Homoclinic orbits for a class of nonperiodic Hamiltonian systems with some twisted conditions

In this paper, by the Masolv index theory, we will study the existence and multiplicity of homoclinic orbits for a class of asymptotically linear nonperiodic Hamiltonian systems with some twisted conditions on the Hamiltonian functions


Introduction and main results
Consider the following first order non-autonomous Hamiltonian systemṡ where z : R → R 2N , J = 0 −I N I N 0 , H ∈ C 1 (R × R 2N , R) and ∇ z H(t, z) denotes the gradient of H(t, z) with respect to z. As usual we say that a nonzero solution z(t) of (HS) is homoclinic (to 0) if z(t) → 0 as |t| → ∞.
As a special case of dynamical systems, Hamiltonian systems are very important in the study of gas dynamics, fluid mechanics, relativistic mechanics and nuclear physics.
While it is well known that homoclinic solutions play an important role in analyzing the chaos of Hamiltonian systems. If a system has the transversely intersected homoclinic solutions, then it must be chaotic. If it has the smoothly connected homoclinic solutions, then it cannot stand the perturbation, its perturbed system probably produces chaotic phenomena. Therefore, it is of practical importance and mathematical significance to consider the existence of homoclinic solutions of Hamiltonian systems emanating from 0.
In the last years, the existence and multiplicity of homoclinic orbits for the first order system (HS) were studied extensively by means of critical point theory, and many results were obtained under the assumption that H(t, z) depends periodically on t (see, e.g., [5,[11][12][13]17,20,[27][28][29][30]). Without assumptions of periodicity the problem is quite different in nature and there is not much work done so far. To the best of our knowledge, the authors in [14] firstly obtained the existence of homoclinic orbits for a class of first order systems without any periodicity on the Hamiltonian function. After this, there were a few papers dealing with the the existence and multiplicity of homoclinic orbits for the first order system (HS) in this situation (see, e.g., [15,16,21]).
In the present paper, with the Maslov index theory of homoclinic orbits introduced by Chen and Hu in [10], we will study the existence and multiplicity of homoclinic orbits for (HS) without any periodicity on the Hamiltonian function. To the best of the author's knowledge, the Maslov index theory of homoclinic orbits is the first time to be used to study the existence of homoclinic solutions. We are mainly interested in the Hamiltonian functions of the form H(t, z) = −L(t)z · z + R(t, z), (1.1) where L is an 2N × 2N symmetric matrix valued function. We assume that (L 1 )L ∈ C(R, R N 2 ), and there are α, c > 0, t 0 ≥ 0 and a constant matrix P , satisfying where I 2N is the identity map on R 2N and for a 2N × 2N matrix M, we say M ≥ 0 if and only if inf ξ∈R 2N ,|ξ|=1 Mξ · ξ ≥ 0.
In (L 1 ), if P = 0 I N I N 0 , then (L 1 ) is similar to the condition (R 0 ) in [15]. But the restrictions on R(t, z) will be different from [15], and we will give some examples in Remark 1.5. If P = ±I 2N or P = I N +m 0 0 −I N −m in condition (L 1 ), for examples, it's quite different from the existing results as authors known. In short, condition (L 1 ) means that the eigenvalues of L(t) will tend to ±∞ with the speed no less than |t| α . But (L 1 ) does not contain all of these cases. For examples, let N = 1 and L(t) = |t| α cos 2t sin 2t sin 2t − cos 2t , we have the eigenvalues of L(t) are ±|t| α , but there is no constant matrix P satisfying Let |F | be the absolute value ofF , and |F | 1/2 be the square root of |F |. D(F ) is a Hilbert space equipped with the norm Let E = D(|F | 1/2 ), and define on E the inner product and norm by where (·, ·) L 2 denotes the usual inner product on L 2 (R, R 2N ). Then E is a Hilbert space.
It is easy to see that E is continuously embedded in H 1/2 (R, R 2N ), and we further have the following lemma.
This lemma is similar to Lemma 2.1-2.3 in [14], and we will prove it in Section 3.
Define the quadratic form Q on E by It's easy to check that Q(u, v) is a bounded quadratic form on E and hence there exists a unique bounded self-adjoint operator F : E → E such that Besides, define a linear operator K : In view of Lemma 1.1, we know that F is a Fredholm operator and K is a compact operator.
Denote by B the set of all uniformly bounded symmetric 2N × 2N matric functions.
That is to say B ∈ B if and only if B T (t) = B(t) for all t ∈ R and B(t) is uniformly bounded in t as the operator on R 2N . For any B ∈ B, it is easy to see B determines a bounded self-adjoint operator on L 2 , by z(t) → B(t)z(t), for any z ∈ L 2 , we still denote this operator by B, then KB : E ⊂ L 2 → E is a self-adjoint compact operator on E and Before presenting the conditions on R(t, z), we need the concept of Maslov index for homoclinic orbits introduced by Chen and Hu in [10] which is equivalent to the relative Morse index. We will give a brief introduction of it by Definition 2.1, where for any B ∈ B, we denote the associated index pair by (µ F (KB), υ F (KB)).
Now we can present the conditions on R(t, z) as follows. For notational simplicity, we , and in what follows the letter c will be repeatedly used to denote various positive constants whose exact value is irrelevant. Besides, for two 2N × 2N symmetric matrices M 1 and M 2 , , and there exists a constant c > 0 such that (R ∞ ) There exists some R 0 > 0 and continuous symmetric matrix functions B 1 , B 2 ∈ B with µ F (KB 1 ) = µ F (KB 2 ) and υ F (KB 2 ) = 0 such that Then we have our first result. Condition (R ∞ ) is a two side pinching condition near the infinity, we can relax (R ∞ ) to condition (R ± ∞ ) as follows.
(R ± ∞ ) There exists some R 0 > 0 and a continuous symmetric matrix function B ∞ ∈ B with υ F (KB ∞ ) = 0 such that Then we have the following results.
, then (HS) has at least one nontrivial homoclinic orbit.

Define
where δ is a smooth cutoff function satisfying δ(|z|) = 1, |z| < 1, 0, |z| > 2. By Proposition 2.6 below, it is easy to verify R satisfies all the conditions in Theorem 1.2. Furthermore, let the constant B ∞ satisfying λ l+i < B ∞ < λ l+i+1 for some l ∈ Z and i ≥ 2 (or i ≤ −2). Define Then R satisfies all the conditions in Theorem 1.3 and Theorem 1.4. However, it is easy to see that some conditions of the main results in [14][15][16]21] does not hold for these examples.
Remark 1.6. Note that the assumption υ F (KB ∞ ) = 0 in (R ± ∞ ) is not essential for our main results. For the case of (R + ∞ ) with υ F (KB ∞ ) = 0, let B ∞ = B ∞ − εI 2N with ε > 0 small enough, where I 2N is the identity map on R 2N , then µ F (K B ∞ ) = µ F (KB ∞ ) and υ F (K B ∞ ) = 0, and hence (R + ∞ ) holds for B ∞ . Therefore Theorems 1.3 and 1.4 still hold in this case. While for the case of (R and υ F (K B ∞ ) = 0, then this case is also reduced to the case of (W

Preliminaries
In this section, we recall the definition of relative Morse index, saddle point reduction, and give the relationship between them. For this propose, the notion of spectral flow will be used.

Relative Morse index
Let H be a separable Hilbert space, for any self-adjoint operator A on H, there is a unique

A-invariant orthogonal splitting
Here and in the sequel, we denote by ind(·) the Fredholm index of a Fredholm operator.

Definition 2.1. For any bounded self-adjoint Fredholm operator F and a compact self-
and

Saddle point reduction
In this subsection, we describe the saddle point reduction in [4,8,24]. Recall that H is a real Hilbert space, and A is a self-adjoint operator with domain D(A) ⊂ H. Let consists of at most finitely many eigenvalues of finite multiplicities.
Consider the solutions of the following equation where {E λ } is the spectral resolution of A, and let Decompose the space V as follows For each u ∈ H, we have the decomposition Define a functional f on H as follows: . The Euler equation of this functional is the system which solves the system (2.5). Let where u 0 (z 0 ) = |A ε | 1/2 z 0 and let z 0 = x, we have . Then, we have the following theorem duo to Amann and Zehnder [4], Chang [8] and Long [24].
Theorem 2.2. Under the assumption (1), (2), (3), there is a one-one correspondence between the critical points of the C 2 -function a ∈ C 2 (H 0 , R) with the solutions of the operator equation Moreover, the functional a satisfies Since H 0 is a finite dimensional space, for every critical point x of a in H 0 , the Morse index and nullity are finite, we denote them by (m − a (x), m 0 a (x)). Now, let the Hilbert space H be L 2 (R, R 2N ), and the operator A beF = −J d dt + L, Proof. Note that Therefore, Next, since where I is the identity map on H 0 , we have Remark 2.4. For z(x), we also have that there is a constant C > 0 dependent of C R , but independent of β, such that If R satisfies the condition (R 1 ), then for any homoclinic orbit z of (HS), ∇ 2 z R(·, z) ∈ B, and hence we have the associated index pair (µ F (KB), υ F (KB)). For notation simplicity, in what follows, we set , z))), and υ F (z) = υ F (K∇ 2 z (R(t, z))).
This theorem shows the relations between the relative Morse index and the Morse index of the saddle point reduction, it will play an important role in the proof of our main results. The proof of this theorem will be postponed in the next subsection where the notion of spectral flow will be used.

The relationship between µ F (T ), spectral flow and the Morse index of saddle point reduction
It is well known that the concept of spectral flow was first introduced by Atiyah, Patodi and Singer in [6], and then extensively studied in [7,18,25,26,31]. Here, we give a brief introduction of the spectral flow as introduced in [ is called a crossing operator, denoted by C r [F θ ]. As mentioned in [26], an eigenvalue crossing at F θ is said to be regular if the null space of C r [F θ ] is trivial. In this case, we A crossing occurs at F θ is called simple crossing if dim H 0 (F θ ) = 1.
As indicated in [31], the spectral flow Sf (F θ ) will remain the same after a small disturbance of F θ , that is, Sf (F θ ) = Sf (F θ + εid) for ε > 0 and small enough, where id is the identity map on H. Furthermore, we can choose suitable ε such that all the eigenvalue crossings occurred in F θ , 0 ≤ θ ≤ 1 are regular [26]. Thus, without loss of generality, we may assume all the crossings are regular. Let D be the set containing all the points in [0, 1] at which the crossing occurs. The set D contains only finitely many points. The spectral flow of F θ is where D * = D ∩ (0, 1). In what follows, the spectral flow of F θ will be simply denoted by Sf (F θ ) when the starting and end points of the flow are clear from the contents. And P F θ will be simply denoted by P θ .

z(t)), letz(t) = z(−t), it's easy to check thatz(t) is a homoclinic solution ofFz(t) = ∇ zR (t,z(t)), and this is a one-one correspondence between the two systems. By the definition of spectral flow and its catenation
property [31] we only consider the case of R + ∞ from now on.

Proof of our main results
where (·, ·) L 2 denotes the usual inner product on L 2 (R, R 2N ). From (L 1 ), there is a Let K ⊂ E be a bounded set. We will show that K is precompact in L p for 1 ≤ p ∈ (2/(1 + α), ∞). We derive the proof into three steps.
Step 1. The case of p = 2. For R > 0, from (L 1 ) and (3.1) we have For any ε > 0, from (3.2), we can choose R 0 large enough, such that |t|>R 0 On the other hand, by the definition of · E , we have Thus, by the Sobolev compact embedding theorem there exist z 1 , z 2 , · · · , z m ∈ K, such that for any z ∈ K there is z i satisfying From (3.3) and (3.5), we have z − z i L 2 < ε, thus, K has a finite ε−net in L 2 , so the embedding E ֒→ L 2 is compact.
Step 2. The case of p > 2. Since E is continuously embedded in H 1/2 , hence by the Sobolev embedding theorem, E is continuously embedded in L p , ∀p > 2. For any p > 2, by the Hölder inequality we have thus, the embedding E ֒→ L p is compact, ∀p > 2.
Step 3. The case of 1 ≤ p ∈ (2/(1 + α), 2). First, we have α 2−p · p > 1, so we can choose α p satisfying α p ∈ (0, α) and αp 2−p · p > 1. Denote by r = αp 2−p . For R > 0 and z ∈ E, denote by E 1 R (z) = {t; |t| ≥ R and |t| r |z(t)| > 1} and E 2 R (z) = {t; |t| ≥ R and |t| r |z(t)| ≤ 1}. Then, from (3.1), and so (3.7) Let K ⊂ E be a bounded set. For any ε > 0, from (3.7), choose R 0 > 0 large enough, such that On the other hand, by the Sobolev compact embedding theorem there are z 1 , z 2 , · · · , z m ∈ K, such that for any z ∈ K, there exists z i satisfying From (3.8) and (3.9), we have that is to say K has a finite ε−net in L p , and the embedding E ֒→ L p is compact. The proof of the lemma is compact.  Let W (t) be the fundamental solution of (3.10), then W (t) is a path in Sp(2N). Let z(t) be a nontrivial homoclinic orbits of (3.10), that is to say z(0) = 0 and satisfies Denote by S 0 the subset of R 2N satisfying then we have dim S = dim(S 0 ). We claim that Jz 0 ∈ S 0 if z 0 ∈ S 0 and z 0 = 0. We prove it indirectly, assume z 0 , Jz 0 ∈ S 0 with z 0 = 0, that is to say By Theorem 2.2, we have a functional a(x) with x ∈ X, whose critical points give rise to solutions of (HS).
Proof. Assume there is a sequence {x n } ⊂ X, satisfying a ′ (x n ) → 0(n → ∞). That is where z n = z(x n ) defined in section 2.1. Since X is a finite dimensional space, and from the definition of z n , it's enough to prove {z n } is bounded in E. For each ε ∈ (0, 1), define (3.14) It is easy to verify that {C n } satisfies where c is the constant in condition (R 1 ) and I is the identity map on R 2N . Since B 1 ≤ B 2 , and υ F (KB 1 ) = υ F (KB 2 ) = 0, we can choose ε small enough, such that for each n ∈ N + , satisfying µ F (KC n ) = µ F (KB 1 ) and υ F (KC n ) = 0. Thus F − KC n is reversible on E and there is a constant δ > 0, such that On the other hand, for b ∈ (0, 1), there is a constant c > 0 depending on b, such that for  [8].
In order to proof Theorem1.3 and Theorem1.4, we need the following lemma which is similar to Lemma 3.4 in [22] and Lemma 3.3 in [23]. Lemma 3.3. Assume (R 1 ), (R 0 ) and (R + ∞ ) hold, then there exists a sequence of functions R k ∈ C 2 (R × R 2N , R), k ∈ N, satisfying the following properties: (1) There exists an increasing sequence of real numbers M k → ∞(k → ∞) such that (3.18) (2) For each k ∈ N, there is a C > 0 independent of k, such that (3) For each k ∈ N, there exists some C k > 0 and a constant γ with γI 2N > B ∞ , where I 2N is the identity map on R 2N .
It's easy to see that η ∈ C 2 ([0, ∞), R). Choose a sequence {M k } of positive numbers such As in [22,23], we can check that R k satisfies (3.18)-(3.21) for each k ∈ N. ✷ For each k ∈ N, we consider the following problem where R k is given in Lemma 3.3. Performing on (HS) k the saddle point reduction. We choose the number β which is used in the projection for the saddle point reduction in section 2.2. First we choose β > max{2(C + 1), 2(γ + 1)}, and β ∈ σ(A 0 ).
Thus for each k and such a β fixed, by Theorem 2.2, we have a functional a k,β (x), x ∈ X β , whose critical points give rise to solutions of (HS) k . Similarly we have a functional a γ,β (x), x ∈ X β , whose critical points give rise to solutions of the following systems (HS) γ For notational simplicity, we denote a k , a γ for a k,β and a γ,β . Define For the functional a k , similar to Lemma 3.2, we have the following lemma. (1) a k satisfies (PS) condition, the critical point set of a k is compact, Proof. The proof is similar to Lemma 3.2. From Theorem 2.5, we have Similar to (3.16), we have for b ∈ (0, 1), there is some c > 0, such that Choose b ∈ ( 1−α 1+α , 1), similar to (3.17), we have From (3.25) and (3.27),we have There exist c > 0, such that for any k ∈ N, and z ∈ L 2 satisfies the systems Proof. We prove it indirectly. Assume there exist R k , z k , satisfies the conditions, and z k L ∞ → ∞, that is z F → ∞. Since |∇ z R k (t, z)| < c|z|, ∀t ∈ R, z ∈ R 2N , we have z k F ≤ c z k L 2 . Denote y k = z k z k F , then we have y k → y in L 2 for some y ∈ L 2 with y L 2 > 0, andF Then for any r > 0, there exist C r > 0, satisfying where I r = [−r, r]. Since y L 2 > 0, there is a r 0 > 0, such that Then from the similar argument in [23], there is a subsequence we may assume {y k } converges in uniform norm to y, and y(t) = 0, ∀t ∈ I r . Therefor |z k (t)| → ∞ uniformly on I r , and there is K(r) depending on r, such that |z k (t)| ≥ R 0 , for any t ∈ I r and k ≥ K(r).
Performing the saddle point reduction on the following systems For β large enough, we have the functional a ∞,β (denote by a ∞ for simplicity) and the function z(x), since υ F (KB ∞ ) = 0, we have the following decomposition where a ′′ ∞ (0) is positive definite on X + β and negative definite on X − β . From Remark2.4, and υ F (KB ∞ ) = 0, there exists α > 0, such that for β large enough From the uniform boundary of ∇ 2 z R k (t, z) and Remark 2.4, we can choose β large enough, such that where I c r = R \ I r , and from the definition of R k , that is ((∇ 2 z R k (t, z k ) − B ∞ )x, x) L 2 (Ir) ≥ 0, ∀x ∈ X − β , k ≥ K(r).
for k large enough. Thus we have That is m − a k (x) ≥ m − a∞ (0), from Theorem2.5, m − a k (x k ) = dim(E − (H 0 )) + µ F (z x (x k )), m − a∞ (0) = dim(E − (H 0 )) + µ F (KB ∞ ), thus µ F (z x ) ≥ µ F (KB ∞ ), which contradicts the assumption. ✷ Proof of Theorem 1.3. As claimed in Remark 2.7, we can only consider the case of (R + ∞ ). Note that z = 0 is a critical point of a k , the morse index of 0 for a k is m − a k (0) = dim(E − (H 0 )) + µ F (KB 0 ), since γ · I 2N > B ∞ , we have (3.37) From proposition (2) in Lemma3.4, use the (m − a k (0)) th and (m − a k (0) + 1) th Morse inequalities, a k has a nontrivial critical point x k with it morse index m − a k (x k ) ≤ m − a k (0) + 1, that is µ F (z k ) ≤ µ F (KB 0 ) + 1 ≤ µ F (KB ∞ ) − 1, then from Lemma3.5, we have {z k } is bounded in L ∞ . Thus z k is a nontrivial solution of (HS) for k large enough. ✷ The proof of Theorem 1.4 is similar to the proof of Theorem 1.3. Instead of Morse theory we make use of minimax arguments for multiplicity of critical points.
Let X be a Hilbert space and assume φ ∈ C 2 (X, R) is an even functional, satisfying the (PS) condition and φ(0) = 0. Denote S c = {u ∈ X| u = c}.
The proof is similar to the case of (R + ∞ ), we omit it here.