Homoclinic Orbits for a Class of Noncoercive Discrete Hamiltonian Systems

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.


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 , where I N is the identity matrix on 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 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 Z 2 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 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 1.5 Let l t denotes the smallest eigenvalue of v * L t v, that is, For later use, we need the following assumptions: 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.

2.1
Define the subspace X of S as another subspace E of X as follows: The space E is a Hilbert space with the inner product and the norm introduced from the inner product as follows: Define a functional F x on E as follows: according to the definition of x , F x can be written in another form as follows: The functional F x is a well-defined C 1 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 then

2.11
Write F x i t ∂F x /∂x i t , i 1, 2, for any given t ∈ Z, there holds

2.12
Then we can draw a conclusion that F x 0 is true if and only if which can be reformed as which is just 1.1 .Therefore, we obtain the following lemma.

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 E 1 ⊕ E 2 with both E 1 and E 2 being infinite dimensional.Suppose ν n resp., ω n is an orthogonal basis for E 1 resp.E 2 , and set and f n f| E n , 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 .
1 with e 1 such that Next we consider the eigenvalue problem.

3.3
Equation 3.3 can be reformed as follows:

3.4
Denote 3 can be expressed by the following: Therefore a standard argument shows that σ A , the spectrum of A, consists of eigenvalues numbered by, counted in their multiplicities the following: with λ k → ±∞ as k → ±∞, and denote the corresponding system of eigenfunctions of A by e k .Let E 0 KerA, E span{e 1 , . . ., e n } and E − E 0 ⊕E ⊥ E , where S ⊥ E stands for the orthogonal complementary subspace of S in E. Then so the functional 2.7 can be rewritten as follows for all t ∈ Z} and their norms are defined by the following: |A| is the absolute value.Give another norm the domain of A by the following: it is easy to get, for all x ∈ E,

3.14
Now we state a fundamental proposition, which will be used in the later.

3.15
Proof.We complete the proof of Proposition 3.2 by 3 steps.
Step 1.When L 1 holds and p 2, we prove that Note that, by L 1 , l t → ∞ as |t| → ∞, that is, l t is bounded from below and so there is a a > 0 such that

3.17
For R > 0, choose a subsequence x k t ∈ E, one has

3.18
For any given > 0, by 3.18 , one can take R 0 so large that

3.19
Without loss of generality, we can assume that I .This together with the uniqueness of the weak limit in l 2 I , we have x k → 0 in E I , so there exists a k 0 such that , ∀k ≥ k 0 .

3.20
Combing 3.19 and 3.20 , we have x k → 0 in l 2 .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 3.21 which together with 3.16 yields 3.15 .
Step 3. Since L 1 implies l t → ∞ as |t| → ∞, by Step 1 and 2, it remains to consider the case for 1

3.24
From 3.24 , we get

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.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.Proof.For any x ∈ E, it is easy to see that there exist two constants 0 < m 0 < M 0 such that By H 1 , H 3 , and 4.5 , for all > 0, there exist a constant C > 0 such that Now by mean value Theorem, 4.5 and 4.6 , for all x ∈ E and t ∈ Z, we have on the other hand,

4.10
By Proposition 3.2 and 4.10 , for any x ∈ E, it holds
Lemma 4.5.Under assumptions of Theorem 4.1, let e ∈ E 2 1 with e 1, there exist r 1 , r 2 > 0 such that where Proof.Let e ∈ E 2 1 with e 1 and F E 1 ⊕ span{e}.For x x − x 0 x ∈ F − {0} and > 0, denote then there exists 1 > 0 such that where Ω x is the number of t in Ω x and • is the greatest integer function.By H 1 , for d 1/2 2 1 m 2 0 , there exists R 1 > 0 such that where m 0 was defined by 4.5 .Then it follows

4.20
Then by 4.19 , for all r 1 > max{ρ, R 1 /m 0 1 }, we have 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 f 1 of Lemma 3.1, we need the following lemma.

4.27
We deduce from 4.27 that for any > 0, there has R > 0 so large that

4.33
Making use of 4.5 and 4.6 , we obtain that there is a constant C 3 such that

Journal of Applied Mathematics
Hence from 4.34 , there holds

4.35
By H ölder inequality and Proposition 3.2, we achieve

4.45
Since E 0 is of finite dimension, using H ölder inequality and 4.45 , for any x 0 k ∈ E 0 , we have x 0 k , z k y k l 2 x 0 k , z k l 2 x 0 k , y k 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.
u Sx t , ∀x ∈ E.

C 5 >
0 is a constant.Hence by 4.45 and 4.46 , there exist positive constants C 6 , C 7 such that 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 , where x 1 , x 2 ∈ 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 .10 respectively.For any given 1 ≤ r < ∞, define l r {x {x t } ∈ S | t∈Z |x t | r < ∞} with the norm Theorem 4.1.Suppose that H satisfies L 1 and H 1 -H 5 .Then the discrete Hamiltonian system 1.1 has a nontrivial homoclinic orbit.
by Proposition 3.2, one can assume that x k → x strongly in l p for p ∈ 1, ∞ .By 4.24 , we have t∈Z∇H t, u Sx t • u Sy t , ∀x, y ∈ E.
∈ N and y ∈ E with y 1.On the hand, it is well known that since x k → x strongly in l 2 , then when x ∈ E I R whereE I R {x | x t ∈ E, t ∈ I R } and I R {t | t ∈ Z, |t| < R}.Thus there is k 0 ∈ N 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 → ∞.
It follows from the arbitrariness of thath x ∈ C 1 E, R .Finally, let us complete the proof of Theorem 4.1 by verifying F x satisfies the Palais-Smale condition.Lemma 4.7.Denote I 1 {t ∈ Z | |x k t | ≥ r/m 0 } and I 1 {t ∈ Z | |x k t | < r/m 0 } for all k ∈ N.