EXISTENCE OF HOMOCLINIC ORBIT FOR SECOND-ORDER NONLINEAR DIFFERENCE EQUATION

: By using the Mountain Pass Theorem, we establish some existence criteria to guarantee the second-order nonlinear diﬀerence equation ∆ [ p ( t )∆ u ( t − 1)] + f ( t, u ( t )) = 0 has at least one homoclinic orbit, where t ∈ Z , u ∈ R .


Introduction
In this paper, we shall be concerned with the existence of homoclinic orbit for the secondorder difference equation: where the forward difference operator ∆u(t) = u(t + 1) − u(t), ∆ 2 u(t) = ∆ (∆u(t)), p(t) > 0, The theory of nonlinear difference equations has been widely used to study discrete models appearing in many fields such as computer science, economics, neural network, ecology, cybernetics, etc.Since the last decade, there has been much literature on qualitative properties of difference equations, those studies cover many of the branches of difference equations, such as [1-3, 10, 11] and references therein.In some recent papers [7][8][9][22][23][24], the authors studied the existence of periodic solutions of second-order nonlinear difference equation by using the critical point theory.These papers show that the critical point method is an effective approach to the study of periodic solutions of second-order difference equations.
In the theory of differential equations, a trajectory which is asymptotic to a constant state as |t| → ∞ (t denotes the time variable) is called a homoclinic orbit.It is well-known that homoclinic orbits play an important role in analyzing the chaos of dynamical systems.
(see, for instance, [5,6,15,[19][20][21], and references therein).If a system has the transversely intersected homoclinic orbits, then it must be chaotic.If it has the smoothly connected homoclinic orbits, then it cannot stand the perturbation, its perturbed system probably produce chaotic phenomenon.
In general, Eq.(1.1) may be regarded as a discrete analogue of the following second-order differential equation Recently, the following second order self-adjoint difference equation has been studied by using variational method ( see [12]).Ma and Guo obtained homoclinic orbits as the limit of the subharmonics for Eq.(1.3) by applying the Mountain Pass theorem , their results are relying on q(t) = 0.If q(t) = 0, the traditional ways in [13] are inapplicable to our case.Some special cases of (1.1) have been studied by many researchers via variational methods, (see, for example, [7] and references therein).However, to our best knowledge, results on homoclinic solutions for Eq.(1.1) has not been studied.Motivated by [6,12], the main purpose of this paper is to give some sufficient conditions for the existence of homoclinic and even homoclinic solutions to Eq.(1.1).
Without loss of generality, we assume that u = 0 is an equilibrium for (1.1), we say that a solution u(t) of (1.1) is a homoclinic orbit if u = 0 and u → 0 as t → ±∞.
Theorem 1.1.Assume that the following conditions are satisfied: , where K, W is T -periodic with respect to t, T > 0, K(t, u), W (t, u) are continuously differentiable in u; (F2) There are constants b 1 , b 2 > 0 such that for all (t, u) (F5) There is a constant µ > 2 such that for every t ∈ Z, u ∈ R\{0}, Then Eq.(1.1) possesses at least one nontrivial homoclinic solution.

Preliminaries
In this section, we will establish the corresponding variational framework for (1.1).
Let S be the vector space of all real sequences of the form u = {u(t)} t∈Z = (..., u(−t), u(−t + 1), ..., u(−1), u(0), u(1), ..., u(t), ...) , by which the norm u k can be induced by It is obvious that E k is a Hilbert space of 2kT -periodic functions on Z with values in R and In what follows, l 2 k denotes the space of functions whose second powers are summable on the interval N[−kT, kT − 1] equipped with the norm Moreover, l ∞ k denotes the space of all bounded real functions on the interval N[−kT, kT − 1] endowed with the norm . (2.2) let where ) and it is easy to check that by (F5), EJQTDE, 2010 No. 72, p. 4 by using we can compute the Fréchet derivative of (2.4) as Thus, u is a critical point of so the critical points of I k in E k are classical 2kT -periodic solutions of (1.1).That is, the functional I k is just the variational framework of (1.1).

Proofs of theorems
At first, let us recall some properties of the function W (t, u) from Theorem 1.1.They are all necessary to the proof of Theorems .
We will obtain a critical point of I k by use of a standard version of the Mountain Pass Theorem(see [17]).It provides the minimax characterization for the critical value which is important for what follows.Therefore, we state this theorem precisely.
By p(t) > 0, P k + I k is positive definite.Suppose that the eigenvalues of P k + I k are λ −kT , λ −kT +1 , ...λ −1 , λ 0 , λ 1 , ...λ kT −1 , then they are all greater than zero.We define By (3.9), we have Take a = 1 4 b1 ρ 2 > 0, we get By Hölder inequality and (3.3), we have ζ ∈ R, ω ∈ E k \{0}, which leads to given by (iii Lemma 3.4.Suppose that the conditions of Theorem 1.1 hold true, then there exists a Lemma 3.5.Suppose that the conditions of Theorem 1.1 hold true, then there exists a constant d independent of k such that the following inequalities are true: Lemmas 3.6.Suppose that (F1) − (F4) are satisfied, then there exists a constant δ such that . By a fashion similar to the proofs in [12], we can prove Lemma 3.4, Lemma 3.5 and Lemma 3.6, respectively.The detailed proofs are omitted.
Proof of Theorem 1.1.We will show that {u k } k∈N possesses a convergent subsequence {u km } in E loc (Z, R) and a nontrivial homoclinic orbit u ∞ emanating from 0 such that u km → u ∞ as k m → ∞.
Since u k = {u k (t)} is well defined on N[−kT, kT − 1] and u k k ≤ d for all k ∈ N, we have the following consequences.

First, let u
Repeat this procedure for all k ∈ N. We obtain sequence {u p km } ⊂ {u p−1 km }, u p ∈ {u p km } and there exists u p ∈ E p such that It shows that By series convergence theorem, u ∞ satisfy u ∞ (t) → 0, △u ∞ (t − 1) → 0, and as m ≥ p, k m ≥ p, where d 1 is independent of k, {k m } ⊂ {k} are chosen as above, we have Letting p → ∞ , by the continuity of F (t, u) and I ′ k , which leads to EJQTDE, 2010 No. 72, p. 10 and Clearly, u ∞ is a solution of (1.1).
To complete the proof of Theorem 1.1, it remains to prove that u ∞ ≡ 0.

Example
In this section, we give an example to illustrate our results.
is a continuous function in the second variables and satisfies f (t + T, u) = f (t, u) for a given positive integer T .As usual, N, Z and R denote the set of all natural, integer and real numbers, respectively.For a, b ∈ Z, denote N(a) = {a, a + 1, ...}, N(a, b) = {a, a + 1, ...b} when a ≤ b.