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<h1>Infinitely homoclinic solutions in discrete Hamiltonian systems without coercive conditions</h1>
<p class="authors">
<span class="author">Fathi Khelifi\(^\ast \)</span>
</p>
<p class="date">October 26, 2019; accepted: March 23, 2020; published online: August 11, 2020.</p>
</div>
<div class="abstract"><p> In this paper, we investigate the existence of infinitely many solutions for the second-order self-adjoint discrete Hamiltonian system </p>
<div class="displaymath" id="a0000000002">
  \[ \Delta \left[p(n)\Delta u(n-1)\right]-L(n)u(n)+\nabla W(n,u(n))=0,\eqno (*) \]
</div>
<p> where \(n\in \mathbb {Z}, u\in \mathbb {R}^{N}, p,L:\mathbb {Z}\rightarrow \mathbb {R}^{N\times N}\) and \(W:\mathbb {Z}\times \mathbb {R}^{N}\rightarrow \mathbb {R}\) are no periodic in \(n\). The novelty of this paper is that \(L(n)\) is bounded in the sense that there two constants \(0{\lt}\tau _1{\lt}\tau _2{\lt}\infty \) such that </p>
<div class="displaymath" id="a0000000003">
  \[  \tau _1\left|u\right|^{2}{\lt}\left(L(n)u,u\right){\lt}\tau _2\left|u\right|^{2},\; \forall n\in \mathbb {Z},\;  u\in \mathbb {R}^{N},  \]
</div>
<p> \(W(t,u)\) satisfies Ambrosetti-Rabinowitz condition and some other reasonable hypotheses, we show that (\(*\)) has infinitely many homoclinic solutions <i class="itshape">via</i> the Symmetric Mountain Pass Theorem. Recent results in the literature are generalized and significantly improved. </p>
<p><b class="bf">MSC.</b> 39A11; 58E05; 70H05. </p>
<p><b class="bf">Keywords.</b> Homoclinic solutions; Discrete Hamiltonian systems; Symmetric Mountain Pass Theorem. </p>
</div>

<p>\(^\ast \)College of Sciences, Department of Mathematics, University of Hail, Arabi Saudi, email: <span class="tt">fathikhlifi77@yahoo.com</span>. </p>
<h1 id="a0000000004">1 Introduction</h1>
<p>Consider the second\(-\)order self-adjoint discrete Hamiltonian system </p>
<div class="equation" id="eq1">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq1} \Delta \left[p(n)\Delta u(n-1)\right]-L(n)u(n)+\nabla W(n,u(n))=0 \end{equation}
  </div>
  <span class="equation_label">1</span>
</p>
</div>
<p> where \(n\in \mathbb {Z}, u\in \mathbb {R}^{N}, p,L:\mathbb {Z}\rightarrow \mathbb {R}^{N\times N}\) and \(W:\mathbb {Z}\times \mathbb {R}^{N}\rightarrow \mathbb {R}\) is continuous differentiable in \(x\), the forward difference operator \(\Delta \) is defined by \(\Delta u(n) = u(n + 1)- u(n)\). As usual, we say a solution \(u(n)\) of <a href="#eq1" class="eqref">1</a> is homoclinic (to 0) if \(u(n)\rightarrow 0\) as \(n\rightarrow \pm \infty \). In addition, if \(u(n)\neq 0\) then \(u(n)\) is called a nontrivial homoclinic solution. It is clear that <a href="#eq1" class="eqref">1</a> can be written as an equivalent first order nonlinear nonautonomous discrete Hamiltonian system </p>
<div class="equation" id="eq2">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq2} \Delta X(t) = J\nabla H_{X}(t,u(t+1),z(t)), \end{equation}
  </div>
  <span class="equation_label">2</span>
</p>
</div>
<p> where \(X(t) = (u(t), z(t))^{T} \); \(z(t)\) is a discrete momentum variable defined by \(z(t) = p(t)\times \Delta u(t-1)\); \(H(t,X(t)) = \frac{1}{2p(t)}z^{2}+\frac{1}{2}q(t)u^{2}(t)- W(t,u(t))\) is called the Hamiltonian function, where \(J\) is the normal symplectic matrix. Moreover, <a href="#eq1" class="eqref">1</a> is a discretization of the following second order differential equation: </p>
<div class="equation" id="eq3">
<p>
  <div class="equation_content">
    \begin{equation} \label{eq3} (p(t)u'(t))'-L(t) u(t)+\nabla W(t,u(t))=0, \end{equation}
  </div>
  <span class="equation_label">3</span>
</p>
</div>
<p> which is also equivalent to a first order nonlinear Hamiltonian system. </p>
<p>Variational methods for difference equations, which allow one to achieve multiplicity results, were introduced by R.P. Agarwal, K. Perera, D. O’Regan see <span class="cite">
	[
	<a href="#APE" >3</a>
	]
</span>. Recently, such methods received considerable attention. We mention here the works of M. Migda, J. Migda, M. Zdanowicz see <span class="cite">
	[
	<a href="#MMM" >16</a>
	]
</span>, A. Pankov see <span class="cite">
	[
	<a href="#APA" >18</a>
	]
</span>, S. Stevic see <span class="cite">
	[
	<a href="#SS" >23</a>
	]
</span>. Moreover, a lot of attention has been devoted in recent years to find periodic solutions of discrete dynamic models, for example, see <span class="cite">
	[
	<a href="#AP" >2</a>
	, 
	<a href="#APE" >3</a>
	, 
	<a href="#AG" >4</a>
	, 
	<a href="#AHP" >5</a>
	, 
	<a href="#GIL" >9</a>
	, 
	<a href="#GUOY" >10</a>
	, 
	<a href="#GYU" >11</a>
	, 
	<a href="#PZ" >19</a>
	]
</span>, other authors studied the existence of positive solutions of discrete fractional systems see <span class="cite">
	[
	<a href="#ACT" >6</a>
	, 
	<a href="#AID" >12</a>
	]
</span>. It is our purpose in the present work to find other types of solutions, namely the doubly asymptotic solutions, first discovered by Poincaré <span class="cite">
	[
	<a href="#POIN" >20</a>
	]
</span> in continuous Hamiltonian systems. In the past 40 years, system (<a href="#eq3">3</a>) has bee widely investigated, see <span class="cite">
	[
	<a href="#ARb" >1</a>
	, 
	<a href="#HW" >13</a>
	, 
	<a href="#OW" >17</a>
	, 
	<a href="#S" >22</a>
	, 
	<a href="#TK" >24</a>
	]
</span> and references therein. System (<a href="#eq3">3</a>) is the special form of the Emden-Fowler equation, appearing in the study of astrophysics, gas dynamics, fluid mechanics, relativistic mechanics, nuclear physics and chemically reacting systems, and many well-known results concerning properties of solutions of (<a href="#eq3">3</a>) are collected in <span class="cite">
	[
	<a href="#W" >25</a>
	]
</span>. When \(W(n, x)\) is an even function on \(x\), there are few result on existence of infinitely many homoclinic orbits for discrete Hamiltonian systems, because it is often very difficult to verify the last condition of the Symmetric Mountain Pass Theorem, different from the Mountain Pass Theorem. </p>
<p>In this paper we show that the Palais-Smale condition is satisfied on the unbounded domain and we use the usual Mountain Pass Theorem to prove the existence of a homoclinic orbit of (<a href="#eq1">1</a>). Moreover, if \(W (n,.)\) is an even function, we prove that (<a href="#eq1">1</a>) possesses an unbounded sequence of homoclinic orbits emanating from 0 by invoking the Symmetric Mountain Pass Theorem. </p>
<p>For the statement of our main result, the potential \(W(t,x)\) is supposed to satisfy the following conditions: </p>
<ul class="itemize">
  <li><p>\(p(n)\) is symmetric and positive definite matrix for all \(n\in \mathbb {Z}\). </p>
</li>
  <li><p>\(L(n)\) is symmetric and positive definite matrix for all \(n\in \mathbb {Z}\) and there are two constants \(0{\lt}\tau _1{\lt}\tau _2{\lt}\infty \) such that </p>
<div class="displaymath" id="a0000000005">
  \[ \tau _1\left|x\right|^{2}{\lt}\left(L(n)x,x\right){\lt}\tau _2\left|x\right|^{2},\quad \forall n\in \mathbb {Z},\;  x\in \mathbb {R}^{N}.  \]
</div>
</li>
  <li><p>there exists a constant \(\mu {\gt}2\) such that </p>
<div class="displaymath" id="a0000000006">
  \[  0{\lt}\mu W(n,x)\leq (\nabla W(n,x),x),\quad \forall n\in \mathbb {Z},\;  x\in \mathbb {R}^N\backslash \{ 0\} .  \]
</div>
</li>
  <li><p>there exists some positive function \(a:\mathbb {Z}\rightarrow \mathbb {R}\) with: </p>
<div class="displaymath" id="12">
  \begin{eqnarray} \label{12} \lim _{\left|n\right|\rightarrow \infty }a(n)=0, \end{eqnarray}
</div>
<p> such that </p>
<div class="displaymath" id="a0000000007">
  \begin{eqnarray*}  \left|\nabla W(n,x),x)\right|\leq a(n)\left|x\right|^{\mu -1},\quad \forall \left(n,x\right)\in \mathbb {Z}\times \mathbb {R}^{n}.\end{eqnarray*}
</div>
</li>
  <li><p>\(\varrho =\sup \left\{ W(n,x): n\in \mathbb {Z}, \left|x\right|=1\right\} {\lt}\frac{1}{2C^{2}_{2}}\), where \(C_2\) is defined in (<a href="#sob">5</a>). </p>
</li>
</ul>
<p><div class="remark_thmwrapper " id="rmk1">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">1</span>
  </div>
  <div class="remark_thmcontent">
  <p> \((i)\) From \((L)\), we assume that \(L(n)\) is bounded. Therefore, the smallest eigenvalue of \(L(n)\) does not tend to \(\infty \) as \(\left|n\right|\rightarrow \infty ,\) <i class="it">i.e.</i>, \(L(n)\) need not satisfy the various coercive conditions in the above mentioned papers. </p>
<p>\((ii)\) As mentioned above, the coercive conditions are used to establish some compact embedding theorems to guarantee that (PS) condition holds, which is the essential step to obtain the existence of homoclinic solutions of (<a href="#eq1">1</a>) <i class="itshape">via</i> Mountain Pass Theorem. In present paper, we assume that \(L(n)\) is bounded and could not obtain some compact embedding theorem. Therefore, one difficulty is to adapt some new technique to overcome this difficulty and test that (PS) condition is verified.<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p>In this paper, we will prove the following Theorems. </p>
<p><div class="theorem_thmwrapper " id="thm1">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">2</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Suppose that \((P)\), \((L)\), \((W_1), (W_2)\) and \((W_3)\) hold. Then <a href="#eq1" class="eqref">1</a> possesses at least one nontrivial homoclinic solution. </p>

  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="thm2">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Assume that \((P)\),\((L)\), \((W_1), (W_2)\) and \((W_3)\) are satisfied. Moreover, assume that \(W(t,x)\) is even in \(x\), that is, </p>
<ul class="itemize">
  <li><p>\(W(n,-x)=W(n,x)\) for all \(n\in \mathbb {Z}\), and \(x\in \mathbb {R}^{N}\), </p>
</li>
</ul>
<p> then <a href="#eq1" class="eqref">1</a> has infinitely many nontrivial homoclinic solutions. </p>

  </div>
</div> </p>
<h1 id="a0000000008">2 Preliminaries</h1>
<p>Let </p>
<div class="displaymath" id="a0000000009">
  \[ S=\left\{ \left\{ u(n)\right\} _{n\in \mathbb {Z}}, u(n)\in \mathbb {R}, n\in \mathbb {Z}\right\} ,  \]
</div>
<div class="displaymath" id="a0000000010">
  \[  E=\bigg\{ u\in S; \sum _{n\in \mathbb {Z}}\left[\left(p(n+1)\Delta u(n),\Delta u(n))+(L(n)u(n),u(n)\right)\right]{\lt}\infty \bigg\} ,  \]
</div>
<p> for \(u, v \in E\), let<br /></p>
<div class="displaymath" id="a0000000011">
  \begin{gather*}  \langle u,v\rangle =\sum _{n\in \mathbb {Z}}\left[(p(n+1)\Delta u(n),\Delta v(n))+(L(n)u(n),v(n))\right]. \end{gather*}
</div>
<p> Then \(E\) is a Hilbert space with the above inner product, and the corresponding norm is: </p>
<div class="displaymath" id="a0000000012">
  \begin{gather*}  \left\| u\right\| =\sum _{n\in \mathbb {Z}}\left[(p(n+1)\Delta u(n),\Delta u(n))+(L(n)u(n),u(n))\right], \quad u\in E. \end{gather*}
</div>
<p> As usual, for \(1\leq p\leq +\infty \), let </p>
<div class="displaymath" id="a0000000013">
  \begin{gather*}  l^{p}(\mathbb {Z},\mathbb {R}^{N})=\bigg\{ u\in S; \sum _{n\in \mathbb {Z}}\left|u(n)\right|^{p}{\lt}\infty \bigg\} , \end{gather*}
</div>
<div class="displaymath" id="a0000000014">
  \begin{gather*}  l^{\infty }(\mathbb {Z},\mathbb {R}^{N})=\bigg\{ u\in S; \sup _{n\in \mathbb {Z}}\left|u(n)\right|{\lt}\infty \bigg\} , \end{gather*}
</div>
<p> and their norms are defined by </p>
<div class="displaymath" id="a0000000015">
  \begin{gather*}  \left\| u\right\| _{q}=\bigg( \sum _{n\in \mathbb {Z}}\left|u(n)\right|^{q}\bigg)^{\frac{1}{q}}, \quad \forall u\in l^{q}(\mathbb {Z},\mathbb {R}^{N}); \end{gather*}
</div>
<div class="displaymath" id="a0000000016">
  \begin{gather*}  \left\| u\right\| _{\infty }=\sup _{n\in \mathbb {Z}}\left|u(n)\right|, \quad \forall u\in l^{\infty }(\mathbb {Z},\mathbb {R}^{N}); \end{gather*}
</div>
<p> respectively. Obviously, \(E\) is continuously embedded into \(l^{q}(\mathbb {Z},\mathbb {R}^{N})\) for \(2\leq q\leq +\infty \) , <i class="it">i.e.</i>, there exists \(C_q{\gt}0\) such that </p>
<div class="displaymath" id="sob">
  \begin{gather} \label{sob} \left\| u\right\| _{q}\leq C_q\left\| u\right\| ,\quad \forall u\in E. \end{gather}
</div>
<p>For any \(n_1,n_2\in \mathbb {Z}\) with \(n_1{\lt}n_2\), we let \(\mathbb {Z}(n_1,n_2)=[n_1,n_2]\cap \mathbb {Z}\), and for function \(f:\mathbb {Z}\rightarrow \mathbb {R}\) and \(a\in \mathbb {R}\), we set</p>
<div class="displaymath" id="a0000000017">
  \[ \mathbb {Z}(f(n)\geq a)=\left\{ n\in \mathbb {Z}: f(n)\geq a\right\} , \; \mathbb {Z}(f(n)\leq a)=\left\{ n\in \mathbb {Z}: f(n)\leq a\right\} .  \]
</div>
<p> Define the functional \(I: E \to \mathbb {R}\) by </p>
<div class="displaymath" id="I">
  \begin{align} \label{I} I(u)= \tfrac {1}{2} \| u\| ^2 - \sum _{n\in \mathbb {Z}} W(n,u(n)). \end{align}
</div>
<p> Under the conditions of <a href="#thm1">theorem 2</a>, we have </p>
<div class="displaymath" id="I'">
  \begin{align}  \label{I'} &  I’(u)v = \\ &  =\sum _{n\in \mathbb {Z}}\left[(p(n+1)\Delta u(n),\Delta v(n))+(L(n)u(n),v(n))-(\nabla W(n,u(n)),v(n))\right]\nonumber \end{align}
</div>
<p> for all \(u,v \in E\). Moreover, \(I\) is a continuously Fréchet- differentiable functional defined on \(E\), <i class="it">i.e.</i>, \(I\in C^{1}(E, \mathbb {R})\). </p>
<p>Observe that for all \(u,v \in E\) </p>
<div class="displaymath" id="delta">
  \begin{align} \label{delta} & \sum _{n\in \mathbb {Z}}\left[(p(n+1)\Delta u(n),\Delta v(n))+(L(n)u(n),v(n))-(\nabla W(n,u(n)),v(n))\right]= \\ & =\sum _{n\in \mathbb {Z}}\left[(-\Delta (p(n)\Delta u(n-1))+L(n)u(n)+\nabla W(n,u(n)),v(n))\right].\nonumber \end{align}
</div>
<p> It follows from (<a href="#I'">7</a>) and (<a href="#delta">8</a>) that \(\langle I'(u),v\rangle =0\) for all \(v\in E\) if only if </p>
<div class="displaymath" id="a0000000018">
  \[ \Delta \left[p(n)\Delta u(n-1)\right]-L(n)u(n)+\nabla W(n,u(n))=0,\quad \forall n\in \mathbb {Z}. \]
</div>
<p> So, the critical points of \(I\) in \(E\) are the solutions of system (<a href="#eq1">1</a>) with \(u(\pm \infty )=0\). </p>
<p>We will obtain the critical points of I by the Mountain Pass Theorem and the Symmetric Mountain Pass Theorem. Therefore, we state the theorems precisely. </p>
<p><div class="lemma_thmwrapper " id="MPT">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">4</span>
    <span class="lemma_thmtitle"><span class="cite">
	[
	<a href="#AR" >21</a>
	]
</span></span>
  </div>
  <div class="lemma_thmcontent">
  <p>  Let \(E\) be a real Banach space and \(I \in C^{1}(E,\mathbb {R})\) satisfying the Palais-Smale condition. If \(I\) satisfies the following conditions: </p>
<ul class="itemize">
  <li><p>\(I(0)=0\), </p>
</li>
  <li><p>there exist constants \(\rho ,\beta {\gt}0\) such that \(I_{/\partial {B_{\rho }}(0)}\geq \beta \), </p>
</li>
  <li><p>there exist \(e\in E\backslash {\overline{B}_{\rho }(0)}\) such that \(I(e)\leq 0\). </p>
</li>
</ul>
<p> Then \(I\) possesses a critical value \(c\geq \beta \) given by </p>
<div class="displaymath" id="a0000000019">
  \[  c=\inf _{g\in \Gamma }\max _{s \in [0,1]}I(g(s)),  \]
</div>
<p> where </p>
<div class="displaymath" id="a0000000020">
  \[  \Gamma =\{  g\in C([0,1],E): g(0)=0,g(1)=e \} .  \]
</div>

  </div>
</div> <div class="lemma_thmwrapper " id="SMPT">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">5</span>
    <span class="lemma_thmtitle"><span class="cite">
	[
	<a href="#AR" >21</a>
	]
</span></span>
  </div>
  <div class="lemma_thmcontent">
  <p>  Let \(E\) be a real Banach space and \(I \in C^{1}(E,\mathbb {R})\) satisfying the Palais-Smale condition. If \(I\) is even and satisfies the following conditions: </p>
<ul class="itemize">
  <li><p>\(I(0)=0\), </p>
</li>
  <li><p>there exist constants \(\rho ,\beta {\gt}0\) such that \(I_{/\partial {B_{\rho }}(0)}\geq \beta \), </p>
</li>
  <li><p>For each finite dimensional \(\tilde{E}\subset E\) there is \(\gamma =\gamma (\tilde{E})\) such that \(I(0)\leq 0\) on \(\tilde{E}\backslash {\rho _{\gamma }}\). </p>
</li>
</ul>
<p> Then \(I\) possesses an unbounded sequence of critical values. </p>

  </div>
</div> </p>
<h1 id="a0000000021">3 Proof of Theorems</h1>
<p> For the Proof of our Theorems we need some technical Lemmas. <div class="lemma_thmwrapper " id="lem3">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">6</span>
  </div>
  <div class="lemma_thmcontent">
  <p> Under the conditions of <span class="rm"><a href="#thm1">theorem 2</a></span>, \(\varphi '\) is compact, <i class="it">i.e.</i>, \(\varphi '(u_k)\rightarrow \varphi '(u)\) if \(u_k\rightharpoonup u\;  in\;  E\), where \(\varphi :E\rightarrow \mathbb {R}\) is defined by </p>
<div class="displaymath" id="3.3">
  \begin{eqnarray} \label{3.3} \varphi (u)=\sum _{n\in \mathbb {Z}}W(n,u).\end{eqnarray}
</div>

  </div>
</div> <div class="proof_wrapper" id="a0000000022">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Assume that \(u_k\rightharpoonup u\;  in\;  E\). Then there exists a constant \(M{\gt}0\) such that</p>
<div class="displaymath" id="a0000000023">
  \[ \left\| u_k\right\| \leq M\; \; and\; \; \left\| u\right\| \leq M \]
</div>
<p>for \(k\in \mathbb {N}\). In addition, from \((W_2)\), for any \(\epsilon {\gt}0\), we can choose \(T_0{\gt}0\) such that </p>
<div class="equation" id="37">
<p>
  <div class="equation_content">
    \begin{equation} \label{37} \left|\nabla W(n,u)\right|\leq \epsilon \left|u\right|^{\mu -1}, \; and\; \left|\nabla W(n,u_k)\right|\leq \epsilon \left|u_k\right|^{\mu -1}, \quad \forall \left|n\right|\geq T_0. \end{equation}
  </div>
  <span class="equation_label">10</span>
</p>
</div>
<p> Consequently, for \(k\) large enough, and using Young inequality we have </p>
<div class="displaymath" id="38">
  \begin{align} \label{38} & \left|\left(\phi ’(u_k)-\phi ’(u)\right)v\right|\leq \nonumber \\ &  \leq \sum _{n\in \mathbb {Z}}\left|\nabla W(n,u_k(n))-\nabla W(n,u(n))\right|\left|v(n)\right|\nonumber \\ & \leq \sum _{\left|n\right|\leq T_0}\left|\nabla W(n,u_k(n))-\nabla W(n,u(n))\right|\left|v(n)\right|\nonumber \\ & \quad +\sum _{\left|n\right|{\gt} T_0}\left|\nabla W(n,u_k(n))\right|\left|v(n)\right|+\sum _{\left|n\right|{\gt} T_0}\left|\nabla W(n,u(n))\right|\left|v(n)\right|\nonumber \\ & \leq \epsilon \left\| v\right\| _\infty +\epsilon \sum _{\left|n\right|{\gt} T_0}\left|u_k(n)\right|^{\mu -1}\left|v\right|+\epsilon \sum _{\left|n\right|{\gt} T_0}\left|u(n)\right|^{\mu -1}\left|v\right|\nonumber \\ & \leq \epsilon \gamma _{\infty }\left\| v\right\| +\epsilon \sum _{\left|n\right|{\gt} T_0}\left(\tfrac {\mu -1}{\mu }\left|u_k(n)\right|^{\mu }+\tfrac {1}{\mu }\left|v\right|^{\mu }\right)+ \epsilon \sum _{\left|n\right|{\gt}T_0}\left(\tfrac {\mu -1}{\mu }\left|u(n)\right|^{\mu }+\tfrac {1}{\mu }\left|v\right|^{\mu }\right)\leq \nonumber \end{align}
</div>
<div class="displaymath" id="a0000000024">
  \begin{align} & \leq \epsilon \gamma _\infty \left\| v\right\| +\epsilon \tfrac {\mu -1}{\mu }\sum _{\left|n\right|{\gt}T_0}\left(\left|u_k\right|^{\mu }+\left|u\right|^{\mu }\right)+\epsilon \tfrac {2}{\mu }\left|v\right|^{\mu }. \end{align}
</div>
<p> Consequently, we obtain that </p>
<div class="displaymath" id="440">
  \begin{eqnarray} \label{440} \left\| \varphi ’(u_k)-\varphi ’(u)\right\| & =& \sup _{\left\| v\right\| =1}\left|\sum _{\mathbb {Z}}\left(\nabla W(n,u_k(n))-\nabla W(n,u(n)),v(n)\right)\right|\nonumber \\ & \leq & \epsilon C_{\infty }+2\epsilon \left(C_{\mu }M\right)^{\mu }\tfrac {\mu -1}{\mu }+\epsilon C^{\mu }_{\mu }\tfrac {2}{\mu }, \end{eqnarray}
</div>
<p> which yields \(\varphi '(u_k)\rightarrow \varphi '(u)\) as \(u_k\rightharpoonup u,\) that is, \(\varphi '\) is compact. <div class="proof_wrapper" id="a0000000025">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> <div class="lemma_thmwrapper " id="leml2">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">7</span>
  </div>
  <div class="lemma_thmcontent">
  <p><span class="cite">
	[
	<a href="#IJ2" >14</a>
	]
</span>  Under the assumption of \((W_1)\), there exist two constants \(d_1{\gt}0,\; d_2{\gt}0\) such that </p>
<ul class="itemize">
  <li><p>\(W(n,u)\leq d_1\left|u\right|^{\mu }\)  for \(n\in \mathbb {Z}\)  and \(0{\lt}\left|u\right|{\lt}1\), </p>
</li>
  <li><p>\(W(n,u)\geq d_2\left|u\right|^{\mu }\)  for \(n\in \mathbb {Z}\)  and \(\left|u\right|\geq 1\). </p>
</li>
</ul>

  </div>
</div> </p>
<p><div class="lemma_thmwrapper " id="lemPS">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">8</span>
  </div>
  <div class="lemma_thmcontent">
  <p> Under the condition of <span class="rm"><a href="#thm1">theorem 2</a></span>, \(I\) satisfies the Palais-Smale condition. </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000026">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Assume that \((u_k)_{k\in \mathbb {N}}\in E\) is a sequence such that \((I(u_k))\) is bounded and \(I'(u_k)\rightarrow 0\) as \(k\rightarrow \infty \). Then there exists a constant \(C_1{\gt}0\) such that </p>
<div class="displaymath" id="3.1">
  \begin{eqnarray} \label{3.1}\left|I(u_k)\right|\leq C_1\; \; and\; \; \left\| I’(u_k)\right\| \leq C_1\end{eqnarray}
</div>
<p> for every \(k\in \mathbb {N}\). We first prove that \((u_k)\) is bounded in \(E\). By (<a href="#I">6</a>), (<a href="#I'">7</a>), \((W_1)\) and\((W_2)\) , we obtain </p>
<div class="displaymath" id="3111">
  \begin{align}  \label{3111} C_1& +\tfrac {C_1}{\mu }\left\| u_k\right\| \geq \nonumber \\ \geq & I(u_k)-\tfrac {1}{\mu }\left\| I’(u_k)\right\| u_k \nonumber \\ =& (\tfrac {1}{2}-\tfrac {1}{\mu })\left\| u_k\right\| ^{2}+\sum _{n\in \mathbb {Z}}\left[W(n,u_k(n))-\tfrac {1}{\mu }(\nabla W(n,u_k(n)),u_k(n))\right] \nonumber \\ \geq & (\tfrac {1}{2}-\tfrac {1}{\mu })\left\| u_k\right\| ^{2}, \;  k\in \mathbb {N}. \end{align}
</div>
<p> Since \(\mu {\gt}2\), the inequality (<a href="#3111">14</a>) shows that \((u_k)\) is bounded in \(E\). So passing to a subsequence if necessary, it can be assumed that \(u_k\rightharpoonup u\) in \(E\), which yields that </p>
<div class="displaymath" id="310">
  \begin{eqnarray} \label{310} \left(I’(u_k)-I’(u)\right)\left(u_k-u\right)\rightarrow 0, \quad as\; k\rightarrow \infty . \end{eqnarray}
</div>
<p> Moreover, according to <a href="#lem3">lemma 6</a>, we have </p>
<div class="displaymath" id="311">
  \begin{eqnarray} \label{311} \phi ’(u_k)\rightarrow \phi ’(u), \quad as\; k\rightarrow \infty . \end{eqnarray}
</div>
<p> It follows from the definition of \(I\) that </p>
<div class="displaymath" id="505">
  \begin{eqnarray} \label{505} & & (I’(u_k)-I’(u))(u_k-u)=\nonumber \\ & =& \left\| u_k-u\right\| ^{2}-\sum _{n\in \mathbb {Z}}\left(\nabla W(n,u_k(n))-\nabla W(n,u(n)),u_k-u\right). \end{eqnarray}
</div>
<p> combining (<a href="#310">15</a>), (<a href="#311">16</a>) with (<a href="#505">17</a>) we obtain that \(u_k\rightarrow u\) in \(E\). So the proof is complete. <div class="proof_wrapper" id="a0000000027">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><b class="bf">Proof of <a href="#thm1">theorem 2</a>.</b> </p>
<p>We will prove the existence of a nontrivial critical point of \(I\).We have already shown that \(I\in C^{1}(E, \mathbb {R})\), \(I(0)=0\) and \(I\) satisfies the Palais-Smale condition. Hence it suffices to prove that \(I\) satisfies \((ii)\) and \((iii)\) of <a href="#MPT">lemma 4</a>. </p>
<p>Let \(\rho =\frac{1}{C_2}\), where \(C_2\) is defined in (<a href="#sob">5</a>). Assume that \(u\in E\) with \(\left\| u\right\| \leq \rho \), we have \(\left\| u\right\| _{\infty }\leq \frac{1}{C_2}.C_2=1\). In consequence, combining this with \((i)\) of <a href="#lem3">lemma 6</a>, we obtain that </p>
<div class="displaymath" id="312">
  \begin{eqnarray} \label{312} I(u)& =&  \tfrac {1}{2} \| u\| ^2-\sum _{n\in \mathbb {Z}} W(n,u(n))\nonumber \\ & \geq &  \tfrac {1}{2} \| u\| ^2-\sum _{n\in \mathbb {Z}} W(n,\tfrac {u(n)}{\left|u(n)\right|})\left|u(n)\right|^{\mu }\nonumber \\ & \geq & \tfrac {1}{2}\left\| u\right\| ^{2}-\varrho \sum _{\mathbb {Z}}\left|u(n)\right|^{2}\nonumber \\ & \geq & \left(\tfrac {1}{2}-\varrho C^{2}_{2}\right)\left\| u\right\| ^{2},\quad \left\| u\right\| \leq \rho , \end{eqnarray}
</div>
<p> where \(\varrho =\sup \left\{ W(n,u): n\in \mathbb {Z}, \left|u\right|=1\right\} \). Since \(\varrho {\lt}\frac{1}{2C^{2}_{2}}\), then we get </p>
<div class="displaymath" id="a0000000028">
  \[ I_{/\partial {B_{\rho }}(0)}\geq \tfrac {1}{C^{2}_{2}}-2\varrho =\alpha {\gt}0.  \]
</div>
<p> It remains to prove that there exists \(e\in E\) such that \(\left\| e\right\| {\gt}\rho \) and \(I(e)\leq 0\), where \(\rho \) is defined above. Take some \(u\in E\) such that \(\left\| u\right\| =1\). Then there exists a nonempty integer interval \(I\subset \mathbb {Z}\) such that \(u(n)\neq 0\) for \(n\in I\). take \(\sigma {\gt}0\) such that \(\sigma \left|u(n)\right|\geq 1\) for \(n\in I\). Then, we obtain </p>
<div class="displaymath" id="F">
  \begin{eqnarray} \label{F} I(\sigma u)& =& \tfrac {\sigma ^{2}}{2} \| u\| ^2-\sum _{n\in \mathbb {Z}}W(n,\sigma u(n))\nonumber \\ & \leq &  \tfrac {\sigma ^{2}}{2} \| u\| ^2-\sigma ^{\mu }\sum _{n\in I}W\left(n,\tfrac {u(n)}{\left|u(n)\right|}\right)\left|u(n)\right|^{\mu }\nonumber \\ & \leq &  \tfrac {\sigma ^{2}}{2} \| u\| ^2-m\sigma ^{\mu }\sum _{n\in I}\left|u(n)\right|^{\mu },\end{eqnarray}
</div>
<p> where \(m=\min \left\{ W(n,u): n\in I, \left|u\right|=1\right\} \). Since \(\mu {\gt}2\), (<a href="#F">19</a>) implies that \(I(\sigma u){\lt}0\) for some \(\sigma {\gt}0\) with \(\sigma \left|u(n)\right|\geq 1\) for \(n\in I\) and \(\left\| \sigma u\right\| {\gt}\rho \). By <a href="#MPT">lemma 4</a>, \(I\) possesses a critical value \(c\geq \beta {\gt}0\) given by </p>
<div class="displaymath" id="a0000000029">
  \[  c=\inf _{g\in \Gamma }\max _{s \in [0,1]}I(g(s)),  \]
</div>
<p> where </p>
<div class="displaymath" id="a0000000030">
  \[  \Gamma =\{  g\in C([0,1],E): g(0)=0,g(1)=e \} .  \]
</div>
<p> Hence there is \(u\in E\) such that \(I(u)=c, I'(u)=0\). <span class="qed">â–¡</span></p>
<p><b class="bf">Proof of <a href="#thm2">theorem 3</a>.</b> </p>
<p>We have already known that \(I\in C^{1}(E, \mathbb {R})\), \(I(0)=0\), \(I\) satisfies the Palais-Smale condition and we have \(I\) is even. To apply the Symmetric mountain Pass Theorem, it suffices to prove that \(I\) satisfies the conditions \((iii)\) of <a href="#SMPT">lemma 5</a>. \((ii)\) is identically the same as in <a href="#thm1">theorem 2</a>, so it is already proved. </p>
<p>Let \(\tilde{E}\) be a finite dimensional subspace of \(E\). Since all norms of a finite dimensional normed space are equivalent, so there is a constant \(c{\gt}0\) such that </p>
<div class="displaymath" id="317">
  \begin{eqnarray} \label{317}\left\| u\right\| {\lt}c\left\| u\right\| _{\infty },\; \; \; for \;  u\in \tilde{E}. \end{eqnarray}
</div>
<p> Assume that \(\dim \tilde{E}=m\) and \(u_1,u_2,\ldots ,u_m\) is the basis of \(\tilde{E}\) such that </p>
<div class="displaymath" id="a0000000031">
  \[  \langle u_i,u_j\rangle = \begin{cases}  c^{2} &  \text{if } i=j,\\ 0 &  \text{if } i\neq j,\; \; \;  i,j=1,2,...,m. \end{cases}  \]
</div>
<p> Since \(u_i\in E\), we can choose an integer \(\bar{T}\) such that </p>
<div class="displaymath" id="319">
  \begin{eqnarray} \label{319} \left|u_i(n)\right|{\lt}\tfrac {1}{m}, \quad \left|n\right|{\gt}\bar{T},\quad i=1,2,...,m.\end{eqnarray}
</div>
<p>Set \(\Sigma =\left\{ u\in \tilde{E},\left\| u\right\| =c\right\} \). Then for \(u\in \Sigma \), there exist \(\lambda _i\in \mathbb {R}, i=1,2,...,m\) such that </p>
<div class="displaymath" id="320">
  \begin{eqnarray} \label{320} u(n)=\sum ^{m}_{i=1}\lambda _iu_i(n), \quad for\; n\in \mathbb {Z}, \end{eqnarray}
</div>
<p> it follows that </p>
<div class="displaymath" id="3200">
  \begin{eqnarray} \label{3200} c^{2}=\left\| u\right\| ^{2}= \sum ^{m}_{i=1}\lambda {_i}^{2}\langle u_i,u_i\rangle =c^{2}\sum ^{m}_{i=1}\lambda {_i}^{2},\end{eqnarray}
</div>
<p> which implies that \(\left|\lambda _{i}\right|\leq 1\) for \(i=1,2,...,m.\) Hence, for \(u\in \Sigma ,\) let \(\left|u(n_0)\right|=\left\| u\right\| _{\infty }\), then by (<a href="#317">20</a>) and (<a href="#320">24</a>) we have </p>
<div class="displaymath" id="321">
  \begin{eqnarray} \label{321} 1\leq \left\| u\right\| _{\infty }=\left|u(n_0)\right|\leq \sum ^{m}_{i=1}\left|\lambda _i\right|\left|u_i(n_0)\right|,\quad u\in \Sigma . \end{eqnarray}
</div>
<p> This shows that there exists \(i_0\in \left\{ 1,2,...,m\right\} \) such that \(\left|u_{i^{0}}(n_0)\right|\geq \frac{1}{m}\), which together with (<a href="#319">23</a>), implies that \(\left|n_0\right|\leq \overline{T}.\) For any \(u\in E\), it follows from (<a href="#319">23</a>), (<a href="#320">24</a>), (<a href="#321">26</a>) and \((W_2)\), we have for \(u\in \sum \) and \(\sigma {\gt}1\) </p>
<div class="displaymath" id="325">
  \begin{eqnarray} \label{325} I(\sigma u)& =& \tfrac {\sigma ^{2}}{2}\left\| u\right\| ^{2}-\sum _{n\in \mathbb {Z}}W(n,\sigma u(n))\nonumber \\ & \leq &  \tfrac {\sigma ^{2}}{2}\left\| u\right\| ^{2}-\sigma ^{\mu }\sum _{n\in \mathbb {Z}}W(n,u(n))\nonumber \\ & =& \tfrac {\sigma ^{2}}{2}\left\| u\right\| ^{2}-\sigma ^{\mu }\sum _{\left|n\right|{\gt}\overline{T}}W(n,u(n))-\sigma ^{\mu }\sum _{\left|n\right|\leq \overline{T}}W(n,u(n))\nonumber \\ & \leq & \tfrac {\sigma ^{2}}{2}\left\| u\right\| ^{2}-d_2\sigma ^{\mu }\left|u(n_0)\right|. \end{eqnarray}
</div>
<p> Since \(\mu {\gt}2\), we deduce that there is \(\sigma _0=\sigma (\widetilde{E}){\gt}1\) such that </p>
<div class="displaymath" id="a0000000032">
  \begin{eqnarray*}  I(\sigma u){\lt}0,\; for\; u\in \sum \; and\;  \sigma \geq \sigma _0. \end{eqnarray*}
</div>
<p> That is </p>
<div class="displaymath" id="a0000000033">
  \begin{eqnarray*}  I( u){\lt}0,\; for\; u\in \widetilde{E}\; and\;  \left\| u\right\| \geq c\sigma _0. \end{eqnarray*}
</div>
<p> This shows that \((iii)\) of <a href="#SMPT">lemma 5</a> holds. By <a href="#SMPT">lemma 5</a>, \(I\) possesses an unbounded sequence \(\left\{ d_k\right\} _{k\in \mathbb {N}}\) of critical values with \(d_k=I(u_k)\), where \(u_k\) is such that \(I'(u_k)=0\) for \(k=1,2,...\) </p>
<p>From (<a href="#I">6</a>), we have </p>
<div class="displaymath" id="1000">
  \begin{eqnarray} \label{1000}\tfrac {1}{2}\left\| u_k\right\| ^{2}& =& d_k+\sum _{n\in \mathbb {Z}}W(n,u_k(n))\nonumber \\ & \geq & d_k,\end{eqnarray}
</div>
<p> since \(\left\{ d_k\right\} _{k\in \mathbb {N}}\) is unbounded, then \(\left\{ \left\| u_k\right\| \right\} _{k\in \mathbb {N}}\) is unbounded. The proof is complete.<span class="qed">â–¡</span></p>
<p><div class="acknowledgement_thmwrapper " id="a0000000034">
  <div class="acknowledgement_thmheading">
    <span class="acknowledgement_thmcaption">
    Acknowledgements
    </span>
  </div>
  <div class="acknowledgement_thmcontent">
  <p>The author thanks the referee for valuable comments and suggestions that improved the paper. </p>

  </div>
</div> </p>
<p><small class="footnotesize">  </small></p>
<div class="bibliography">
<h1>Bibliography</h1>
<dl class="bibliography">
  <dt><a name="ARb">1</a></dt>
  <dd><p><a href ="https://doi.org/10.1016/0022-1236(73)90051-7"> <i class="sc">A. Ambrosetti, P.H. Rabinowitz</i>, <i class="it">Dual variational methods in critical point theory and applications</i>, J. Funct. Anal., <b class="bf">14</b> (1973), pp.&#160;349-–381. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="AP">2</a></dt>
  <dd><p><a href ="https://doi.org/10.1016/0895-7177(95)00096-k"> <i class="sc">R.P. Agarwal, J. Popenda</i>, <i class="it">Periodic solution of first order linear difference equations</i>, Math. Comput. Modelling, <b class="bf">22</b> (1995) no. 1, pp.&#160;11-–19. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="APE">3</a></dt>
  <dd><p><a href ="https://doi.org/10.1155/ade.2005.93"> <i class="sc">R.P. Agarwal, K. Perera, D. O’Regan</i>, <i class="it">Multiple positive solutions of singular discrete p-Laplacian problems via variational methods</i>, Adv. Difference Eq., <b class="bfseries">2005</b> (2005) no. 2, pp.&#160;93–-99. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="AG">4</a></dt>
  <dd><p><a href ="https://doi.org/10.1201/9781420027020"> <i class="sc">R.P. Agarwal</i>, <i class="it">Difference Equations and Inequalities. Theory, Methods and Applications</i>, 2nd ed., Dekker,New York, 2000. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="AHP">5</a></dt>
  <dd><p><i class="sc">C.D. Ahlbrandt, A.C. Peterson</i>, <i class="it">Discrete Hamiltonian Systems: Difference Equations, Continued Fraction and Riccati Equations</i>, Kluwer Academic, Dordrecht, 1996. </p>
</dd>
  <dt><a name="ACT">6</a></dt>
  <dd><p><a href ="https://doi.org/10.7494/opmath.2018.38.1.31"> <i class="sc">A. Chidouh, D. Torres</i>, <i class="it">Existence of positive solutions to a discrete fractional boundary value problem and corresponding Lyapunov-type inequalities</i>, Opuscula Math., <b class="bf">38</b> (2018) no. 1, pp.&#160;31–40. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="CZR">7</a></dt>
  <dd><p><a href ="https://doi.org/10.1090/s0894-0347-1991-1119200-3"> <i class="sc">V. Coti Zelati, P.H. Rabinowitz</i>, <i class="it">Homoclinic orbits for a second order Hamiltonian systems possessing superquadratic potentials</i>, J. Amer. Math. Soc., <b class="bf">4</b> (1991), pp.&#160;693–727. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="DLI">8</a></dt>
  <dd><p><a href ="https://doi.org/10.1006/jmaa.1995.1037"> <i class="sc">Y. Ding, S. Li</i>, <i class="it">Homoclinic orbits for the first-order Hamiltonian systems</i>, J. Math. Anal. Appl., <b class="bf">189</b> (1995), pp.&#160;585–601. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="GIL">9</a></dt>
  <dd><p><i class="sc">M. Gil’</i>, <i class="it">Periodic solutions of abstract difference equation</i>, Appl. Math. E-Notes, 1 (2001), pp.&#160;18–23. </p>
</dd>
  <dt><a name="GUOY">10</a></dt>
  <dd><p><i class="sc">Z.M. Guo, J.S. Yu</i>, <i class="it">The existence of periodic and subharmonic solutions of subquadratic second order difference equations</i>, J. London Math. Soc., <b class="bf">68</b> (2003) no. 2, pp.&#160; 419–430. </p>
</dd>
  <dt><a name="GYU">11</a></dt>
  <dd><p><a href ="https://doi.org/10.1112/s0024610703004563"> <i class="sc">Z.M. Guo, J.S. Yu</i>, <i class="it">The existence of periodic and subharmonic solutions for second order superlinear difference equations</i>, Sci. China Ser. A, <b class="bfseries">33</b> (2003), pp.&#160;226–235 (in Chinese). <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="AID">12</a></dt>
  <dd><p><i class="sc">A. Iannizzotto, V.D. Radulescu</i>, <i class="it">Positive homoclinic solutions for the discrete \(p\)-Laplacian with a coercive weight function</i>, Diff. Integral Eq., <b class="bf">27</b> (2014) no. 1-2, pp.&#160;35–44. </p>
</dd>
  <dt><a name="HW">13</a></dt>
  <dd><p><a href ="https://doi.org/10.1007/bf01444543"> <i class="sc">H. Hofer, K. Wysocki</i>, <i class="it">First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems</i>, Math. Ann., <b class="bf">288</b> (1990), pp.483–503. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="IJ2">14</a></dt>
  <dd><p><a href ="https://doi.org/10.1016/j.jmaa.2007.02.038"> <i class="sc">M. Izydorek, J. Janczewska</i>, <i class="it">Homoclinic solutions for nonautonomous second order Hamiltonian systems with a coercive potential</i>, J. Math. Anal. Appl., <b class="bf">335</b> (2007), pp.&#160;1119–1127. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="MW">15</a></dt>
  <dd><p><a href ="https://doi.org/10.1007/978-1-4757-2061-7"> <i class="sc">J. Mawhin, M. Willem</i>, <i class="it">Critical Point Theory and Hamiltonian System</i>, Springer-Verlag, New York, 1989. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="MMM">16</a></dt>
  <dd><p><a href ="https://doi.org/10.7494/opmath.2019.39.1.61"> <i class="sc">M. Migda, J. Migda, M. Zdanowicz</i>, <i class="it">On the convergence of solutions to secondorder neutral difference equations</i>, Opuscula Math., <b class="bf">39</b> (2019) no. 1, pp.&#160;61–75. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="OW">17</a></dt>
  <dd><p><i class="sc">W. Omana, M. Willem</i>, <i class="it">Homoclinic orbits for a class of Hamiltonian systems</i>, Diff. Integral Eq., <b class="bf">5</b> (1992), pp.&#160;1115–1120. </p>
</dd>
  <dt><a name="APA">18</a></dt>
  <dd><p><a href ="https://doi.org/10.1515/anona-2017-0041"> <i class="sc">A. Pankov</i>, <i class="it">Homoclinics for strongly indefinite almost periodic second order Hamiltonian systems</i>, Adv. Nonlinear Anal., <b class="bf">8</b> (2019) no. 1, pp.&#160;372–385. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="PZ">19</a></dt>
  <dd><p><a href ="https://doi.org/10.1023/a:1010655000447"> <i class="sc">A. Pankov, N. Zakharchenko</i>, <i class="it">On some discrete variational problems</i>, Acta Appl. Math., <b class="bf">65</b> (2001) nos. 1–3, pp.&#160;295–303. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="POIN">20</a></dt>
  <dd><p><i class="sc">H. Poincaré</i>, <i class="it">Les méthodes nouvelles de la mécanique céleste</i>, Gauthier–Villars, Paris, 1899. </p>
</dd>
  <dt><a name="AR">21</a></dt>
  <dd><p><a href ="https://doi.org/10.1090/cbms/065"> <i class="sc">P.H. Rabinowitz</i>, <i class="it">Minimax Methods in Critical Point Theory with Applications in Differential Equations</i>, CBMS Reg. Conf. Ser. Math., <b class="bf">35</b>, Amer. Math. Soc., Providence, RI, 1986. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="S">22</a></dt>
  <dd><p><a href ="https://doi.org/10.1007/bf02570817"> <i class="sc">E. Séré</i>, <i class="it">Existence of infinitely many homoclinic systems</i>, Math. Z., <b class="bf">209</b> (1992), pp.&#160;27–42. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="SS">23</a></dt>
  <dd><p><i class="sc">S. Stevic</i>, <i class="it">Solvability of a product-type system of difference equations with six parameters</i>, Adv. Nonlinear Anal., <b class="bf">8</b> (2019) no. 1, pp.&#160;29–51. </p>
</dd>
  <dt><a name="TK">24</a></dt>
  <dd><p><a href ="https://doi.org/10.1016/0022-0396(91)90095-q"> <i class="sc">K. Tanaka</i>, <i class="it">Homoclinic orbits in a first-order superquadratic Hamiltonian system: Convergence of subharmonic orbits</i>, J. Diff. Eq., <b class="bf">94</b> (1991), pp.&#160;315–339. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="W">25</a></dt>
  <dd><p><a href ="https://doi.org/10.1137/1017036"> <i class="sc">J.S.W. Wong</i>, <i class="it">On the generalized Emden–Fowler equation</i>, SIAM Rev., <b class="bf">17</b> (1975), pp.&#160;339–360. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="YG">26</a></dt>
  <dd><p><a href ="https://doi.org/10.1112/s0024610704005939"> <i class="sc">J.S. Yu, Z.M. Guo, X.F. Zou</i>, <i class="it">Positive periodic solutions of second order self-adjoint difference equations</i>, J. London Math. Soc., <b class="bf">71</b> (1975) no. 2, pp.&#160;146–160 <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
</dl>


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