Homoclinic Orbits for a Class of Noncoercive Discrete Hamiltonian Systems

Abstract

A class of first-order noncoercive discrete Hamiltonian systems are considered. Based on a generalized mountain pass theorem, some existence results of homoclinic orbits are obtained when the discrete Hamiltonian system is not periodical and need not satisfy the global Ambrosetti-Rabinowitz condition.

1. Introduction

Let 𝐍, 𝐙, and 𝐑 denote the set of all natural numbers, integers, and real numbers, respectively. Throughout this paper, without special statement, |⋅| denotes the usual norm in 𝐑^𝑁 with 𝑁∈𝐍, 𝑢⋅𝑣 denotes the inner product of 𝑢∈𝐑^𝑁 and 𝑣∈𝐑^𝑁.

Consider the noncoercive discrete Hamiltonian systems 𝐽Δ𝑥(𝑡)−𝑀(𝑡)𝑆𝑥(𝑡)+𝑣^∗×𝑣^∗𝐻^′(𝑡,𝑣×𝑣(𝑆𝑥(𝑡)))=0,𝑡∈𝐙, (1.1) where 𝑣∶𝐑^𝑁→𝐑^𝑚(1≤𝑚≤𝑁) is a nontrivial linear operator, 𝑣^∗ is its adjoint, 𝑣×𝑣 is the tensorial product of 𝑣, 𝑣∶(𝑣×𝑣)(𝑝,𝑞)=(𝑣(𝑝),𝑣(𝑞))𝑀(𝑡)=0𝑣^∗𝐿(𝑡)𝑣𝑣^∗𝐿(𝑡)𝑣0, (1.2) with 𝐿(𝑡) is an (𝑁×𝑁) symmetric matrix valued function and 𝐻∶𝐑×𝐑^𝑁, (𝑡,𝑦)↦𝐻(𝑡,𝑦) is a continuous function, differentiable with respect to the second variable with continuous derivative 𝐻′(𝑡,𝑦)=(𝜕𝐻/𝜕𝑦)(𝑡,𝑦). 𝑆 is the shift operator defined as 𝑆𝑥(𝑡)=^𝑥1_𝑥^(𝑡+1)_2(𝑡) and 𝑥(𝑡)=^𝑥1_𝑥^(𝑡)_2(𝑡), where 𝑥₁,𝑥₂∈𝐑^𝑁. Δ𝑥_𝑖(𝑡)=𝑥_𝑖(𝑡+1)−𝑥_𝑖(𝑡), 𝑖=1,2, is the forward difference operator. 𝐽 is the standard symplectic matrix 𝐽=^0−𝐼𝑁_𝐼𝑁0, where 𝐼_𝑁 is the identity matrix on 𝐑^𝑁.

As usual, assuming that a solution 𝑥(𝑡)=0 an equilibrium for (1.1), we say that a solution 𝑥(𝑡) is homoclinic to 0 if 𝑥(𝑡) satisfies 𝑥(𝑡)≠0, and the asymptotic condition 𝑥(𝑡)→0 as |𝑡|→+∞. Such solutions have been found in various models of continuous dynamical systems and frequently have tremendous effects on the dynamics of such nonlinear systems. So the homoclinic orbits have been extensively studied since the time of Poincaré, see [1–7] and references therein.

In recent years, there has been much research activity concerning the theory of difference equations. To a large extent, this due to the realization that difference equations are important in applications. New applications that involve difference equations continue to arise with frequency in the modelling of computer science, economics, neural network, ecology, cybernetics, and so forth, we can refer to [8–13] for detail. Many scholars have investigated discrete Hamiltonian systems independently main for two reasons. The first one is that the behaviour of discrete Hamiltonian systems is sometimes sharply different from the behaviour of the corresponding continuous systems. The second one is that there is a fundamental relationship between solutions of continuous systems and the corresponding discrete systems by employing discrete variable methods (see [8] for detail).

The general form of (1.1) is 𝐽Δ𝑥(𝑡)−𝐻^′(𝑡,𝑥(𝑡))=0,𝑡∈𝐙, (1.3) which was studied by many scholars in various fields. By making use of minimax theory and geometrical index theory, [14] gave results on subharmonic solutions with prescribed minimal periods. When (1.3) are superquadratic systems, Guo and Yu [15] obtained some existence and multiplicity results by 𝑍₂ index theory and linking theorem. In [16], when 𝐻 is subquadratic at infinity, the authors gave some existence results of periodic solutions. As to homoclinic orbits for discrete systems, [17–19] studied the second order discrete systems by critical point theory recently. While for the first order discrete systems, such as (1.1) or (1.3), to the authors' best knowledge, it seems there exists no similar results.

Moreover, we may regard (1.1) as being a discrete analogue of Hamiltonian systems 𝐽𝑥̇(𝑡)−𝑀(𝑡)𝑥+𝐻^′(𝑡,𝑥(𝑡))=0,𝑡∈𝐑. (1.4) Equation (1.1) is the best approximation of (1.4) when one lets the step size not be equal to 1 but the variable's step size go to zero, so solutions of (1.1) can give some desirable numerical features for (1.4). (1.4) is one form of classical Hamiltonian systems appearing in the study of various fields and many well-known results were given.

In view of above reasons, the goal of this paper is to study the existence of homoclinic orbits for the first order discrete Hamiltonian system (1.1) when 𝐻 satisfies superquadratic conditions and need not satisfy the global Ambrosetti-Rabinowitz (AR) condition:

(AR): there exist two constants 𝜇>2 and 𝑟>0 such that for all 𝑡∈𝐙 and 𝑥∈𝐑^2𝑁, |𝑥|≥𝑟0<𝜇𝐻(𝑡,𝑥)≤𝐻^′(𝑡,𝑥(𝑡))⋅𝑥. (1.5)

Let 𝑙(𝑡) denotes the smallest eigenvalue of 𝑣^∗𝐿(𝑡)𝑣, that is, 𝑙(𝑡)=inf_{𝜉∈𝐑𝑁,𝜉=1}𝑣^∗𝐿(𝑡)𝑣(𝜉)⋅𝜉,∀𝑡∈𝐙. (1.6) For later use, we need the following assumptions:

(𝐿₁) there exists 1<𝛾<2 such that 𝑙(𝑡)|𝑡|^𝛾−2→+∞ as |𝑡|→+∞;

(𝐻₁)𝐻(𝑡,𝑦)/|𝑦|²→+∞ as |𝑦|→+∞, 𝑡∈𝐙;

(𝐻₂)|𝐻′(𝑡,𝑦)|/|𝑦|→0 as |𝑦|→+0 and 𝑡∈𝐙;

(𝐻₃) there exist 𝑎>0 and 𝛼>1 such that 𝐻^′𝑡,𝑦≤𝑎𝑦^𝛼+1,∀𝑡∈𝐙,∀𝑦∈𝐑^2𝑁; (1.7)

(𝐻₄) there exist 𝛽>𝛼, 𝑏>0 and 𝑟>0 such that 𝐻^′𝑡,𝑦⋅𝑦−2𝐻𝑡,𝑦≥𝑏𝑦^𝛽,∀𝑡∈𝐙,∀𝑦≥𝑟; (1.8)

(𝐻₅) for all 𝑡∈𝐙 and all 𝑦∈𝐑^2𝑁𝐻^′𝑡,𝑦𝑦≥2𝐻𝑡,𝑦≥0. (1.9)

Remark 1.1. By assumption (𝐻₁) and (𝐻₂), we know that 𝐻(𝑡,𝑦) satisfies the superquadratic condition at both infinity and 0 respect to the second variable 𝑦.

The rest of the paper is organized as follows. In Section 2, we shall establish the variational structure for (1.1) and turn the problem of looking for homoclinic orbits for (1.1) to the problem for seeking critical points of the corresponding functional. In order to apply the generalized mountain pass theorem, we give some preliminary results in Section 3. In Section 4, we shall state our main result and complete the proof of our result.

2. Variational Structure

Set 𝑆={𝑥={𝑥(𝑡)}∣𝑥(𝑡)=^𝑥1_𝑥^(𝑡)_2(𝑡)∈𝐑^2𝑁,𝑥_𝑗(𝑡)∈𝐑^𝑁,𝑗=1,2,𝑡∈𝐙} is a space which is composed of the following vectors, 𝑥={𝑥(𝑡)}_𝑡∈𝐙=…,𝑥^𝑇(−𝑡),…,𝑥^𝑇(−1),𝑥^𝑇(0),𝑥^𝑇(1),…,𝑥^𝑇(𝑡),…. (2.1) Define the subspace 𝑋 of 𝑆 as 𝑋=𝑥∈𝑆∣_𝑡∈𝐙[(𝐽Δ𝑆𝑥(𝑡−1),𝑥(𝑡))+(𝑀(𝑡)𝑆𝑥(𝑡),𝑥(𝑡))]<+∞. (2.2) Denote 𝑢=𝑣×𝑣, 𝑢^∗=𝑣^∗×𝑣^∗, define another subspace 𝐸 of 𝑋 as follows: 𝐸=𝑥∈𝑋∣𝑥(𝑡)∈(Keru)^⟂. (2.3) The space 𝐸 is a Hilbert space with the inner product 𝑥,𝑦=_𝑡∈𝐙𝐽Δ𝑆𝑥(𝑡−1),𝑦(𝑡)+𝑀(𝑡)𝑆𝑥(𝑡),𝑦(𝑡),∀𝑥,𝑦∈𝐸, (2.4) and the norm introduced from the inner product as follows: ‖𝑥‖²=⟨𝑥,𝑥⟩=_𝑡∈𝐙[(𝐽Δ𝑆𝑥(𝑡−1),𝑥(𝑡))+(𝑀(𝑡)𝑆𝑥(𝑡),𝑥(𝑡))],∀𝑥∈𝐸. (2.5)

Define a functional 𝐹(𝑥) on 𝐸 as follows: 𝐹(𝑥)1=−2_𝑡∈𝐙[(𝐽Δ𝑆𝑥(𝑡−1),𝑥(𝑡))+(𝑀(𝑡)𝑆𝑥(𝑡),𝑥(𝑡))]+_𝑡∈𝐙𝐺(𝑡,𝑢(𝑆𝑥(𝑡−1))), (2.6) according to the definition of ‖𝑥‖, 𝐹(𝑥) can be written in another form as follows: 𝐹(𝑥)1=−2‖𝑥‖²+_𝑡∈𝐙𝐺(𝑡,𝑢(𝑆𝑥(𝑡−1))). (2.7) The functional 𝐹(𝑥) is a well-defined 𝐶¹ on 𝐸, and next we prove that the problem of looking for homoclinic orbits for (1.1) can be turned to the problem for seeking critical points of the corresponding functional 𝐹(𝑥) (see (2.6) or (2.7)).

Let 𝐹₁(𝑥)1=−2_𝑡∈𝐙(()(𝑡1𝐽Δ𝑆𝑥𝑡−1,𝑥))=−2_𝑡∈𝐙0−𝐼_𝑁𝐼_𝑁0Δ𝑥₁(𝑡)Δ𝑥₂(),𝑥𝑡−1¹(𝑡)𝑥₂(𝑡)1=−2_𝑡∈𝐙𝑥₂(𝑡−1),𝑥₁(𝑡)−𝑥₂(𝑡),𝑥₁(𝑡)+𝑥₁(𝑡+1),𝑥₂(𝑡)−𝑥₁(𝑡),𝑥₂(𝑡), (2.8) while _𝑡∈𝐙𝑥₂(𝑡−1),𝑥₁(𝑡)−𝑥₂(𝑡),𝑥₁(𝑡)=_𝑡∈𝐙𝑥₂(𝑡−1),𝑥₁(𝑡)−_𝑡∈𝐙𝑥₂(𝑡),𝑥₁(𝑡)=_𝑡∈𝐙𝑥₂(𝑡),𝑥₁(𝑡+1)−_𝑡∈𝐙𝑥₂(𝑡),𝑥₁(𝑡)=_𝑡∈𝐙𝑥₂(𝑡),Δ𝑥₁(𝑡), (2.9) then 𝐹₁(𝑥)=−_𝑡∈𝐙Δ𝑥₁(𝑡),𝑥₂(𝑡), (2.10) it follows that 𝐹(𝑥)=−_𝑡∈𝐙Δ𝑥₁(𝑡),𝑥₂(𝑡−1)2_𝑡∈𝐙(𝑀(𝑡)𝑆𝑥(𝑡),𝑥(𝑡))+_𝑡∈𝐙𝐺(𝑡,𝑢(𝑆𝑥(𝑡−1))). (2.11) Write 𝐹^′_{𝑥𝑖(𝑡)}=𝜕𝐹(𝑥)/𝜕𝑥_𝑖(𝑡), 𝑖=1,2, for any given 𝑡∈𝐙, there holds 𝐹^′_𝑥1(𝑡)=𝑥₂(𝑡−1)−𝑥₂(𝑡)−𝑣^∗𝐿𝑣𝑥₂(𝑡−1)+𝑢^∗𝐺_𝑥1(𝑡)(((,𝐹𝑡−1,𝑢𝑆𝑥𝑡−1)))_{′𝑥2(𝑡)}=𝑥₁(𝑡+1)−𝑥₁(𝑡)−𝑣^∗𝐿𝑣𝑥₁(𝑡+1)+𝑢^∗𝐺_𝑥2(𝑡)(𝑡,𝑢(𝑆𝑥(𝑡))). (2.12) Then we can draw a conclusion that 𝐹^′(𝑥)=0 is true if and only if 𝐹^′_𝑥1(𝑡)𝐹=0,_{′𝑥2(𝑡)}=0, (2.13) so 𝑥₂(𝑡−1)−𝑥₂(𝑡)−𝑣^∗𝐿𝑣𝑥₂(𝑡−1)+𝑢^∗𝐺_𝑥1(𝑡)(((𝑥𝑡−1,𝑢𝑆𝑥𝑡−1)))=0,₁(𝑡+1)−𝑥₁(𝑡)−𝑣^∗𝐿𝑣𝑥₁(𝑡+1)+𝑢^∗𝐺_𝑥2(𝑡)(𝑡,𝑢(𝑆𝑥(𝑡)))=0, (2.14) which can be reformed as 0−𝐼_𝑁𝐼_𝑁0Δ𝑥₁(𝑡)Δ𝑥₂(𝑡)−0𝑣^∗𝐿(𝑡)𝑣𝑣^∗𝐿(𝑡)𝑥𝑣0¹()𝑥𝑡+1₂(𝑡)+𝑢^∗𝐺_𝑥1(𝑡)(((𝑡𝐺𝑡−1,𝑢𝑆𝑥)))_𝑥2(𝑡)(𝑡,𝑢(𝑆𝑥(𝑡)))=0, (2.15) that is 𝐽Δ𝑥(𝑡)−𝑀(𝑡)𝑆𝑥(𝑡)+𝑢^∗𝐻^′(𝑡,𝑢(𝑆𝑥(𝑡)))=0, (2.16) which is just (1.1). Therefore, we obtain the following lemma.

Lemma 2.1. 𝑥={𝑥(𝑡)}≠0 is a homoclinic orbit of (1.1) if and only if 𝑥 is a critical point of functional 𝐹(𝑥) in 𝐸.

3. Preliminary Results

In order to apply the critical point theory to look for critical points for (2.6), we give some lemmas which will be of fundamental importance in proving our main result.

Let 𝐸 be a real Hilbert space with the norm ‖⋅‖. Suppose that 𝐸 has an orthogonal decomposition 𝐸=𝐸¹⊕𝐸² with both 𝐸¹ and 𝐸² being infinite dimensional. Suppose (𝜈_𝑛) (resp., (𝜔_𝑛)) is an orthogonal basis for 𝐸¹ (resp. 𝐸²), and set 𝐸¹_𝑛=span{𝜈₁,𝜈₂,…,𝜈_𝑛}, 𝐸¹_𝑛=span{𝜔₁,𝜔₂,…,𝜔_𝑛}, 𝐸_𝑛=𝐸¹_𝑛⊕𝐸²_𝑛, and 𝑓_𝑛=𝑓∣_𝐸𝑛, the restriction of 𝑓 on 𝐸_𝑛. We say that 𝑓 satisfies the (PS)^∗ condition if any sequence (𝑥^(𝑛)) in 𝐸, (𝑥^(𝑛))∈𝐸_𝑛 such that 𝑓_𝑛(𝑥^(𝑛))≤𝐶 a constant, and 𝑓^′_𝑛(𝑥^(𝑛))→0 possesses a convergent subsequence.

We state a basic theorem introduced in [20] by Rabinowitz which is used to obtain the critical points of the functional 𝐹(𝑥).

Lemma 3.1 (Generalized mountain pass lemma). Let 𝑓∈𝐶¹(𝐸,𝐑) satisfy
(𝑓₁) the (PS)^∗condition;
(𝑓₂) there are 𝜌,𝛿>0 such that 𝑓(𝑥)≥𝛿, (3.1) for all 𝑥∈𝑆_𝜌={𝑥∈𝐸²∣‖𝑥‖=𝜌};
(𝑓₃) there are 𝑟>𝜌, 𝑀>0, 𝑒∈𝐸²₁ with ‖𝑒‖=1 such that 𝑓∣_𝜕𝑄≤0,𝑓∣_𝑄≤𝑀, (3.2) where 𝑄={(𝐵_𝑟⋂𝐸¹)}⊕{𝑠𝑒0≤𝑠≤𝑟}.
Then 𝑓 has a critical point 𝑥 with 𝑓(𝑥)≥𝛿.

Next we consider the eigenvalue problem. 𝐽Δ𝑆𝑥(𝑡−1)+𝑀(𝑡)𝑆𝑥(𝑡)=𝜆𝑥(𝑡). (3.3) Equation (3.3) can be reformed as follows: 𝑥₁(𝑡+1)=𝐼+𝑣^∗𝐿𝑣⁻¹𝑥₁(𝑡)+𝜆𝐼+𝑣^∗𝐿𝑣⁻¹𝑥₂(𝑡),𝑥₂(𝑡+1)=𝜆𝑣^∗𝐿𝑣−𝐼⁻¹𝐼+𝑣^∗𝐿𝑣⁻¹𝑥₁(𝑡)+𝜆²𝑣^∗𝐿𝑣−𝐼⁻¹𝐼+𝑣^∗𝐿𝑣⁻¹−𝐼𝑥₂(𝑡), (3.4) Denote 𝐴(𝜆)=𝐼+𝑣^∗𝐿𝑣⁻¹𝜆𝐼+𝑣^∗𝐿𝑣⁻¹𝜆𝑣^∗𝐿𝑣−𝐼⁻¹𝐼+𝑣^∗𝐿𝑣⁻¹𝜆²𝑣^∗𝐿𝑣−𝐼⁻¹𝐼+𝑣^∗𝐿𝑣⁻¹−𝐼, (3.5) then (3.3) can be expressed by the following: 𝑥(𝑡+1)=𝐴(𝜆)𝑥(𝑡). (3.6) Therefore a standard argument shows that 𝜎(𝐴), the spectrum of 𝐴, consists of eigenvalues numbered by, (counted in their multiplicities) the following: ⋯𝜆₋₂≤𝜆₋₁≤0≤𝜆₁≤𝜆₂⋯, (3.7) with 𝜆_𝑘→±∞ as 𝑘→±∞, and denote the corresponding system of eigenfunctions of 𝐴 by (𝑒_𝑘).

Let 𝐸⁰= Ker𝐴, 𝐸⁺=span{𝑒₁,…,𝑒_𝑛} and 𝐸⁻=(𝐸⁰⊕𝐸⁺)^⟂𝐸, where 𝑆^⟂𝐸 stands for the orthogonal complementary subspace of 𝑆 in 𝐸. Then 𝐸=𝐸⁻⊕𝐸⁰⊕𝐸⁺, (3.8) so the functional (2.7) can be rewritten as follows 𝐹(𝑥)1=−2𝑥⁺²−‖𝑥⁻‖²+_𝑡∈𝐙𝐺(𝑡,𝑢(𝑆𝑥(𝑡−1))), (3.9) for all 𝑥=𝑥⁻+𝑥⁰+𝑥⁺∈𝐸⁻+𝐸⁰+𝐸⁺.

Set 𝑙²={𝑥={𝑥(𝑡)}∈𝑆∣∑_𝑡∈𝐙|𝑥(𝑡)|²<+∞} and 𝑙^∞={𝑥={𝑥(𝑡)}∈𝑆∣|𝑥(𝑡)|<+∞, for all 𝑡∈𝐙} and their norms are defined by the following: ‖𝑥‖²_𝑙2=_𝑡∈𝐙|𝑥(𝑡)|²=(𝑥,𝑥)_𝑙²,‖𝑥‖_𝑙^∞=sup_𝑡∈𝐙|𝑥(𝑡)|, (3.10) respectively. For any given 1≤𝑟<+∞, define 𝑙^𝑟={𝑥={𝑥(𝑡)}∈𝑆∣∑_𝑡∈𝐙|𝑥(𝑡)|^𝑟<+∞} with the norm ‖𝑥‖^𝑟_𝑙^𝑟=_𝑡∈𝐙|𝑥(𝑡)|^𝑟=(𝑥,𝑥)_𝑙^𝑟. (3.11)

Define a selfadjoint operator 𝐴 on 𝐸 by the following: (𝐴𝑥,𝑥)=_𝑡∈𝐙(𝐽Δ𝑆𝑥(𝑡−1)+𝑀(𝑡)𝑆𝑥(𝑡),𝑥(𝑡))=‖𝑥‖²,(𝐴𝑥,𝐽𝑥)_𝑙²=_𝑡∈𝐙(𝑀(𝑡)𝑆𝑥(𝑡),𝐽𝑥(𝑡)), (3.12)|𝐴| is the absolute value. Give another norm the domain of 𝐴 by the following: ‖𝑥‖₁=‖(𝐼+|𝐴|)𝑥‖_𝑙2, (3.13) it is easy to get, for all 𝑥∈𝐸, (𝐴𝑥,𝐽𝑥)_𝑙²≤‖𝑥‖²₁(𝑥,𝑥)_𝑙²≤‖𝑥‖²₁. (3.14)

Now we state a fundamental proposition, which will be used in the later.

Proposition 3.2. Let 𝐿 satisfy (𝐿₁). Then for all 1≤𝑝∈(2/(3−𝛾),+∞) there exists a constant 𝜆_𝑝>0 such that ‖𝑥‖_𝑙^𝑝≤𝜆_𝑝‖𝑥‖,∀𝑥∈𝐸. (3.15)

Proof. We complete the proof of Proposition 3.2 by 3 steps.
Step  1. When (𝐿₁) holds and 𝑝=2, we prove that ‖𝑥‖_𝑙2≤𝜆₂‖𝑥‖,∀𝑥∈𝐸, (3.16)
Note that, by (𝐿₁), 𝑙(𝑡)→+∞ as |𝑡|→+∞, that is, 𝑙(𝑡) is bounded from below and so there is a 𝑎>0 such that 𝑣^∗𝐿(𝑡)𝑣+𝑎𝐼_𝑁𝛽(𝑅)≥0,𝑡∈𝐙,=inf_{|𝑡|≥𝑅}𝑙(𝑡)→∞,as|𝑡|→∞. (3.17) For 𝑅>0, choose a subsequence 𝑥^(𝑘)(𝑡)∈𝐸, one has _{|𝑡|≥𝑅}𝑥^(𝑘)2≤_{|𝑡|≥𝑅}𝐽𝑥^(𝑘)(𝑡)⋅𝑣^∗𝐿(𝑡)𝑣+𝑎𝐽𝑥^(𝑘)(𝑡)𝑙(𝑡)≤1𝛽(𝑅)_𝑡∈𝐙𝐽𝑥^(𝑘)(𝑡)⋅𝑣^∗𝐿(𝑡)𝑣+𝑎𝐽𝑥^(𝑘)(𝑡)=1𝛽(𝑅)_𝑡∈𝐙𝐽𝑥^(𝑘)(𝑡)⋅𝑣^∗𝐿(𝑡)𝑣⋅𝑥^(𝑘)(𝑡)+𝑎𝐽𝑥^(𝑘)(𝑡)⋅𝐽𝑥^(𝑘)(𝑡)=1𝛽(𝑅)𝐴𝑥^(𝑘),𝐽𝑥^(𝑘)_𝑙2+1𝛽(𝑅)𝑎𝐽𝑥^(𝑘),𝐽𝑥^(𝑘)_𝑙2≤1𝛽(𝑅)𝑥^(𝑘)2₁. (3.18) For any given 𝜖>0, by (3.18), one can take 𝑅₀ so large that _{|𝑡|≥𝑅0}𝑥^(𝑘)2≤𝜖²4. (3.19) Without loss of generality, we can assume that 𝑥^(𝑘)⇀0 in 𝐸. Define 𝐸_𝐼={𝑥∈𝑆_𝐼∣∑_𝑡∈𝐙[(𝐽Δ𝑆𝑥(𝑡−1),𝑥(𝑡))+(𝑀(𝑡)𝑆𝑥(𝑡),𝑥(𝑡))]<+∞}, 𝑆_𝐼={𝑥={𝑥(𝑡)}∣𝑥(𝑡)∈𝐸,𝑡∈𝐼} and 𝐼={𝑡∣|𝑡|≤𝑅₀}. So 𝑥^(𝑘) is bounded in 𝐸_𝐼, which implies that 𝑥^(𝑘) is bounded in 𝑙²_𝐼. This together with the uniqueness of the weak limit in 𝑙²_𝐼, we have 𝑥^(𝑘)→0 in 𝐸_𝐼, so there exists a 𝑘₀ such that _𝑡∈𝐼𝑥^(𝑘)2≤𝜖²4,∀𝑘≥𝑘₀. (3.20) Combing (3.19) and (3.20), we have 𝑥^(𝑘)→0 in 𝑙². It follows that (3.16) is true.
Step  2. For all 𝑝>2, there exists a constant 𝜆_𝑝>0 such that (3.15) holds.
For any 𝑝>2 and 𝑥∈𝐸, by the Hölder inequality, we have _𝑡∈𝐙|𝑥|^𝑝=_𝑡∈𝐙|𝑥|⋅|𝑥|^𝑝−1≤_𝑡∈𝐙|𝑥|^21/2⋅_𝑡∈𝐙|𝑥|^{2(𝑝−1)(𝑝−1)/2(𝑝−1)}≤𝐶‖𝑥‖^𝑝−1‖𝑥‖_𝑙2, (3.21) which together with (3.16) yields (3.15).
Step  3. Since (𝐿₁) implies 𝑙(𝑡)→+∞ as |𝑡|→∞, by Step 1 and 2, it remains to consider the case for 1≤𝑝∈(2/(3−𝛾),2).
Let 𝛽(𝑅)=inf_|𝑡|>𝑅𝑙(𝑡)|𝑡|^𝛾−2. (3.22) By (𝐿₁), 𝛽(𝑅)→∞ as 𝑅→∞.
Write 𝛼=(2−𝛾)/(2−𝑝), then 𝛼𝑝>1. Set for 𝑅>0 and 𝑥∈𝐸, 𝐸¹_𝑅(𝑥)={𝑡∣|𝑡|≥𝑅 and |𝑡|^𝛼⋅|𝑥(𝑡)|>1} and 𝐸²_𝑅(𝑥)={𝑡∣|𝑡|≥𝑅 and |𝑡|^𝛼⋅|𝑥(𝑡)|≤1}. Then _{𝑡∈𝐸1𝑅(𝑥)}|𝑥|^𝑝=_{𝑡∈𝐸1𝑅(𝑥)}|𝑡|^𝛼|𝑥(𝑡)|^𝑝⋅|𝑡|^−𝛼𝑝≤_{𝑡∈𝐸1𝑅(𝑥)}|𝑡|^𝛼|𝑥(𝑡)|²⋅|𝑡|^−𝛼𝑝=_{𝑡∈𝐸1𝑅(𝑥)}|𝑡|^2𝛼|𝑥(𝑡)|²|𝑡|^−𝛼𝑝=_{𝑡∈𝐸1𝑅(𝑥)}|𝑡|^{2𝛼−𝛼𝑝}|𝑥(𝑡)|²=_{𝑡∈𝐸1𝑅(𝑥)}|𝑡|^2−𝛾|𝑥(𝑡)|²,_𝑡∈𝐙(𝑀(𝑡)𝑆𝑥(𝑡),𝐽𝑥(𝑡))=(𝐴𝑥,𝐽𝑥)_𝑙²≤|𝐴|^1/2𝑥_𝑙2⋅|𝐴|^1/2𝐽𝑥_𝑙2≤𝐶²_{𝑡∈𝐸1𝑅(𝑥)}‖𝑥‖|𝑥|²≤_{𝑡∈𝐸1𝑅(𝑥)}𝐽𝑥⋅(𝑀+𝑎𝐽)𝑥𝑙(𝑡)≤1𝛽(𝑅)|𝑡|^𝛾−2((𝑀+𝑎𝐽)𝑥,𝐽𝑥)_𝑙², (3.23)(3.24) From (3.24), we get _{𝑡∈𝐸1𝑅(𝑥)}|𝑥|^𝑝≤_{𝑡∈𝐸1𝑅(𝑥)}|𝑥|²|𝑡|^2−𝛾≤1𝛽(𝑅)((𝑀+𝑎𝐽)𝑥,𝐽𝑥)_𝑙²≤1𝛽(𝑅)𝐶‖𝑥‖² (3.25) and so _{|𝑡|≥𝑅}|𝑥|^𝑝=_{𝑡∈𝐸1𝑅(𝑥)}|𝑥|^𝑝+_{𝑡∈𝐸2𝑅(𝑥)}|𝑥|^𝑝=_{𝑡∈𝐸1𝑅(𝑥)}|𝑡|^𝛼|𝑥(𝑡)|^𝑝|𝑡|^−𝛼𝑝+_{𝑡∈𝐸2𝑅(𝑥)}|𝑡|^𝛼|𝑥(𝑡)|^𝑝|𝑡|^−𝛼𝑝≤1𝛽(𝑅)𝐶‖𝑥‖²+_{𝑡∈𝐸2𝑅(𝑥)}|𝑡|^𝛼|𝑥(𝑡)|^𝑝|𝑡|^−𝛼𝑝≤1𝛽(𝑅)𝐶‖𝑥‖²+_{𝑡∈𝐸2𝑅(𝑥)}|𝑡|^−𝛼𝑝. (3.26) Since 𝛼𝑝>1 and ∑_𝑡∈𝐸²_𝑅(𝑥)|𝑡|^−𝛼𝑝=∑_{|𝑡|≥𝑅}|𝑡|^−𝛼𝑝 then there exists a constant 𝑠 such that _{|𝑡|≥𝑅}|𝑡|^−𝛼𝑝=𝑠, (3.27) which together with (3.26) yields _{|𝑡|≥𝑅}|𝑥|^𝑝≤1𝛽(𝑅)𝐶‖𝑥‖²+𝑠. (3.28) Give 𝜖>0, by (3.28), choose 𝑅₀>0 so large that _{|𝑡|≥𝑅0}|𝑥|^𝑝<𝜖^𝑝2. (3.29)
Denote 𝐼={𝑡∈𝐙∣|𝑡|≤𝑅₀}, 𝐸_𝐼={𝑥∣𝑥(𝑡)∈𝐸,𝑡∈𝐼}. Any given subsequence (𝑥^(𝑘))∈𝐸, we can suppose 𝑥^(𝑘)⇀0 on 𝐸, now 𝑥^(𝑘) is bounded in 𝑙^𝑝_𝐼. This together with the uniqueness of the weak limit in 𝑙^𝑝_𝐼 on 𝐼. For any 𝑥∈𝐸_𝐼, we have ‖𝑥‖^𝑝_𝑙𝑝𝐼<𝜖^𝑝2. (3.30)
Combining (3.29) and (3.30), it follows _𝑡∈𝐙|𝑥|^𝑝<𝜖, (3.31) that is, ‖𝑥‖_𝑙^𝑝<𝜖, there has a constant 𝜆_𝑝>0 such that ‖𝑥‖_𝑙^𝑝≤𝜆_𝑝‖𝑥‖, (3.32) while 1≤𝑝∈(2/(3−𝛾),2).

4. Main Results and Proofs

In the previous section, we turned the homoclinic orbits problem of (1.1) to the corresponding critical point problem of the functional (2.6) or (2.7). Next, we state our main results and complete their proofs by Lemma 3.1.

Our main result is as follows.

Theorem 4.1. Suppose that 𝐻 satisfies (𝐿₁) and (𝐻₁)–(𝐻₅). Then the discrete Hamiltonian system (1.1) has a nontrivial homoclinic orbit.

Remark 4.2. Observe that if 𝑥(𝑡) is a homoclinic solution of (1.1), then 𝑦(𝑡)=𝑥(−𝑡) is a homoclinic solution of the following: 𝐽Δ𝑦(𝑡)+𝑀(−𝑡)𝑆𝑦(𝑡)−𝑣^∗×𝑣^∗𝐻^′(−𝑡,𝑣×𝑣(𝑆𝑥(𝑡)))=0. (4.1) Moreover, −𝐻(−𝑡,𝑦) satisfies (𝐻₁)‒(𝐻₅) whenever 𝐻(𝑡,𝑦) satisfies (𝐻^′₁), (𝐻₂), (𝐻₃), (𝐻^′₄) and (𝐻^′₅), where (𝐻^′₁)𝐻(𝑡,𝑦)/|𝑦|²→−∞ as |𝑦|→+∞, 𝑡∈𝐙; (𝐻^′₄) there exist 𝛽>𝛼, 𝑏>0 and 𝑟>0 such that 𝐻^′𝑡,𝑦⋅𝑦−2𝐻𝑡,𝑦≤−𝑏𝑦^𝛽,∀𝑡∈𝐙,∀𝑦≥𝑟; (4.2)(𝐻^′₅) for all 𝑡∈𝐙 and all 𝑦∈𝐑^2𝑁𝐻^′𝑡,𝑦𝑦≤2𝐻𝑡,𝑦≤0. (4.3)So in the following, we will give another theorem and can omit its proof.

Theorem 4.3. The conclusion of Theorem 4.1 holds when replacing (𝐻₁), (𝐻₄), and (𝐻₅) with (𝐻^′₁), (𝐻^′₄), and (𝐻^′₅).

With the aid of previous sections, we will verify that 𝐹(𝑥) satisfies the assumptions of Lemma 3.1. We will proceed by successive lemmas.

Lemma 4.4. If 𝐻 satisfies assumptions of Theorem 4.1, then there are constants 𝜌>0, 𝛿>0 such that 𝐹|_𝐵≥𝛿, (4.4) where 𝐵={𝑥∈𝐸²∣‖𝑥‖=𝜌}.

Proof. For any 𝑥∈𝐸, it is easy to see that there exist two constants 0<𝑚₀<𝑀₀ such that 𝑚₀|𝑥|≤|𝑢(𝑥)|≤𝑀₀|𝑥|. (4.5) By (𝐻₁), (𝐻₃), and (4.5), for all 𝜖>0, there exist a constant 𝐶_𝜖>0 such that 𝐻^′((𝑥≤1𝑡,𝑢))𝑀₀2𝜖|𝑥|+(𝛼+1)𝐶_𝜖|𝑥|^𝛼,∀𝑥∈𝐸. (4.6) Now by mean value Theorem, (4.5) and (4.6), for all 𝑥∈𝐸 and 𝑡∈𝐙, we have |𝐻(𝑡,𝑢(𝑥))|=¹₀𝐻^′((𝑥(𝑥)|𝑥|𝑡,𝑠𝑢))⋅𝑢𝑑𝑠≤𝜖²+𝐶_𝜖|𝑥|^𝛼+1, (4.7) on the other hand, ‖𝑆𝑥‖²_𝑙2=_𝑡∈𝐙|𝑆𝑥(𝑡)|²=_𝑡∈𝐙𝑥₁(𝑡+1)²+𝑥₂(𝑡)²=‖𝑥‖²_𝑙2, (4.8) similarly, ‖𝑆𝑥‖_𝑙^𝛼+1_𝛼+1=‖𝑥‖_𝑙^𝛼+1_𝛼+1, (4.9) it follows that _𝑡∈𝐙|𝐻(𝑡,𝑢(𝑆𝑥(𝑡)))|≤_𝑡∈𝐙𝜖|𝑆𝑥|²+𝐶_𝜖|𝑆𝑥|^𝛼+1=𝜖_𝑡∈𝐙|𝑆𝑥|²+𝐶_{𝜖𝑡∈𝐙}|𝑆𝑥|^𝛼+1=𝜖‖𝑥‖²_𝑙2+𝐶_𝜖‖𝑥‖_𝑙^𝛼+1_𝛼+1, (4.10) By Proposition 3.2 and (4.10), for any 𝑥∈𝐸, it holds 𝐹(𝑥)=12‖𝑥‖²−_𝑡∈𝐙𝐻(((𝑡≥1𝑡,𝑢𝑆𝑥)))2‖𝑥‖²−𝜖‖𝑥‖²_𝑙2−𝐶_𝜖‖𝑥‖_𝑙^𝛼+1_𝛼+1≥12‖𝑥‖²−𝜖𝜆²₂‖𝑥‖²−𝐶_𝜖𝜆^𝛼+1_𝛼+1‖𝑥‖^𝛼+1, (4.11) Choosing 𝜖 such that 𝜖𝜆²₂=1/4, we obtain, for any 𝑥∈𝐸², 𝐹(𝑥)≥14‖𝑥‖²−𝐶_𝜖𝜆^𝛼+1_𝛼+1‖𝑥‖^𝛼+1. (4.12) Since 𝛼>1, then there are constants 𝜌>0, 𝛿>0 such that 𝐹∣_𝐵≥𝛿, (4.13) which completes the proof of Lemma 4.4.

Lemma 4.5. Under assumptions of Theorem 4.1, let 𝑒∈𝐸²₁ with ‖𝑒‖=1, there exist 𝑟₁,𝑟₂>0 such that 𝐹(𝑥)≤0,𝑥∈𝜕𝑄, (4.14) where 𝑄={𝑠𝑒∣0≤𝑠≤𝑟₁}⊕{𝑥∈𝐸¹∣‖𝑥‖≤𝑟₂}.

Proof. Let 𝑒∈𝐸²₁ with ‖𝑒‖=1 and 𝐹=𝐸¹⊕span{𝑒}. For 𝑥=𝑥⁻+𝑥⁰+𝑥⁺∈𝐹−{0} and 𝜖>0, denote Ω_𝑥=𝑡∈𝐙∣|𝑥(𝑡)|≥𝜖‖𝑥‖, (4.15) then there exists 𝜖₁>0 such that ♯Ω_𝑥=♯𝑡∈𝐙∣|𝑥(𝑡)|≥𝜖₁‖𝑥≥1‖𝜖₁+1, (4.16) where ♯Ω_𝑥 is the number of 𝑡 in Ω_𝑥 and [⋅] is the greatest integer function.
By (𝐻₁), for 𝑑=1/2𝜖²₁𝑚²₀, there exists 𝑅₁>0 such that 𝐻𝑡,𝑦≥𝑑𝑦²,∀𝑦≥𝑅₁,𝑡∈𝐙, (4.17) where 𝑚₀ was defined by (4.5). Then it follows 𝐻(𝑡,𝑢(𝑆𝑥(𝑡)))≥𝑑|𝑢(𝑆𝑥(𝑡))|²≥12𝜖³₁𝑚²₀|𝑢(𝑆𝑥(𝑡))|²≥12𝜖³₁|𝑆𝑥(𝑡)|²≥12𝜖₁‖𝑆𝑥(𝑡)‖²=12𝜖₁‖𝑥‖², (4.18) for all 𝑥∈𝐹−{0} with ‖𝑥‖≥𝑅₁/𝑚₀𝜖₁ and 𝑡∈Ω_𝑥. Hence, from (4.16) and (4.18), one has 𝐹(𝑥)=12𝑥⁺²−12‖𝑥⁻‖²−_𝑡∈𝐙𝐻(((𝑡≤1𝑡,𝑢𝑆𝑥)))2𝑥⁺²−_{𝑡∈Ω𝑥}𝐻(((𝑡≤1𝑡,𝑢𝑆𝑥)))2𝑥⁺²−12𝜖₁‖𝑆𝑥(𝑡)‖²1𝜖₁≤1+12𝑥⁺²−12‖𝑥‖²≤0, (4.19) is true for all 𝑥∈𝐹−{0} with ‖𝑥‖≥𝑅₁/𝑚₀𝜖₁. Let 𝑟₁>0 and denote 𝑄=𝑠𝑒∣0≤𝑠≤𝑟₁⊕𝑥∈𝐸¹‖𝑥‖≤𝑟₂. (4.20) Then by (4.19), for all 𝑟₁>max{𝜌,𝑅₁/𝑚₀𝜖₁}, we have 𝐹(𝑥)≤0,∀𝑥∈𝜕𝑄, (4.21) where 𝜌 is defined by Lemma 4.5, this is just (4.14). We completed the proof of Lemma 4.5.

In order to verify that 𝐹(𝑥) satisfies (𝑓₁) of Lemma 3.1, we need the following lemma.

Lemma 4.6. Write ℎ(𝑥)=_𝑡∈𝐙𝐻(𝑡,𝑢(𝑆𝑥(𝑡))),∀𝑥∈𝐸. (4.22) Then ℎ(𝑥)∈𝐶¹(𝐸,𝐑).

Proof. Let 𝜑(𝑧)=ℎ(𝑡,𝑥+𝑧𝑦), 0≤𝑧≤1, for all 𝑥,𝑦∈𝐸, since 𝐻∈𝐶¹(𝐸,𝐑)𝜑^′(0)=lim_𝑧→0𝜑(𝑧)−𝜑(0)𝑧=lim_𝑧→0ℎ𝑡,𝑥+𝑧𝑦−ℎ(𝑡,𝑥)𝑧=lim_𝑧→01𝑧_𝑡∈𝐙𝐻𝑡,𝑢𝑆𝑥+𝑧𝑦(𝑡)−𝐻(𝑡,𝑢(𝑆𝑥(𝑡)))=lim_{𝑧→0𝑡∈𝐙}∇𝐻𝑡,𝑢𝑆𝑥_𝑡+𝜃_𝑡𝑧𝑦_𝑡⋅𝑢𝑆𝑦_𝑡=_𝑡∈𝐙∇𝐻𝑡,𝑢𝑆𝑥_𝑡+𝜃_𝑡𝑧𝑦_𝑡⋅𝑢𝑆𝑦_𝑡, (4.23) where 0<𝜃_𝑡<1, then ℎ(𝑥) is Gâteaux differential on 𝐸 and ℎ^′(𝑥)𝑦=_𝑡∈𝐙∇𝐻(𝑡,𝑢(𝑆𝑥(𝑡)))⋅𝑢𝑆𝑦(𝑡),∀𝑥,𝑦∈𝐸. (4.24)
Let 𝑥^(𝑘)⇀𝑥 weakly in 𝐸, by Proposition 3.2, one can assume that 𝑥^(𝑘)→𝑥 strongly in 𝑙^𝑝 for 𝑝∈[1,+∞). By (4.24), we have ℎ^′𝑥^(𝑘)−ℎ^′(𝑥)=sup_{𝑦=1𝑡∈𝐙}∇𝐻𝑡,𝑢𝑆𝑥^(𝑘)(𝑡)−∇𝐻(𝑡,𝑢(𝑆𝑥(𝑡)))⋅𝑢𝑆𝑦(𝑡). (4.25) By (4.5) and (4.6), there exists a constant 𝐶₁>0 such that for any 𝑅>0, there holds _{|𝑡|≥𝑅}∇𝐻𝑡,𝑢𝑆𝑥^(𝑘)(𝑡)−∇𝐻(𝑡,𝑢(𝑆𝑥(𝑡)))⋅𝑢𝑆𝑦(𝑡)≤_{|𝑡|≥𝑅}∇𝐻𝑡,𝑢𝑆𝑥^(𝑘)(𝑡)+|(((𝑡⋅𝑢(𝑡)≤∇𝐻𝑡,𝑢𝑆𝑥)))|𝑆𝑦_{|𝑡|≥𝑅}𝐶₁𝑥^(𝑘)+|𝑥|+𝑥^(𝑘)𝛼+|𝑥|^𝛼𝑦≤𝐶_{1|𝑡|≥𝑅}𝑥^(𝑘)+|𝑥|^21/2_{|𝑡|≥𝑅}𝑦^21/2+𝐶_{1|𝑡|≥𝑅}𝑥^(𝑘)+|𝑥|^{𝛼+1𝛼/(𝛼+1)}_{|𝑡|≥𝑅}𝑦^{𝛼+11/(𝑎+1)}≤𝐶₁𝑦_𝑙²_{|𝑡|≥𝑅}𝑥^(𝑘)+|𝑥|^21/2+𝐶₁𝑦_𝑙^𝛼+1_{|𝑡|≥𝑅}𝑥^(𝑘)+|𝑥|^{𝛼+1𝛼/(𝛼+1)}, (4.26) so by Proposition 3.2, there exists a constant 𝐶₂>0 such that for any ‖𝑦‖=1, _{|𝑡|≥𝑅}∇𝐻𝑡,𝑢𝑆𝑥^(𝑘)(𝑡)−∇𝐻(𝑡,𝑢(𝑆𝑥(𝑡)))⋅𝑢𝑆𝑦(𝑡)≤𝐶_{2|𝑡|≥𝑅}𝑥^(𝑘)2+|𝑥|^21/2+_{|𝑡|≥𝑅}𝑥^(𝑘)𝛼+1+|𝑥|^{𝛼+1𝛼/(𝛼+1)}. (4.27) We deduce from (4.27) that for any 𝜖>0, there has 𝑅>0 so large that _{|𝑡|≥𝑅}∇𝐻𝑡,𝑢𝑆𝑥^(𝑘)(𝑡)(((𝑡(𝑡)<𝜖−∇𝐻𝑡,𝑢𝑆𝑥)))⋅𝑢𝑆𝑦2, (4.28) for all 𝑘∈𝐍 and 𝑦∈𝐸 with ‖𝑦‖=1. On the hand, it is well known that since 𝑥^(𝑘)→𝑥 strongly in 𝑙², then when 𝑥∈𝐸_𝐼𝑅∇𝐻𝑡,𝑢𝑆𝑥^(𝑘)−∇𝐻(𝑡,𝑢𝑆(𝑥))_𝑙2→0, (4.29) as 𝑘→∞, where 𝐸_𝐼𝑅={𝑥∣𝑥(𝑡)∈𝐸,𝑡∈𝐼_𝑅} and 𝐼_𝑅={𝑡∣𝑡∈𝐙,|𝑡|<𝑅}. Thus there is 𝑘₀∈𝐍 such that _{|𝑡|≤𝑅}∇𝐻𝑡,𝑢𝑆𝑥^(𝑘)(𝑡)(((𝑡(𝑡)<𝜖−∇𝐻𝑡,𝑢𝑆𝑥)))⋅𝑢𝑆𝑦2, (4.30) is true for all integer 𝑘≥𝑘₀ and all 𝑦∈𝐸 with ‖𝑦‖=1. Combining (4.28) and (4.30), it is easy to see ∇𝐻𝑡,𝑢𝑆𝑥^(𝑘)−∇𝐻(𝑡,𝑢(𝑆𝑥))<𝜖,𝑘≥𝑘₀. (4.31) It follows from the arbitrariness of 𝜖 that ℎ(𝑥)∈𝐶¹(𝐸,𝐑).
Finally, let us complete the proof of Theorem 4.1 by verifying 𝐹(𝑥) satisfies the Palais-Smale condition.

Lemma 4.7. With assumptions of Theorem 4.1, 𝐹(𝑥) satisfies the (PS)^∗ condition.

Proof. Let (𝑥^(𝑘)) be a (PS)^∗ sequence, that is, 𝑥^(𝑘)∈𝐸_𝑛, for all 𝑘∈𝐍, and 𝐹(𝑥^(𝑘))≤𝐶, 𝐹^′(𝑥^(𝑘))→0, as 𝑘→∞. We claim that (𝑥^(𝑘)) is bounded. If not, passing to a subsequence if necessary, we may assume that ‖𝑥^(𝑘)‖→∞ as 𝑘→∞.
Denote 𝐼₁={𝑡∈𝐙∣|𝑥^(𝑘)(𝑡)|≥𝑟/𝑚₀} and 𝐼₁={𝑡∈𝐙∣|𝑥^(𝑘)(𝑡)|<𝑟/𝑚₀} for all 𝑘∈𝐍. By (𝐻₄), (𝐻₅), (4.5), and Lemma 4.6, we have 2𝐹𝑥^(𝑘)−𝐹^′𝑥^(𝑘)⋅𝑥^(𝑘)=_𝑡∈𝐙𝐻^′𝑡,𝑢𝑆𝑥^(𝑘)(𝑡)⋅𝑢𝑆𝑥^(𝑘)(𝑡)−2𝐻𝑡,𝑢𝑆𝑥^(𝑘)(𝑡)≥_{{𝑡∈𝐙|𝑢(𝑆𝑥(𝑘)(𝑡))|≥𝑟}}𝑏𝑢𝑆𝑥^(𝑘)𝑥(𝑡)^𝛽≥_𝑡∈𝐼1𝑚^𝛽₀𝑥^(𝑘)𝛽, (4.32) which implies that 1𝑥^(𝑘)_𝑡∈𝐼1𝑥^(𝑘)𝛽→0,as𝑘→∞. (4.33) Making use of (4.5) and (4.6), we obtain that there is a constant 𝐶₃ such that 𝐻^′𝑡,𝑢𝑆𝑥^(𝑘)(𝑡)≤𝐶₃𝑥^(𝑘)+𝑥^(𝑘)𝛼,∀𝑥∈𝐸. (4.34) Hence from (4.34), there holds 𝐹^′_𝑛𝑥^(𝑘)⋅𝑥⁺_(𝑘)=12𝑥⁺_(𝑘)²−_𝑡∈𝐙𝐻𝑡,𝑢𝑆𝑥^(𝑘)(𝑡)⋅𝑢𝑆𝑥⁺_(𝑘)(𝑡≥1)2𝑥⁺_(𝑘)²−_𝑡∈𝐙𝐻𝑡,𝑢𝑆𝑥^(𝑘)(𝑡)⋅𝑢𝑆𝑥⁺_(𝑘)(𝑡)≥12𝑥⁺_(𝑘)²−_𝑡∈𝐙𝐶₃𝑥^(𝑘)+𝑥^(𝑘)𝛼𝑢𝑆𝑥⁺_(𝑘)(𝑡)≥12𝑥⁺_(𝑘)²−_𝑡∈𝐙𝐶₃𝑀₀𝑥^(𝑘)𝑥⁺_(𝑘)−_𝑡∈𝐙𝐶₃𝑀₀𝑥^(𝑘)𝛼𝑥⁺_(𝑘). (4.35) By Hölder inequality and Proposition 3.2, we achieve _𝑡∈𝐙𝑥⁺_(𝑘)^𝛼𝑥⁺_(𝑘)=_𝑡∈𝐼1𝑥^(𝑘)𝛼𝑥⁺_(𝑘)+_𝑡∈𝐼2𝑥^(𝑘)𝛼𝑥⁺_(𝑘)≤𝑟𝑚₀^𝛼_𝑡∈𝐼1𝑥⁺_(𝑘)+_𝑡∈𝐼2𝑥^{(𝑘)𝛼𝛽/𝛼𝛼/𝛽}⋅_𝑡∈𝐼2𝑥⁺_(𝑘)^{𝛽/(𝛽−𝛼)(𝛽−𝛼)/𝛽}≤𝑟𝑚₀^𝛼_𝑡∈𝐙𝑥⁺_(𝑘)+_𝑡∈𝐼2𝑥^{(𝑘)𝛽𝛼/𝛽}⋅_𝑡∈𝐙𝑥⁺_(𝑘)^{𝛽/(𝛽−𝛼)(𝛽−𝛼)/𝛽}=𝑟𝑚₀^𝛼𝑥⁺_(𝑘)𝑙1+_𝑡∈𝐼2𝑥^{(𝑘)𝛽𝛼/𝛽}𝑥⁺_{(𝑘)𝑙𝛽/(𝛽−𝛼)}≤𝑟𝑚₀^𝛼𝜆₁𝑥⁺_(𝑘)+_𝑡∈𝐼2𝑥^{(𝑘)𝛽𝛼/𝛽}𝜆_{𝛽/(𝛽−𝛼)}𝑥⁺_(𝑘)=𝑟𝑚₀^𝛼𝜆₁+𝜆_{𝛽/(𝛽−𝛼)𝑡∈𝐼2}𝑥^{(𝑘)𝛽𝛼/𝛽}𝑥⁺_(𝑘). (4.36) Similarly, 𝑥^(𝑘)𝑥⁺_(𝑘)≤𝑟𝑚₀𝜆₁+𝜆_{𝛽/(𝛽−1)𝑡∈𝐼2}𝑥^{(𝑘)𝛽1/𝛽}𝑥⁺_(𝑘). (4.37) Combing (4.35)–(4.37), yields 𝐹^′_𝑛𝑥^(𝑘)⋅𝑥⁺_(𝑘)≥12𝑥⁺_(𝑘)²−𝜆₁𝐶₃𝑀₀𝑟𝑚₀^𝛼+𝑟𝑚₀𝑥⁺_(𝑘)−𝐶₃𝑀₀𝜆_{𝛽/(𝛽−𝛼)𝑡∈𝐼2}𝑥^{(𝑘)𝛽𝛼/𝛽}𝑥⁺_(𝑘)−𝐶₃𝑀₀𝜆_{𝛽/(𝛽−1)𝑡∈𝐼2}𝑥^{(𝑘)𝛽1/𝛽}𝑥⁺_(𝑘), (4.38) Since 1<𝛼<𝛽, we deduce from (4.33) and (4.38) that 𝑥⁺_(𝑘)𝑥^(𝑘)→0,as𝑘→∞, (4.39) similarly, 𝑥⁻_(𝑘)𝑥^(𝑘)→0,as𝑘→∞. (4.40) Now, let 𝑦^(𝑘)(𝑡)=𝑥^(𝑘)(𝑡),if𝑥^(𝑘)(𝑡≤𝑟)𝑚₀0,if𝑥^(𝑘)(𝑡>𝑟)𝑚₀,𝑧^(𝑘)(𝑡)=𝑥^(𝑘)(𝑡)−𝑦^(𝑘)(𝑡),∀𝑘∈𝐍,𝑡∈𝐙. (4.41)(4.42) Then _𝑡∈𝐙𝑧^(𝑘)𝛽=_𝑡∈𝐙𝑥^(𝑘)−𝑦^(𝑘)𝛽=_𝑡∈𝐼1𝑥^(𝑘)−𝑦^(𝑘)𝛽+_𝑡∈𝐼2𝑥^(𝑘)−𝑦^(𝑘)𝛽=_𝑡∈𝐼1𝑥^(𝑘)𝛽≤_𝑡∈𝐙𝑥^(𝑘)𝛽, (4.43) that is 𝑧^(𝑘)𝛽_𝑙𝛽≤𝑥^(𝑘)𝛽_𝑙𝛽. (4.44) For 𝛽>1, by (4.32), (4.41) and Proposition 3.2, there exists a constant 𝐶₄>0 such that 𝐶₄1+𝑥^(𝑘)≥𝑧^(𝑘)𝛽_𝑙𝛽,∀𝑘∈𝐍. (4.45) Since 𝐸⁰ is of finite dimension, using Hölder inequality and (4.45), for any 𝑥⁰_(𝑘)∈𝐸⁰, we have 𝑥⁰_(𝑘)²_𝑙2=𝑥⁰_(𝑘),𝑥^(𝑘)_𝑙2=𝑥⁰_(𝑘),𝑧^(𝑘)+𝑦^(𝑘)_𝑙2=𝑥⁰_(𝑘),𝑧^(𝑘)_𝑙2+𝑥⁰_(𝑘),𝑦^(𝑘)_𝑙2≤_𝑡∈𝐙𝑥⁰_(𝑘)𝑧^(𝑘)+_𝑡∈𝐙𝑥⁰_(𝑘)𝑦^(𝑘)≤_𝑡∈𝐙𝑥⁰_(𝑘)𝑟𝑚₀+_𝑡∈𝐙𝑥⁰_(𝑘)^{𝛽/(𝛽−1)(𝛽−1)/𝛽}_𝑡∈𝐙𝑧^{(𝑘)𝛽1/𝛽}=𝑟𝑚₀𝑥^0(𝑘)_𝑙1+𝑥⁰_{(𝑘)𝑙𝛽/(𝛽−1)}𝑧^(𝑘)_𝑙𝛽≤𝐶₅𝑥⁰_(𝑘)𝑙21+𝑧^(𝑘)_𝑙𝛽, (4.46) where 𝐶₅>0 is a constant.
Hence by (4.45) and (4.46), there exist positive constants 𝐶₆, 𝐶₇ such that 𝑥⁰_(𝑘)≤𝐶₆𝐶₅1+𝑧^(𝑘)_𝑙𝛽≤𝐶₇1+𝑧^(𝑘)1/𝛽, (4.47) which implies 𝑥⁰_(𝑘)𝑥^(𝑘)→0,as𝑘→∞, (4.48) when 𝛽>1.
By (4.39), (4.40), and (4.48), it follows 𝑥1=^(𝑘)𝑥^(𝑘)=𝑥⁺_(𝑘)+𝑥⁻_(𝑘)+𝑥⁰_(𝑘)𝑥^(𝑘)→0,as𝑘→∞, (4.49) which is a contradiction. Therefore, (𝑥^(𝑘)) must be bounded. That is, 𝐹(𝑥) satisfies the (PS)^∗ condition.

By Lemma 3.1, 𝐹(𝑥) possesses a critical point 𝑥∈𝐸 such that 𝐹(𝑥)≥𝛿>0 and (1.1) has a nontrivial homoclinic orbit.

Acknowledgment

This work was supported by the National Natural Science Foundation of China (no. 11101098) and the Xinmiao Program of Guangzhou University.