7}, $\lambda_1, \varepsilon$ are
as in Lemma 2.1 and $\eta \in (0, 1)$ is chosen small so that
 $V(s)\geq v(s)$ and to be decided in the following
such that $v(s)$ is a lower solution.
Clearly, we have $0\leq v(s) \leq V(s)\leq x^+$ and
$v(s)\not\equiv 0$ for $s\in \mathbb{R}$.
\end{definition}

\begin{lemma} \label{lem2.2}
 For any $c<c_\ast<0$,  $V$ is an upper solution and $v$ is a
lower solution of \eqref{e1.5}.
\end{lemma}

\begin{proof}
If $s\geq 0$,  $V(s)=x^+$. Note that $f(u)\leq 1$ for any
$u\in\mathbb{R}$, then we have
\[
 c V'(s)-V(s)+\alpha f(V(s))+\beta\int_0^\tau K(u)f(V(s+1+cu)){\rm d} u
\leq   0-x^++\alpha+\beta=0.
\]
If $s \leq 0$, $V(s)= x^+{\rm e}^{\lambda_1 s}$. Note that
$V(s)\leq x^+{\rm e}^{\lambda_1 s}$ for $s\in \mathbb{R}$.
Therefore, if $x^+{\rm e}^{\lambda_1 s}>1$, one has
$f(V(s))=1<x^+{\rm e}^{\lambda_1 s} \leq \mu x^+{\rm e}^{\lambda_1 s}$;
if  $x^+{\rm e}^{\lambda_1 s}\leq 1$, one also has
$f(V(s))\leq\mu V(s)=\mu x^+{\rm e}^{\lambda_1 s}$
from the assumption (3) in (H1). According to Lemma 2.1, we have
\begin{align*}
& cV'(s)-V(s)+\alpha f(V(s))+\beta\int_0^\tau K(u)f(V(s+1+cu)){\rm d} u\\
&\leq  c\lambda_1x^+{\rm e}^{\lambda_1 s}-x^+{\rm e}^{\lambda_1 s}+\mu\alpha x^+
{\rm e}^{\lambda_1 s}+\mu\beta\int_0^\tau K(u)x^+{\rm e}^{\lambda_1(s+1+cu)}{\rm d} u\\
&=   -x^+{\rm e}^{\lambda_1 s}\Delta(\lambda_1, c, x^0)=0.
\end{align*}
So $V(s)$ is an upper solution of \eqref{e1.5}.

Next we show that $v$ is a lower solution of \eqref{e1.5}. That is,
\begin{equation}
c v'(s)-v(s)+\alpha f(v(s))+\beta \int_0^\tau K(u)f(v(s+1+cu)){\rm d} u\geq 0.
\label{e2.4}
\end{equation}
If $s\geq 0$, it is obviously that \eqref{e2.4} holds because
\begin{align*}
& c v'(s)-v(s)+\alpha f(v(s))+\beta \int_0^\tau K(u)f(v(s+1+cu)){\rm d} u\\
=&0-0+0+\left\{\begin{array}{lll}\beta{\int_{-\frac{s+1}{c}}^\tau }K(u)\eta(1-{\rm e}^{\varepsilon(s+1+cu)}){\rm e}^{\lambda_1(s+1+cu)}{\rm d} u & \mbox{if} -\frac{s+1}{c}\in [0, \tau)\\
0& \mbox{if} -\frac{s+1}{c}\geq\tau\end{array}\right.\\
\geq &0.
\end{align*}
 If $s< 0$, we will consider the following three cases, namely,
$-(c\tau+1)<s<0$, $-1<s\leq -(c\tau+1)$, and $s\leq -1$.
\smallskip

\noindent{\bf Case 1.}  $-(c\tau+1)<s<0$. In this case, for any
$u\in [0, \tau]$, one  has $s+1+cu>0$ which implies $v(s+1+cu)=0$.
Moreover, if $s<0, 0<\eta<1$, one  has
$0<\eta{\rm e}^{\lambda_1 s}(1-{\rm e}^{\varepsilon s})<1$. So we can obtain
\begin{align*}
& cv'(s)-v(s)+\alpha f(v(s))+\beta\int_0^\tau K(u)f(v(s+1+cu))
{\rm d} u\\
&= -\eta{\rm e}^{\lambda_1
s}(-c\lambda_1+1)+\eta{\rm e}^{(\lambda_1+\varepsilon)s}\left(-c(\lambda_1
+\varepsilon)+1\right)\\
&\quad +\alpha f(\eta(1-{\rm e}^{\varepsilon s}){\rm e}^{\lambda_1 s})\\
&= -\eta {\rm e}^{\lambda_1 s}\Delta(\lambda_1, c, x^0)+\eta
{\rm e}^{(\lambda_1+\varepsilon)s}\Delta(\lambda_1+\varepsilon, c,
 x^0) +G_1,
\end{align*}
where
\begin{align*}
G_1&=-\mu\beta\eta\int_0^\tau K(u){\rm e}^{\lambda_1(s+1+cu)} [1-{\rm e}^{\varepsilon(s+1+cu)}]{\rm d}
u-\alpha
{\rm e}^{\lambda_1s}(1-{\rm e}^{\varepsilon s})|o(\eta)|\\
&\geq \eta\Big(\mu\beta\int^\tau_0
K(u){\rm e}^{\lambda_1(s+1+cu)}\;[{\rm e}^{\varepsilon(s+1+cu)}-1\;]{\rm d}
u-\alpha{\rm e}^{\lambda_1s}(1-{\rm e}^{\varepsilon s})\frac{|o(\eta)|}{\eta}\Big).
\end{align*}
Thus, $G_1\geq 0$  if $\eta$ is small enough.
>From Lemma 2.1, $\Delta(\lambda_1, c,x^0)=0$,
$\Delta(\lambda_1+\varepsilon, c, \ x^0)>0$, hence \eqref{e2.4} holds.
\smallskip

\noindent{\bf Case 2.} $-1<s\leq -(c\tau+1)$. In this case, for
$u\in [0, -\frac{s+1}{c}]$, one has $s+1+cu\geq0$, and for
 $u\in(-\frac{s+1}{c},\tau]$ one has $s+1+cu<0$. Choose
$\eta\in (0,1)$, we have
\begin{align*}
& cv'(s)-v(s)+\alpha f(v(s))+\beta\int_0^\tau K(u)f(v(s+1+cu))
{\rm d} u\\
&= -\eta{\rm e}^{\lambda_1
s}(-c\lambda_1+1)+\eta{\rm e}^{(\lambda_1+\varepsilon)s}\left[-c(\lambda_1
+\varepsilon)+1\right]+\alpha f(\eta  (1-{\rm e}^{\varepsilon s})
{\rm e}^{\lambda_1 s})\\
&\quad +\beta\int_{-\frac{s+1}{c}}^\tau K(u) f(\eta  (1-{\rm e}^{\varepsilon
(s+1+cu)}) {\rm e}^{\lambda_1 (s+1+cu)}){\rm d} u\\
&= -\eta{\rm e}^{\lambda_1 s}\Delta(\lambda_1, c,
x^0)+\eta{\rm e}^{(\lambda_1+\varepsilon)s}\Delta(\lambda_1+\varepsilon,
c, x^0) +G_2\\
&= \eta{\rm e}^{(\lambda_1+\varepsilon)s}\Delta(\lambda_1+\varepsilon, c, x^0)
+G_2,
\end{align*}
where
\begin{align*}
G_2&=\mu\beta\eta\int_0^\tau K(u){\rm e}^{\lambda_1(s+1+cu)}
[{\rm e}^{\varepsilon(s+1+cu)} -1]{\rm d} u-\alpha
{\rm e}^{\lambda_1s}(1-{\rm e}^{\varepsilon s})|o(\eta)|\\
&\quad +\beta(\mu\eta-|o(\eta)|)\int^\tau_{-\frac{s+1}{c}}K(u)
{\rm e}^{\lambda_1(s+1+cu)}
(1-{\rm e}^{\varepsilon(s+1+cu)}){\rm d} u\\
&=\mu\beta\eta\int_0^{-\frac{s+1}{c}} K(u){\rm e}^{\lambda_1(s+1+cu)}
[{\rm e}^{\varepsilon(s+1+cu)}-1]{\rm d} u
-\alpha {\rm e}^{\lambda_1s}(1-{\rm e}^{\varepsilon s})|o(\eta)|\\
&\quad  -\beta|o(\eta)|\int^\tau_{-\frac{s+1}{c}}K(u) {\rm e}^{\lambda_1(s+1+cu)}
(1-{\rm e}^{\varepsilon(s+1+cu)}){\rm d} u\\
&\geq\mu\beta\eta\int_0^{-\frac{s}{c}} K(u){\rm e}^{\lambda_1(s+1+cu)}
[{\rm e}^{\varepsilon(s+1+cu)}-1]{\rm d} u
-\alpha {\rm e}^{\lambda_1s}(1-{\rm e}^{\varepsilon s})|o(\eta)|\\
&\quad -\beta|o(\eta)|\int^\tau_{-\frac{s+1}{c}}K(u){\rm e}^{\lambda_1
(s+1+cu)}(1-{\rm e}^{\varepsilon(s+1+cu)}){\rm d} u\\
&\geq\mu\beta\eta\Big[{\rm e}^{\lambda_1}({\rm e}^{\varepsilon}-1)
\int_0^{-\frac{s}{c}}
K(u){\rm d} u -\alpha{\rm e}^{\lambda_1s}(1-{\rm e}^{\varepsilon s})
\frac{|o(\eta)|}{\eta} \\
&\quad -\frac{|o(\eta)|}{\eta}\int^\tau_{-\frac{s+1}{c}}K(u)
{\rm e}^{\lambda_1(s+1+cu)}
(1-{\rm e}^{\varepsilon(s+1+cu)}){\rm d} u\Big].
\end{align*}
Therefore, $G_2\geq 0$  if $\eta$ is small enough, and similar to case 1,
we have \eqref{e2.4}.
\smallskip

\noindent {\bf Case 3.} $s\leq -1$.
In this case for any $u\in [0, \tau]$, $s+1+cu\leq 0$. Hence
\begin{align*}
& cv'(s)-v(s)+\alpha f(v(s))+\beta\int_0^\tau K(u)f(v(s+1+cu))
{\rm d} u\\
&= -\eta{\rm e}^{\lambda_1
s}(-c\lambda_1+1)+\eta{\rm e}^{(\lambda_1+\varepsilon)s}\left[-c(\lambda_1+\varepsilon)
+1\right]+\alpha f(\eta{\rm e}^{\lambda_1s}(1-{\rm e}^{\varepsilon s}))\\
 &\quad +\beta\int_0^\tau K(u)f(\eta{\rm e}^{\lambda_1(s+1+cu)}(1-{\rm e}^{\varepsilon
 (s+1+cu)})){\rm d} u\\
&= -\eta{\rm e}^{\lambda_1 s}\Delta(\lambda_1, c,
x^0)+\eta{\rm e}^{(\lambda_1+\varepsilon)s}\Delta(\lambda_1+\varepsilon,
c, x^0) +G_3\\
&= \eta{\rm e}^{(\lambda_1+\varepsilon)s}\Delta(\lambda_1+\varepsilon,
c, x^0) +G_3,
\end{align*}
where
\begin{align*}
G_3=-\Big[\alpha{\rm e}^{\lambda_1s}(1-{\rm e}^{\varepsilon s})+\beta\int_0^\tau
K(u){\rm e}^{\lambda_1(s+1+cu)}(1-{\rm e}^{\varepsilon(s+1+cu)}){\rm d} u\Big]|o(\eta)|.
\end{align*}
 So we can choose $\eta$ small enough such that
\[
\eta{\rm e}^{(\lambda_1+\varepsilon)s}\Delta(\lambda_1+\varepsilon, c, x^0)
+G_3>0.
\]
 Hence \eqref{e2.4} still holds in this case.
According to the above discussion, we know that  $v(s)$ is a lower
solution of \eqref{e1.5}.
This completes the proof.
\end{proof}

Now we let ${\bf C}={\bf C}(\mathbb{R}, [x^0, x^+])$, and
\begin{align*}
S_1=\big\{\phi\in{\bf C}:
&{\rm (i)}\; \phi(s) \mbox{ is  nondecreasing  for $s$ in }\mathbb{R};\\
&{\rm (ii)}\; \lim_{s\to -\infty}\phi(s)=x^0, \;
\lim_{s\to \infty}\phi(s)=x^+. \big\}.
\end{align*} %{e2.5}
Assume that $c<c_\ast<0$. Consider the following equivalent form
of equation \eqref{e1.5},
\begin{equation}
\frac{{\rm d}\phi(s)}{{\rm d} s}+\gamma\phi(s)=F(\phi)(s), \label{e2.6}
\end{equation}
where
\[
F(\phi)(s)=(\gamma+\frac{1}{c})\phi(s)-\frac{\alpha}{c}f(\phi(s))
-\frac{\beta}{c}\int_0^\tau K(u)f(\phi(s+1+cu)){\rm d} u .
\]
Note that $c<0$ and $f$ is
non-decreasing, so we can choose $\gamma>-\frac{1}{c}>0$ such that\,
$ F(\phi)(s)\geq F(\psi)(s)$ provided that $\phi(s)\geq \psi(s)$ for
$s\in \mathbb{R}$. It is easy to show that \eqref{e2.6} is equivalent
to
$$
\phi(s)={\rm e}^{-\gamma s}\int_{-\infty}^s{\rm e}^{\gamma u }
F(\phi)(u){\rm d} u. %{e2.7}
$$
Define an operator $T:\ S_1\to{\bf C}$ by
\begin{equation}
(T\phi)(s)={\rm e}^{-\gamma s}\int_{-\infty}^s{\rm e}^{\gamma u }
F(\phi)(u){\rm d} u, \quad \phi\in S_1, \; s\in \mathbb{R}. \label{e2.8}
\end{equation}
 Then we have  the following result.

\begin{lemma} \label{lem2.3}
 Assume that $c<c_\ast<0$. Then $T$ defined in \eqref{e2.8}
satisfies
\begin{enumerate}

\item  if $\phi \in S_1$, then $T\phi \in S_1$;

\item if $\phi$ is an upper ( resp. a lower) solution of \eqref{e1.5},
 then $\phi(s)\geq (T\phi)(s)$ (resp. $\phi(s)\leq (T\phi)(s)$)
for $s\in \mathbb{R}$;

\item if $\phi(s)\geq \psi(s)$ for $s\in \mathbb{R}$, then
$(T\phi)(s)\geq (T\psi)(s)$ for $s\in\mathbb{R}$;

\item if $ \phi $ is an upper (a lower) solution of \eqref{e1.5},
then $T\phi$ is also an upper (a lower) solution of \eqref{e1.5}.

\end{enumerate}
\end{lemma}

\begin{proof}  (1) is a direct verification by L'Hospital rule.
On the other hand,  the monotonicity of $F$ leads to the conclusion (3).
In the following, we only show that (2) and (4) hold. In fact,
if $\phi(s)$ is an upper solution
of \eqref{e1.5}, then
\[
\frac{{\rm d}\phi(s)}{{\rm d} s}+\gamma\phi(s)\geq F(\phi)(s).
\]
This leads to
\[
\frac{{\rm d}({\rm e}^{\gamma s}\phi(s))}{{\rm d} s}\geq {\rm e}^{\gamma s}
F(\phi)(s).
\]
Integrating  the  inequality above from $-\infty $ to $s$ ,
we obtain (2).
Noting that $F(\phi)(s)\geq F((T\phi)(s))$ from (2), we have
\[
\frac{{\rm d} (T\phi)(s))}{{\rm d} s}+\gamma(T\phi)(s)= F(\phi)(s)
\geq F((T\phi)(s)).
\]
This means  that $(T\phi)(s) $  is also an upper solution of \eqref{e1.5}.
The proof is similar if $\phi(s)$ is a lower solution.
Thus the proof is complete.
\end{proof}

 Consider the iterative scheme
\[
V_0=V\quad \mbox{and}\quad  V_n=TV_{n-1},\quad n=1,2,\cdots.
\]
By Lemma 2.2, we have
\[
x^0\leq v(s)\leq V_n(s)\leq V_{n-1}(s)\leq \cdots\leq V(s)\leq x^+.
\]
By Lebesgue's dominated convergence theorem, the limit exists which
allows defining the function
$\phi(s)=\lim_{n\to\infty}V_n(s)\geq v(s)$ exists and is  a
fixed point of $T$. Therefore, $\phi$ is a
solution of \eqref{e1.2} and satisfies
\[
\lim_{s\to -\infty}\phi(s)=x^0, \quad \lim_{s\to\infty}\phi(s)
=x^+.
\]
So we can obtain the following existing theorem of travelling waves.


\begin{theorem} \label{thm2.1}
 For any $c<c_\ast<0$, there exists a wave solution
$\phi(s)$ of \eqref{e1.2} which is increasing and satisfies
$$
\lim_{s\to-\infty}\phi(s)=x^0, \ \lim_{s\to\infty}\phi(s)=x^+.
$$
\end{theorem}

Note that $f$ is an odd function, let $\psi=-\phi$, then \eqref{e1.5}
is changed to
$$
-c\psi'(s)=-\psi(s)+\alpha f(\psi(s))+\beta \int_0^\tau K(u)f(\psi(s+1+u)){\rm d} u.
 %{e2.9}
$$
which is exactly of the same form as \eqref{e1.5}, so we have the
following result.

\begin{theorem} \label{thm2.2}
 For any $c<c_\ast<0$, there exists a wave solution
$\phi(s)$ of \eqref{e1.2} which is decreasing and satisfies
$$
\lim_{s\to-\infty}\phi(s)=x^0, \quad
 \lim_{s\to\infty}\phi(s)=x^-.
$$
\end{theorem}
In the rest of this section, we will discuss the existence of
monotone travelling waves of \eqref{e1.2} for $c>0$. By the facts
\begin{gather*}
\Delta(0, c, x^0)=1-\mu(\alpha+\beta)<0; \quad
 \lim_{\lambda\to -\infty} \Delta(\lambda, c, x^0)=+\infty;
\\
\frac{\partial\Delta}{\partial\lambda}(\lambda, c, x^0)<0, \lambda\in\mathbb{R};
\quad
\frac{\partial^2\Delta}{\partial\lambda^2}(\lambda, c, x^0)<0,
\lambda\in\mathbb{R}.
\end{gather*}
Thus, we know,  for any fixed $c>0$, the equation
$\Delta(\lambda, c, x^0)=0$ has a
unique real root $\lambda_2=\lambda_2(c)<0$. Furthermore,
there is $\varepsilon_2>0$ such
that for $0<\varepsilon<\varepsilon_2$, one has
\begin{equation}
\Delta(\lambda_2-\varepsilon, c, x^0)>0. \label{e2.10}
\end{equation}
We note that \eqref{e1.5} becomes
\begin{equation}
\phi'(s)=\frac{1}{c}(\phi(s)-\alpha-\beta) \label{e2.11}
\end{equation}
if $\phi(s)\geq 1$ for large $|s|$, and
$$
\phi(s)=(\phi(0)-\alpha-\beta){\rm e}^{\frac{s}{c}}+\alpha+\beta. %{e2.12}
$$
Therefore, \eqref{e2.11} has no monotone solution satisfying \eqref{e1.8},
so we consider monotone solutions with boundary conditions
\[
\lim_{s\to -\infty}\phi(s)=x^+, \quad  \lim_{s\to\infty}=x^0.
\]
Let
\begin{align*}
S_2=\big\{&\phi\in {\bf C}:
{\rm (i)} \phi(s)  \text{ is  nonincreasing  for }  s\in \mathbb{R};\\
&{\rm (ii)} \lim_{s\to -\infty}\phi(s)=x^+,  \lim_{s\to
\infty}\phi(s)=x^0.\},
\end{align*}
 \[
\bar{V}(s)=\begin{cases}
x^+,&s\leq 0,\\
x^+{\rm e}^{\lambda_2 s},& s  \geq0,
\end{cases}
\]
and
\[
\bar{v}(s)=\begin{cases}
0,&s\leq 0,\\
\eta(1-{\rm e}^{-\varepsilon s}){\rm e}^{\lambda_2 s},& s\geq 0,
\end{cases}
\]
 where $\lambda_2, {\varepsilon}$ are given in \eqref{e2.10}.
We can show that $\bar{V}(s)$ is an
upper solution and $\bar{v}(s)$ is a lower solution of \eqref{e1.5}
while ${\eta}$ is appropriately chosen, with the argument similar
to that for the situation where
$c<c_\ast<0$.
 Let
\[
H(\phi)(s)=(\frac{1}{c}-\gamma)\phi(s)-\frac{\alpha}{c}f(\phi(s))
-\frac{\beta}{c} \int_0^\tau K(u)f(\phi(s+1+cu)){\rm d} u .
\]
We choose
$\gamma>1/c$ such that $ H(\phi)(s) \leq H(\psi)(s)$
provided that $\phi(s)\geq \psi(s)$ for $ s\in \mathbb{R}$.
Note that \eqref{e1.5} is equivalent to
$$
\phi(s)=-{\rm e}^{\gamma s}\int_s^{\infty}{\rm e}^{-\gamma u }
H(\phi)(u){\rm d} u. %{e2.13}
$$
Define an operator $Q: S_2\to {\bf C}$ by
\begin{equation}
(Q\phi)(s)=-{\rm e}^{\gamma s}\int^{\infty}_s{\rm e}^{-\gamma u }
H(\phi)(u){\rm d} u, \quad
\phi\in S_2, s\in \mathbb{R}. \label{e2.14}
\end{equation}
Similar to the proof of Lemma 2.3 we have the following result.


\begin{lemma} \label{lem2.4}
Let $Q$ be defined in \eqref{e2.14}.
Then
\begin{enumerate}
\item  if $\phi \in S_2$, then $Q\phi \in S_2$;
\item  if $\phi$ is an upper (a lower) solution of \eqref{e1.5},
then $\phi(s)\geq (Q\phi)(s)\ (\phi(s)\leq (Q\phi)(s))$
for $s\in \mathbb{R}$;
\item if $\phi(s)\geq
\psi(s)$ for $s\in \mathbb{R}$, then $(Q\phi)(s) \geq (Q\psi)(s)$
for $s\in\mathbb{R}$;
\item  if $ \phi $ is an upper (a lower) solution of \eqref{e1.5},
then $Q\phi$ is also an upper (a lower) solution of \eqref{e1.5}.
\end{enumerate}
\end{lemma}

Then we can show the existence of a monotone solution in $S_2$ of
 \eqref{e1.5} by monotone iteration method, with the argument
similar to that of the situation where $c<c_\ast<0$.
In particular,  we have the following.

\begin{theorem} \label{thm2.3}
 For any $c>0$, we have the following conclusions.
\begin{enumerate}
\item There exists a wave solution $\phi(s)$ of \eqref{e1.2} which is
decreasing and satisfies
$$
\lim_{s\to-\infty}\phi(s)=x^+, \quad
\lim_{s\to\infty}\phi(s)=x^0.
$$
\item There exists a wave solution $\phi(s)$ of \eqref{e1.2} which
is increasing and satisfying
$$\lim_{s\to-\infty}\phi(s) =x^-, \quad
\lim_{s\to\infty}\phi(s)=x^0.
$$
\end{enumerate}
\end{theorem}

 Finally,  we shall briefly discuss the existence of monotone waves
 of  CNN model with some explicit output function $f$.


\begin{example} \label{ex2.1} \rm
 Let the output function $f$ be defined as \eqref{e1.3}.
Obviously, the assumption (H1) is satisfied with $\mu=1$.
Then \eqref{e1.2} has monotone waves by Theorem 2.1-2.3,
which leads to \cite[Theorem 2.1-2.2]{l1}.
\end{example}

\begin{example} \label{ex2.2} \rm
 Let the output function $f$ be defined as \eqref{e1.9}
[see \cite{w1}). Then the assumption (H1) is satisfied with
$\mu=\frac{\pi}{2}>1$. Hence
\eqref{e1.2} has monotone waves by Theorem 2.1-2.3.
\end{example}


\begin{example} \label{ex2.3} \rm
 Let the output function $f$ be defined by
$$
f(x)=\begin{cases}
1 & \mbox {if }  x\geq 1,\\
2x-x^2& \mbox {if }  0\leq x\leq 1,\\
 2x+x^2& \mbox {if }  -1\leq x\leq 0,\\
-1 & \mbox {if }  x\leq -1.
\end{cases}%{e2.15}
$$
Obviously the assumption (H1) is satisfied with $\mu=2$, which
leads to  \eqref{e1.2}
having  monotone waves by Theorem 2.1-2.3.
\end{example}

\subsection*{Acknowledgments}
The authors are grateful to the anonymous referee for her/his
suggestions and comments on the original manuscript.


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\end{document}