\(^\ast \)College of Sciences, Department of Mathematics, University of Hail, Arabi Saudi, email: fathikhlifi77@yahoo.com.

1 Introduction

Consider the second\(-\)order self-adjoint discrete Hamiltonian system

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 1 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 1 can be written as an equivalent first order nonlinear nonautonomous discrete Hamiltonian system

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, 1 is a discretization of the following second order differential equation:

which is also equivalent to a first order nonlinear Hamiltonian system.

Variational methods for difference equations, which allow one to achieve multiplicity results, were introduced by R.P. Agarwal, K. Perera, D. O’Regan see [ 3 ] . Recently, such methods received considerable attention. We mention here the works of M. Migda, J. Migda, M. Zdanowicz see [ 16 ] , A. Pankov see [ 18 ] , S. Stevic see [ 23 ] . Moreover, a lot of attention has been devoted in recent years to find periodic solutions of discrete dynamic models, for example, see [ 2 , 3 , 4 , 5 , 9 , 10 , 11 , 19 ] , other authors studied the existence of positive solutions of discrete fractional systems see [ 6 , 12 ] . It is our purpose in the present work to find other types of solutions, namely the doubly asymptotic solutions, first discovered by Poincaré [ 20 ] in continuous Hamiltonian systems. In the past 40 years, system (3) has bee widely investigated, see [ 1 , 13 , 17 , 22 , 24 ] and references therein. System (3) 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 (3) are collected in [ 25 ] . 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.

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 (1). Moreover, if \(W (n,.)\) is an even function, we prove that (1) possesses an unbounded sequence of homoclinic orbits emanating from 0 by invoking the Symmetric Mountain Pass Theorem.

For the statement of our main result, the potential \(W(t,x)\) is supposed to satisfy the following conditions:

• \(p(n)\) is symmetric and positive definite matrix for all \(n\in \mathbb {Z}\).

• \(L(n)\) 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

  \[ \tau _1\left|x\right|^{2}{\lt}\left(L(n)x,x\right){\lt}\tau _2\left|x\right|^{2},\quad \forall n\in \mathbb {Z},\; x\in \mathbb {R}^{N}. \]

• there exists a constant \(\mu {\gt}2\) such that

  \[ 0{\lt}\mu W(n,x)\leq (\nabla W(n,x),x),\quad \forall n\in \mathbb {Z},\; x\in \mathbb {R}^N\backslash \{ 0\} . \]

• there exists some positive function \(a:\mathbb {Z}\rightarrow \mathbb {R}\) with:

  \begin{eqnarray} \label{12} \lim _{\left|n\right|\rightarrow \infty }a(n)=0, \end{eqnarray}

  such that

  \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*}

• \(\varrho =\sup \left\{ W(n,x): n\in \mathbb {Z}, \left|x\right|=1\right\} {\lt}\frac{1}{2C^{2}_{2}}\), where \(C_2\) is defined in (5).

In this paper, we will prove the following Theorems.

2 Preliminaries

Let

for \(u, v \in E\), let

Then \(E\) is a Hilbert space with the above inner product, and the corresponding norm is:

As usual, for \(1\leq p\leq +\infty \), let

and their norms are defined by

respectively. Obviously, \(E\) is continuously embedded into \(l^{q}(\mathbb {Z},\mathbb {R}^{N})\) for \(2\leq q\leq +\infty \) , i.e., there exists \(C_q{\gt}0\) such that

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

Define the functional \(I: E \to \mathbb {R}\) by

Under the conditions of theorem 2, we have

for all \(u,v \in E\). Moreover, \(I\) is a continuously Fréchet- differentiable functional defined on \(E\), i.e., \(I\in C^{1}(E, \mathbb {R})\).

Observe that for all \(u,v \in E\)

It follows from (7) and (8) that \(\langle I'(u),v\rangle =0\) for all \(v\in E\) if only if

So, the critical points of \(I\) in \(E\) are the solutions of system (1) with \(u(\pm \infty )=0\).

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.

3 Proof of Theorems

For the Proof of our Theorems we need some technical Lemmas.

for \(k\in \mathbb {N}\). In addition, from \((W_2)\), for any \(\epsilon {\gt}0\), we can choose \(T_0{\gt}0\) such that

Consequently, for \(k\) large enough, and using Young inequality we have

Consequently, we obtain that

which yields \(\varphi '(u_k)\rightarrow \varphi '(u)\) as \(u_k\rightharpoonup u,\) that is, \(\varphi '\) is compact.

for every \(k\in \mathbb {N}\). We first prove that \((u_k)\) is bounded in \(E\). By (6), (7), \((W_1)\) and\((W_2)\) , we obtain

Since \(\mu {\gt}2\), the inequality (14) 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

Moreover, according to lemma 6, we have

It follows from the definition of \(I\) that

combining (15), (16) with (17) we obtain that \(u_k\rightarrow u\) in \(E\). So the proof is complete.

Proof of theorem 2.

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 lemma 4.

Let \(\rho =\frac{1}{C_2}\), where \(C_2\) is defined in (5). 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 lemma 6, we obtain that

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

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

where \(m=\min \left\{ W(n,u): n\in I, \left|u\right|=1\right\} \). Since \(\mu {\gt}2\), (19) 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 lemma 4, \(I\) possesses a critical value \(c\geq \beta {\gt}0\) given by

where

Hence there is \(u\in E\) such that \(I(u)=c, I'(u)=0\). â–¡

Proof of theorem 3.

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 lemma 5. \((ii)\) is identically the same as in theorem 2, so it is already proved.

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

Assume that \(\dim \tilde{E}=m\) and \(u_1,u_2,\ldots ,u_m\) is the basis of \(\tilde{E}\) such that

Since \(u_i\in E\), we can choose an integer \(\bar{T}\) such that

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

it follows that

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 (20) and (24) we have

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 (23), implies that \(\left|n_0\right|\leq \overline{T}.\) For any \(u\in E\), it follows from (23), (24), (26) and \((W_2)\), we have for \(u\in \sum \) and \(\sigma {\gt}1\)

Since \(\mu {\gt}2\), we deduce that there is \(\sigma _0=\sigma (\widetilde{E}){\gt}1\) such that

That is

This shows that \((iii)\) of lemma 5 holds. By lemma 5, \(I\) 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 \(k=1,2,...\)

From (6), we have

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.â–¡