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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 260, pp. 1--9.\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-2017/260\hfil Heteroclinic orbits]
{Heteroclinic orbits of a second order nonlinear difference equation}

\author[H. Shi, X. Liu, T. Zhou \hfil EJDE-2017/260\hfilneg]
{Haiping Shi, Xia Liu, Tao Zhou}

\address{Haiping Shi \newline
Modern Business and Management Department,
Guangdong Construction Polytechnic,
Guangzhou 510440, China}
\email{shp7971@163.com}

\address{Xia Liu \newline
Oriental Science and Technology College, 
Hunan Agricultural University, 
Changsha 410128, China.\newline
Science College, Hunan Agricultural University, 
Changsha 410128, China}
\email{xia991002@163.com}

\address{Tao Zhou \newline
School of Business Administration, 
South China University of Technology,
Guangzhou 510640, China}
\email{zhoutaoscut@hotmail.com}


\dedicatory{Communicated by Paul H. Rabinowitz}

\thanks{Submitted August 23, 2017. Published October 16, 2017.}
\subjclass[2010]{34C37, 37J45, 39A12, 47J30, 58E05}
\keywords{Heteroclinic orbits; difference equations; critical point theory}

\begin{abstract}
 This article concerns a second-order nonlinear difference equation.
 By using critical point theory, the existence of two heteroclinic orbits
 is obtained. The main method used is variational.
\end{abstract}

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

\section{Introduction}

Let $\mathbb{N}$, $\mathbb{Z}$ and $\mathbb{R}$ denote the sets of all natural
numbers, integers and real numbers respectively.
For $a$, $b$ $\in \mathbb{Z}$, we define
 $\mathbb{Z}(a,b)=\{n\in\mathbb{Z}|a<n<b\}$,
$\mathbb{Z}[a,b]=\{n\in\mathbb{Z}|a\leq n\leq b\}$. For a set
 $M\subset\mathbb{R}, r>0$, $B_r(M)$ is denoted by
 $$
B_r(M)=\{u\in\mathbb{R}: \inf_{v\in M}|u-v|<r\}.
$$
In this article we consider the existence of heteroclinic orbits of the
 second-order nonlinear difference equation
\begin{equation}\label{e1.1}
 \Delta^2 u_{n-1}+p_nf(u_n)=0, \quad n\in \mathbb{Z},
\end{equation}
 where $\Delta$ is the forward difference operator
 $\Delta u_n=u_{n+1}-u_n$, $\Delta^2 u_n=\Delta(\Delta u_n)$,
 $\{p_n\}_{n\in \mathbb{Z}}$ is a positive real
 sequence, $f\in C(\mathbb{R}, \mathbb{R})$.
Moreover, $p$ and $f$ satisfy the conditions:
\begin{itemize}
\item[(A1)] $0<\underline{p}=\inf_{n\in \mathbb{Z}}\{p_n\}
\leq\bar{p}=\sup_{n\in \mathbb{Z}}\{p_n\}<+\infty$;

\item[(A2)] there exists a function $F\in C^1(\mathbb{R}, \mathbb{R})$ with
$F(0)=0$, $F(u+T)=F(u)$, $F'(u)=f(u)$ and
 $F$ has a maximum 0 on $\mathbb{R}$.
 Denote $\Psi=\{u\in\mathbb{R}: F(u)=0\}$.
\item[(A3)] $\Psi$ consists only of isolated points and $0\in\Psi$.
\end{itemize}

 As usual, a solution $u$ of \eqref{e1.1} is called a heteroclinic orbit
 (or heteroclinic solution) if there exist
 two constants $\mu, \nu\in \mathbb{R}, \mu\neq\nu$ such that $u$
joins $\mu$ to $\nu$, i.e.,
\begin{gather*}
 u_{-\infty}=\lim_{n\to -\infty}u_n=\mu, \\
 u_{+\infty}=\lim_{n\to +\infty}u_n=\nu.
\end{gather*}
 Such orbits and homoclinic orbits have been found in various models of
continuous and discrete dynamical  systems and frequently have tremendous
effects on the dynamics of such nonlinear systems. So the heteroclinic orbits
and homoclinic orbits have been extensively studied, the reader is referred to
 \cite{ACE,Ch,ChC,ChHT,ChW,GAWO,GOXA1,GOXA2,GOWA,HT,HZ1,HZ2,HZ3,R1,R2,R3,
T,X,XLS,XY,Y,ZL,ZM,ZY1,ZY2,ZYC}.

 In 1989, Rabinowitz \cite{R3} considered the following second-order Hamiltonian 
system
\begin{equation}\label{e1.2}
 \ddot{q}+V'(q)=0
\end{equation}
 where $q=(q_1,\dots,q_n)$, $V$ is periodic in $q_i, 1\leq i\leq n$, and proved
the existence and multiple heteroclinic orbits joining maxima of $V$.

 By using variational method and a delicate analysis technique, 
Xiao and Yu \cite{XY} showed that there indeed exist
 heteroclinic orbits of discrete pendulum equation
\begin{equation}\label{e1.3}
 \Delta^2 u_{n-1}+A\sin u_n=0, n\in \mathbb{Z},
\end{equation}
 joining every two adjacent points of $\{2k\pi+\pi: k\in \mathbb{Z}\}$.

When $p_n\equiv1$, Xiao, Long and Shi \cite{XLS} in 2010 investigated the 
existence and multiplicity of heteroclinic orbits of the system
\begin{equation}\label{e1.4}
 \Delta^2 u_{n-1}+V'(u_n)=0, \quad n\in \mathbb{Z},
\end{equation}
by using the critical point theory. Zhang and Li \cite{ZL} using variational 
method proved some existence results of heteroclinic orbits and heteroclinic 
chains for a second order discrete Hamiltonian system of \eqref{e1.4}.

 However, to the best of our knowledge, the results on heteroclinic orbits of
 discrete systems are very scarce in the literature \cite{XLS,XY,ZL}. 
The difficulty is the idea of continuous systems
 depend heavily on the continuity of the solutions and therefore they can 
not be applied directly to discrete systems. Motivated by the recent 
papers \cite{ChC,GAWO}, the purpose of this paper is to consider 
problem \eqref{e1.3} in a more general sense. It is obvious that \eqref{e1.3} 
is a special of \eqref{e1.1} with $p_n\equiv A$ and $f(u_n)=\sin u_n$.
Our main result is as follows.

 \begin{theorem} \label{thm1.1}
 Suppose that {\rm (A1)--(A3)} are satisfied.
 Then \eqref{e1.1} possesses two heteroclinic orbits joining $0$
 to some $\tau\in\Psi\backslash \{0\}$, one of which originates from
$0$ and one of which terminates at $0$.
\end{theorem}

For basic knowledge of variational methods, we refer the reader to the
 monographs \cite{MW,R4}.


\section{Variational structure and some lemmas}

To apply the critical point theory, we shall establish the corresponding
 variational functional associated with \eqref{e1.1} and give some lemmas 
which will  be used in proving our main results. We  firstly introduce some 
basic notation.

 Let $S$ be the set of bi-infinite convergent sequences 
$u=\{u_n\}_{n=-\infty}^{+\infty}$, that is
\[
S:=\big\{\{u_n\}|\lim_{n\to +\infty}u_n \text{ and }
\lim_{n\to -\infty}u_n\text{ exist, }u_n\in \mathbb{R}, n\in \mathbb{Z}\big\}.
\]
Define
 $$
E:=\big\{u\in S:\sum_{n=-\infty}^{+\infty}|\Delta u_n|^2<+\infty\big\},
$$
 with the inner product
\begin{equation}\label{e2.1}
 \langle u,v\rangle=\sum_{n=-\infty}^{+\infty}\Delta u_n\Delta v_n+u_0v_0, 
\quad \forall u,v\in E.
\end{equation}
 Then $E$ is a Hilbert space with the norm
\begin{equation}\label{e2.2}
 \|u\|^2=\sum_{n=-\infty}^{+\infty}|\Delta u_n|^2+|u_0|^2, \quad \forall u\in E.
\end{equation}
For $1<s<+\infty$, the spaces $l^s$ and $l^\infty$ are defined by
\begin{gather*}
l^s:=\big\{\{u_n\}:\sum_{n=-\infty}^{+\infty}|u_n|^s<+\infty,\;
 u_n\in \mathbb{R}, n\in \mathbb{Z}\}, \\
l^\infty:=\big\{\{u_n\}:\sup_{n\in \mathbb{Z}}|u_n|<+\infty,\; 
u_n\in \mathbb{R}, n\in \mathbb{Z}\big\}.
\end{gather*}
 For any $u\in E$, define the functional $J$ associated with \eqref{e1.1} 
on $E$ as follows:
\begin{equation}\label{e2.3}
 J(u):=\frac{1}{2}\sum_{n=-\infty}^{+\infty}|\Delta u_n|^2
-\sum_{n=-\infty}^{+\infty}p_nF(u_n).
\end{equation}
By (A2) and (A3), we have
 $$
\delta:=\frac{1}{3}\inf_{\rho,\varrho\in\Psi, \rho\neq\varrho}|\rho-\varrho|>0.
$$
For  $\rho\in\Psi$ and $0<\epsilon<\delta$, let the set 
$\Gamma_\epsilon(\rho)$ satisfy
\begin{itemize}
\item[(i)] $u_{-\infty}$=0,

\item[(ii)] $u_{+\infty}=\rho$,

\item[(iii)] $u_n\not\in B_\epsilon(\Psi\setminus\{0,\rho\})$ for all 
$n\in \mathbb{Z}$.
\end{itemize}
It is easy to see that $\Gamma_\epsilon(\rho)$ is nonempty for all 
$\rho\in\Psi\setminus\{0\}$ and $0<\epsilon<\delta$. Denote
 \begin{gather*}
c_\epsilon(\rho):=\inf_{u\in\Gamma_\epsilon(\rho)}J(u), \\
\varphi_\epsilon:=\inf_{u\not\in B_\epsilon(\Psi)}[-F(u)].
\end{gather*}

\begin{remark} \label{rem2.1}\rm
 From (A2) and (A3) it follows that $\varphi_\epsilon>0$ for all 
$0<\epsilon<\delta$. As a matter of fact, $\varphi_\epsilon\neq 0$. 
If not, there is $v\in\mathbb{R}\not\in B_\epsilon(\Psi)$ such that
 $F(0)=0$ implies that $v\in\Psi$. This is a contradiction. 
From $F(u+T)=F(u)$ and $u\not\in B_\epsilon(\Psi)$ it follows that 
$\varphi_\epsilon>0$.
\end{remark}

\begin{lemma}\label{lem2.2}
For any $a\leq b$, assume that $u\in E$ such that $u_n\not\in B_\epsilon(\Psi)$, 
then
 $$
\frac{1}{2}\sum_{n=a}^b|\Delta u_n|^2-\sum_{n=a}^bp_nF(u_n)
\geq\sqrt{2\underline{p}\varphi_\epsilon}|u_{b+1}-u_a|.
$$
\end{lemma}

\begin{proof}
By the definition of $\varphi_\epsilon$ and H\"{o}lder inequality, we have
 $$
|u_{b+1}-u_a|\leq\sqrt{b+1-a}\Big(\sum_{n=a}^b|\Delta u_n|^2\Big)^{1/2}.
$$
 Then
 $$
\sum_{n=a}^b|\Delta u_n|^2\geq\frac{|u_{b+1}-u_a|^2}{b+1-a}.
$$
 Thus,
\begin{align*}
\frac{1}{2}\sum_{n=a}^b|\Delta u_n|^2-\sum_{n=a}^bp_nF(u_n)
 &\geq\frac{1}{2}\sum_{n=a}^b|\Delta u_n|^2+\underline{p}\sum_{n=a}^b[-F(u_n)]\\
 &\geq\frac{|u_b-u_a|^2}{2(b+1-a)}+\underline{p}(b+1-a)\varphi_\epsilon\\
 &\geq\sqrt{2\underline{p}\varphi_\epsilon}|u_{b+1}-u_a|.
\end{align*}
 The desired results are obtained.
\end{proof}

\begin{remark} \label{rem2.3}\rm
 For all $\rho\in\Psi\setminus\{0\}$ and $0<\epsilon<\delta$, it follows 
immediately from Lemma \ref{lem2.2} that $c_\epsilon(\rho)>0.$
\end{remark}

\begin{lemma}\label{lem2.4}
Assume that $u\in E$ and $J(u)<+\infty$, then there are two constants 
$\mu, \nu\in\Psi$ such that $u_{-\infty}=\mu, u_{+\infty}=\nu$.
\end{lemma}

\begin{proof}
To prove $\mu\in\Psi$, arguing by contradiction, we suppose that there exists
 $\theta>0$ such that $u_n\not\in B_\theta(\Psi)$ for all $n$ near $-\infty$. 
Then, we have
 $$
J(u)\geq\sum_{n=-\infty}^n[-p_nF(u_n)]
\geq\underline{p}\sum_{n=-\infty}^n\varphi_\theta, \forall n\in \mathbb{Z},
$$
 which contradicts with $J(u)<+\infty$. Thus, $\mu\in\Psi$.
 The proof of $\nu\in\Psi$ is similar to the proof of $\mu\in\Psi$.
\end{proof}

By using the ideas developed in \cite{XLS,ZL}, we can easily obtain the 
following three lemmas, but for the sake of completeness, we give the proofs.

\begin{lemma}\label{lem2.5}
For any given $\rho\in\Psi\backslash\{0\}$, assume that 
$\{u^{(k)}\}_{k=1}^\infty$ is a minimizing sequence for \eqref{e1.1} 
restricted to $\Gamma_\epsilon(\rho)$ such that
 $u_n^{(k)}\to  u\in E$ and $J(u)<+\infty$, then $u\in\Gamma_\epsilon(\rho)$.
\end{lemma}

\begin{proof}
First, $u_n\not\in B_\epsilon(\Psi\setminus\{0,\rho\})$ for all $n\in \mathbb{Z}$. 
Otherwise, there is $n_0$ and $\psi\in\Psi\setminus\{0,\rho\}$ such that 
$u_{n_0}\in B_\epsilon(\psi)$. Therefore, for sufficiently large $k$, we have
 $$
|u_{n_0}^{(k)}-\psi|\leq|u_{n_0}^{(k)}-u_{n_0}|  +|\psi-u_{n_0}|<\epsilon,
$$
which is a contradiction.

Then $u_{-\infty}=\mu\in\{0,\rho\}, u_{+\infty}=\nu\in\{0,\rho\}$. 
Otherwise, for sufficiently large $k_1$ and
 $k_2$, we have
 $$
|u_{-k_1}^{(k)}-\mu|\leq|u_{-k_1}^{(k)}-u_{-k_1}|
 +|u_{-k_1}-\mu|<\epsilon,
$$
 and
 $$
|u_{k_2}^{(k)}-\nu|\leq|u_{k_2}^{(k)}-u_{k_2}|
 +|u_{k_2}-\nu|<\epsilon,
$$
 which are contradictions.

 Next, $u_{-\infty}=0$. From $u^{(k)}\in\Gamma_\epsilon(\rho)$, 
$u_n^{(k)}\in B_\epsilon(0)$ and $u_n^{(k)}\in \bar{B}_\epsilon(0)$ for $n<0$. 
Therefore, $\mu\in\bar{B}_\epsilon(0)\cap \{0,\rho\}=\{0\}$.

 Finally, $u_{+\infty}=\rho$. Otherwise, $u_{+\infty}=0$. 
If $u_1^{(k)}\in B_\epsilon(0)$, then $|\Delta u_0^{(k)}|\geq\delta$. Thus,
\begin{equation}\label{e2.4}
 J\big(u^{(k)}\big)\geq\frac{\delta^2}{2}
+\frac{1}{2}\sum_{n=2}^{+\infty}|\Delta u_n^{(k)}|^2
-\sum_{n=2}^{+\infty}p_nF\big(u_n^{(k)}\big).
\end{equation}
 If $u_1^{(k)}\not\in B_\epsilon(0)$, then there is an $n^{(k)}\leq 1$ such that
 $u_n^{(k)}\not\in B_{\frac{\epsilon}{2}}\{\Psi\}$, 
$n=n^{(k)}, n^{(k)}+1, \dots, 1$. It follows from Lemma \ref{lem2.2} that
\begin{equation}\label{e2.5}
\begin{aligned}
& J\big(u^{(k)}\big)\\
&\geq\frac{1}{2}\sum_{n=n^{(k)}}^1|\Delta u_n^{(k)}|^2
 -\sum_{n=n^{(k)}}^1p_nF\big(u_n^{(k)}\big)
 +\frac{1}{2}\sum_{n=2}^{+\infty}|\Delta u_n^{(k)}|^2 
  -\sum_{n=2}^{+\infty}p_nF\big(u_n^{(k)}\big)\\
&\geq\frac{\sqrt{2\underline{p}\varphi_{\frac{\epsilon}{2}}}\epsilon}{2}
 +\frac{1}{2}\sum_{n=2}^{+\infty}|\Delta u_n^{(k)}|^2
-\sum_{n=2}^{+\infty}p_nF\big(u_n^{(k)}\big).
\end{aligned}
\end{equation}
 Set
 $$
M=\min\big\{\frac{\delta^2}{2},\frac{\sqrt{2\underline{p}
\varphi_{\frac{\epsilon}{2}}}\epsilon}{2}\big\}.
$$
 By \eqref{e2.4} and \eqref{e2.5}, we have
\begin{equation}\label{e2.6}
 J\big(u^{(k)}\big)\geq M+\frac{1}{2}\sum_{n=2}^{+\infty}|\Delta u_n^{(k)}|^2
-\sum_{n=2}^{+\infty}p_nF\big(u_n^{(k)}\big).
\end{equation}
 Since $u_{+\infty}=0$, there is $\tilde{n}\geq1$ such that
 $$
u_n^2\leq\frac{M}{16}, \quad \forall n\geq\tilde{n}.
$$
 For $k$ large enough, we have
 $$
\Big(u_{\tilde{n}}^{(k)}\Big)^2 \leq\frac{M}{12}, \quad
\Big(u_{{\tilde{n}+1}}^{(k)}\Big)^2\leq\frac{M}{12}.
$$
 Denote
 $$
v_n^{(k)}=\begin{cases}
  0, & n<\tilde{n}+1,\\[3pt]
 u_n^{(k)}, & n\geq\tilde{n}+1.
 \end{cases}
$$
 Thus,
\begin{equation}\label{e2.7}
\begin{aligned}
 |\Delta v_{\tilde{n}}^{(k)}|^2
&=|u_{\tilde{n}+1}^{(k)}|^2
 =|\Delta u_{\tilde{n}}^{(k)}+u_{\tilde{n}}^{(k)}|^2 \\
&\leq|\Delta u_{\tilde{n}}^{(k)}|^2+4|u_{\tilde{n}}^{(k)}|^2
 +2|u_{\tilde{n}+1}^{(k)}|^2
 \leq|\Delta u_{\tilde{n}}^{(k)}|^2+\frac{M}{2}.
\end{aligned}
\end{equation}
By \eqref{e2.6} and \eqref{e2.7}, we have
\begin{equation}\label{e2.8}
\begin{aligned}
 J\big(v^{(k)}\big)
&=\frac{1}{2}\sum_{n=\tilde{n}+1}^{+\infty}|\Delta v_n^{(k)}|^2
 -\sum_{n=\tilde{n}+1}^{+\infty}p_nF\big(v_n^{(k)}\big)\\
 &\leq\frac{1}{2}\sum_{n=2}^{+\infty}|\Delta u_n^{(k)}|^2
 -\sum_{n=2}^{+\infty}p_nF\big(u_n^{(k)}\big)+\frac{M}{2} \\
&\leq J\big(u^{(k)}\big)-\frac{M}{2}.
\end{aligned}
\end{equation}
 From \eqref{e2.8}, we have
 $$
\inf_{s\in\Gamma_\epsilon(\rho)}J(s)
\leq\inf_{s\in\Gamma_\epsilon(\rho)}J(s)-\frac{M}{2},
$$
 which is a contradiction. The proof is complete.
\end{proof}

\begin{lemma}\label{lem2.6}
For any given $\rho\in\Psi\backslash\{0\}$ and $0<\epsilon<\delta$, 
there is $\bar{u}=u_{\epsilon,\rho}\in\Gamma_\epsilon(\rho)$ such that 
$J(u_{\epsilon,\rho})=c_{\epsilon,\rho}$.
\end{lemma}

\begin{proof}
Assume that $\{u^{(k)}\}_{k=1}^\infty$ is a minimizing sequence for 
\eqref{e1.1} restricted to $\Gamma_\epsilon(\rho)$. There is a constant 
$K>0$ such that $J\big(u^{(k)}\big)\leq K$.

 On one hand, $\{u_0^{(k)}\}_{k=1}^\infty$ is a bounded sequence. If not, 
$\lim_{i\to \infty}u_0^{(k_i)}=\infty$ and there is $i_0\in \mathbb{N}$ such that
 $u_0^{(k_i)}\not\in B_\epsilon (\rho), i\geq i_0$. 
Consider $\{u_j^{(k_i)}\}_{i=i_0}^\infty$.
\smallskip

\noindent\textbf{Case 1.} If $u_j^{(k_i)}\in \bar{B}_\epsilon (\rho)$, then
 $J(u^{(k_i)})\geq\frac{|u_0^{(k_i)}-\rho-\epsilon|^2}{2}$
 and
 $J(u^{(k_i)})\to  \infty$, $i\to  +\infty$,
 which is a contradiction.
\smallskip

\noindent\textbf{Case 2.}
 If $u_j^{(k_i)}\not\in \bar{B}_\epsilon (\rho)$. Set
 $$
n_i=\{n>0:u_{n+j}^{(k_i)}\in \bar{B}_\epsilon (\rho), u_j^{(k_i)}
\not\in \bar{B}_\epsilon (\rho), \forall j\in\mathbb{Z}[0,n]\}.
$$
 Then 
\[
J(u^{(k_i)})\geq \sqrt{2\underline{p}\varphi_\epsilon}|u_0^{(k_i)}-u_{n_i}^{(k_i)}|+
 \frac{1}{2}|u_{n_i+j}^{(k_i)}-u_{n_i}^{(k_i)}|^2
\]
 and  $J(u^{(k_i)})\to  \infty$ as $i\to  +\infty$,
 which is also a contradiction.

 By the definition of the norm on $E$, $\{u^{(k)}\}_{k=1}^\infty$ 
is a bounded sequence. Thus, passing to a subsequence if necessary, 
there is $\bar{u}\in E$ such that $u^{(k)}$ weakly converges to $\bar{u}$.

 On the other hand, $J(\bar{u})<\infty$. As a matter of fact, for 
$-\infty<a<b<+\infty$, let
 $$
J(a,b,u)\geq\frac{1}{2}\sum_{n=a}^b|\Delta u_n|^2-\sum_{n=a}^b p_nF(u_n), u\in E.
$$
 Thus,
 $$
J(a,b,\bar{u})\leq c_{\epsilon,\rho}\leq K,
$$
 which implies that $J(\bar{u})\leq\inf_{u\in\Gamma_\epsilon(\rho)}J(u)$. 
It follows from Lemma \ref{lem2.5} that
 $\bar{u}\in\Gamma_\epsilon(\rho)$. 
Therefore, $J(u_{\epsilon,\rho})=c_{\epsilon,\rho}$.
\end{proof}

Set
 $$
c_\epsilon=\inf_{\rho\in\Psi\backslash\{0\}}c_{\epsilon,\rho}.
$$

\begin{lemma}\label{lem2.7}
For any given $\rho\in\Psi\backslash\{0\}$ and $0<\epsilon<\delta$, $c_\epsilon$ 
can be achieved by some $c_{\epsilon,\tau}=J(u_{\epsilon,\tau})$ with 
$\tau=\tau_{\epsilon}$ and $u=u_{\epsilon}=u_{\epsilon,\tau}$ is an interior 
point of $\Gamma_\epsilon(\tau)$.
\end{lemma}

\begin{proof}
Let $0<\epsilon^{(i)}<\delta$ is a sequence converging to 0. By $(A3)$, 
$\{\tau_{\epsilon^{(i)}}\}$ consists of finite elements. 
Thus, for larger $i$, $\tau_{\epsilon^{(i)}}=\tau$ independent of $i$. 
Denote $u^{(i)}=u_{\epsilon^{(i)},\tau}$. For each $i\in\mathbb{N}$, 
there is $N_i>0$ such that
 $$
u_{-n}^{(i)}\in B_\epsilon(0), u_n^{(i)}\in B_{\epsilon^{(i)}}(\tau), \quad
\forall n\geq N_i.
$$
 Assume that for all $i\in\mathbb{N}$, $u^{(i)}$ is not an interior point of 
$\Gamma_\epsilon(\tau)$. Thus, there is $n^{(i)}\in [-N_i,N_i]$ such that 
$u_{n^{(i)}}^{(i)}\in \overline{B_{\epsilon^{(i)}}(\Psi\setminus\{0,\tau\})}$. 
Then, there is $\omega^{(i)}\in\Psi\setminus\{0,\tau\}$ such that 
$u_{n^{(i)}}^{(i)}\in \overline{B_{\epsilon^{(i)}}(\omega^{(i)})}$ and 
$\omega_i=\omega$ independent of $i$. Set
 $$
\Omega_n^{(i)}=\begin{cases}
 u_n^{(i)},& n\leq n^{(i)},\\
 \omega, & n>n^{(i)}.
 \end{cases}
$$
 Therefore, we have $\Omega^{(i)}\in \Gamma_{\epsilon^{(i)}}(\omega)$ and
\begin{equation}\label{e2.9}
\begin{aligned}
&J(u^{(i)})-J(\Omega^{(i)}) \\
&=\frac{1}{2}\sum_{n=n^{(i)}+1}^{+\infty}|\Delta u_n^{(i)}|^2
-\sum_{n=n^{(i)}+1}^{+\infty}p_nF(u_n^{(i)})
 -\frac{1}{2}|\omega-u_{n^{(i)}}^{(i)}|^2.
\end{aligned}
\end{equation}
If there is $n>n^{(i)}$ such that $|\Delta u_n^{(i)}|>|\omega-u_{n^{(i)}}^{(i)}|$, 
then  $J(\Omega^{(i)})<J(u^{(i)})=c_{\epsilon^{(i)}}$ which is a contradiction 
to the definition of  $c_{\epsilon^{(i)}}$. 
Thus, 
\[
|\Delta u_n^{(i)}|\leq|\omega-u_{n^{(i)}}^{(i)}|\leq\epsilon^{(i)},quad
 \forall n>n^{(i)}.
\]
 From $u_\infty^{(i)}=\tau$, there is $m^{(i)}$ such that 
$u_{m^{(i)}}^{(i)}\in B_{\epsilon^{(0)}}(\tau),
 m^{(i)}>n^{(i)}$ and $u_n^{(i)}\not\in B_{\epsilon^{(0)}}(\Psi), n^{(i)}<n<m^{(i)}$. 
Since $u_{m^{(i)}}^{(i)}\in B_{\epsilon^{(0)}}(\tau),
 u_{m^{(i)}-1}^{(i)}\not\in B_{\epsilon^{(0)}}(\tau), 
|\Delta u_n^{(i)}|\leq|\omega-u_{m^{(i)}}^{(i)}|$, for $i$ large enough, we have
 $u_{m^{(i)}}^{(i)}\in B_{\epsilon^{(0)}}(\tau)\setminus 
\overline{B_{\frac{\epsilon^{(0)}}{2}}(\tau)}$.
 It follows from \eqref{e2.9} and Lemma \ref{lem2.2} that
\begin{equation}\label{e2.10}
\begin{aligned}
 &J(u^{(i)})-J(\Omega^{(i)}) \\
&\geq\frac{1}{2}\sum_{n=n^{(i)}+1}^{m^{(i)}-1}|\Delta u_n^{(i)}|^2
 -\sum_{n=n^{(i)}+1}^{m^{(i)}-1}p_nF(u_n^{(i)})
 -\frac{(\epsilon^{(i)})^2}{2}\\
 &\geq\sqrt{2\underline{p}\varphi_{\frac{\epsilon^{(0)}}{2}}}
 \sum_{n=n^{(i)}+1}^{m^{(i)}-1}|\Delta u_n^{(i)}|
 -\sqrt{2\underline{p}\varphi_{\frac{\epsilon^{(0)}}{2}}}|\Delta u_{m^{(i)}-1}^{(i)}|
 -\frac{(\epsilon^{(i)})^2}{2}\\
 &\geq\sqrt{2\underline{p}\varphi_{\frac{\epsilon^{(0)}}{2}}}|u_{m^{(i)}}^{(i)}
 -u_{n^{(i)}}^{(i)}|
 -\sqrt{2\underline{p}\varphi_{\frac{\epsilon^{(0)}}{2}}}\epsilon^{(i)}
 -\frac{(\epsilon^{(i)})^2}{2} \\
& \geq\sqrt{2\underline{p}\varphi_{\frac{\epsilon^{(0)}}{2}}}\epsilon^{(0)}
 -\sqrt{2\underline{p}\varphi_{\frac{\epsilon^{(0)}}{2}}}\epsilon^{(i)}
 -\frac{(\epsilon^{(i)})^2}{2}.
\end{aligned}
\end{equation}
 Since $\epsilon^{(i)}$ is a sequence converging to 0, for $i$ large enough, 
we have 
\[
J(\Omega^{(i)})\leq J(u^{(i)})-\sqrt{2\underline{p}
\varphi_{\frac{\epsilon^{(0)}}{2}}}\epsilon^{(0)},
\]
 which contradicts 
$J(u^{(i)})=J(u_{\epsilon^{(i)},\tau})
=\inf_{\rho\in\Psi\setminus\{0\}}c_\epsilon(\rho)$.
 The proof is complete.
\end{proof}


\section{Proof of main result}
In this section, we  proof Theorem \ref{thm1.1} 
using a variational method.

\begin{proof}[Proof of Theorem \ref{thm1.1}]
For any $n\in \mathbb{Z}$, it follows from Lemma \ref{lem2.7} that
\begin{equation}\label{e3.1}
 \frac{d}{d u_n}J|_{\Gamma_\epsilon(\tau)}(u)=0.
\end{equation}
 By \eqref{e1.1}, we have
\begin{equation}\label{e3.2}
 \frac{d}{d u_n}J|_{\Gamma_\epsilon(\tau)}(u)=\frac{d}{d u_n}J(u)
=-\Delta^2 u_{n-1}-p_nf(u_n).
\end{equation}
 From \eqref{e3.1} and \eqref{e3.2}, we know that $u=u_{\epsilon}=u_{\epsilon,\tau}$
 is a heteroclinic orbit of \eqref{e1.1} connecting 0 to $\tau$, which originates
 from $0$. And $\omega_{(\cdot)}=u_{(-\cdot)}$ is also a heteroclinic orbit
 of \eqref{e1.1} connecting $\tau$ to 0, which terminates at $\tau$. 
The proof is complete.
\end{proof}



\subsection*{Acknowledgments} This project is supported by the
National Natural Science Foundation of China (No. 11401121),
 and by the Hunan Provincial Natural Science Foundation of China (No. 2015JJ2075).

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\end{document}