Homoclinic Orbits for a Class of Subquadratic Second Order Hamiltonian Systems

and q(t) → 0 as |t| → ∞ (see [1]). In the following, (⋅, ⋅) : R × R 󳨃→ R denotes the standard inner product in R and | ⋅ | is the induced norm. Hamiltonian system theory, a classical as well as a modern study area, widely consists in mathematical sciences, life sciences, and various aspects of social science. Lots of mechanical and field theory models even exist in the form of Hamiltonian system. Solutions of Hamiltonian system can be divided into periodic solution, subharmonic solution, homoclinic orbit, heteroclinic orbit, and so on. Homoclinic orbit was discovered among the models of nonlinear dynamical system and affects the nature of the whole system significantly. Starting from the Poincaré era, the study of homoclinic orbits in the nonlinear dynamical system exploits Perturbation method mainly. It is not until the recent decade that variational principle has been widely used. The existence and multiplicity of homoclinic orbits for the second order Hamiltonian systems have been extensively investigated in many recent papers (see [1–16]). The main feature of the problem is the lack of global compactness due to unboundedness of domain. To overcome the difficulty, many authors have considered the periodic case, autonomous case, or asymptotically periodic case (see [1–4]). Some papers treat the coercive case (see [5–8]). Recently, the symmetric case has been dealt with (see [9–11]). Compared with the superquadratic case, the case that W(t, x) is subquadratic as |x| → +∞ has been considered only by a few authors. As far as the author is aware, the author in [5] first discussed the subquadratic case. Later, the authors in [12] dealt with this case by use of a standard minimizing argument, which is the following theorem.


Introduction and Main Results
In this paper, we will study the existence and multiplicity of homoclinic orbits for the second order Hamiltonian systems of the type: q () −  ()  () + ∇ (,  ()) = 0, where  ∈ (R, R N 2 ) is a symmetric matrix valued function and  ∈  1 (R × R N , R).As usual, we say that  is a nontrivial homoclinic orbit (to 0) if  ∈  2 (R, R N ),  ̸ = 0 and () → 0 as || → ∞ (see [1]).In the following, (⋅, ⋅) : R N × R N  → R denotes the standard inner product in R N and | ⋅ | is the induced norm.
Hamiltonian system theory, a classical as well as a modern study area, widely consists in mathematical sciences, life sciences, and various aspects of social science.Lots of mechanical and field theory models even exist in the form of Hamiltonian system.Solutions of Hamiltonian system can be divided into periodic solution, subharmonic solution, homoclinic orbit, heteroclinic orbit, and so on.Homoclinic orbit was discovered among the models of nonlinear dynamical system and affects the nature of the whole system significantly.Starting from the Poincaré era, the study of homoclinic orbits in the nonlinear dynamical system exploits Perturbation method mainly.It is not until the recent decade that variational principle has been widely used.The existence and multiplicity of homoclinic orbits for the second order Hamiltonian systems have been extensively investigated in many recent papers (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]).The main feature of the problem is the lack of global compactness due to unboundedness of domain.To overcome the difficulty, many authors have considered the periodic case, autonomous case, or asymptotically periodic case (see [1][2][3][4]).Some papers treat the coercive case (see [5][6][7][8]).Recently, the symmetric case has been dealt with (see [9][10][11]).Compared with the superquadratic case, the case that (, ) is subquadratic as || → +∞ has been considered only by a few authors.As far as the author is aware, the author in [5] first discussed the subquadratic case.Later, the authors in [12] dealt with this case by use of a standard minimizing argument, which is the following theorem.
Subsequently, the condition ( 2 ) was generalized, respectively, in [13,14] by the following conditions: The authors in [13,14] got infinitely many homoclinic orbits by using the variant fountain theorem in [17].
Motivated by these papers, we also consider the subquadratic case.Main results are the following theorems.
Theorem 2. Assume that ( 1 ) is satisfied and the following conditions hold: Then, (1) has at least one nontrivial homoclinic orbit.
Remark 5. Theorem 4 generalizes Theorem 1.2 in [13].It is easy to see that the condition ( 3 ) is stronger than our conditions ( 1 )-( 5 ).On the other hand, there are functions  satisfying Theorem 4 and not satisfying the corresponding result in [13].For example, let where Theorem 6. Assume that (A 1 ), (W 3 )-(W 5 ) are satisfied and the following conditions hold: Then, (1) has infinitely many homoclinic orbits.Remark 7. Theorem 6 generalizes Theorem 1.2 in [14].It is obvious that the condition ( 4 ) is stronger than our conditions ( 3 )-( 7 ).Besides, there are functions  satisfying our Theorem 6 and not satisfying the corresponding result in [14].For example, let where Remark 8. We should point out that there exist errors in the proofs of [13,14].On one hand, in Lemma 2.2, in [13], ‖  − ‖ 2 → 0 can not imply that ∑ ∞ =1 ‖  − ‖ 2 < +∞ since the convergence of the general term is only the necessary but not the sufficient condition for the convergence of the progression.On the other hand, the proof of Lemma 2.2 in [14] is not like the proof of Lemma 2.2 in [12] as the authors said.It is easy to see that  ∈  2 (R, R) plays an important role in the proof of Lemma 2.2 in [12], however, ,  in condition ( 4 ) do not belong to  2 (R, R).In the proofs of our results, we take some methods to avoid such mistakes.

Let
Then,  is a Hilbert space with the norm given by where   > 0. For  ∈ , let ISRN Mathematical Analysis 3 In the following, we always denote by   ( ∈ N) any suitable positive constant.
Therefore, for any  > 0, since proving the weak continuity of .Moreover,   is compact for  is weakly continuous (see [5]).
Lemma 11.Under the conditions of (W 3 ), (W 6 ), and (W 7 ), one has that  ∈  1 (, R) and that Proof.Since the proof is exactly similar to the proof of Lemma 10, we omit it here.
By Lemmas 10 and 11, it is routine to verify that any critical point of  on  is a classical solution of (1) with (±∞) = 0. Now, we state the critical point theorem used in [12].
Lemma 12 (see [18]).Let  be a real Banach space and let us have  ∈  1 (, R) satisfying the (PS) condition.If  is bounded from below, then is a critical value of .
Proof of Theorem 2. We divide our proof into three steps.
In order to prove Theorems 4 and 6, we need the Dual fountain theorem.For the reader's convenience, we recall it here.
Lemma 13 (see [19]).Let  be a Banach space with the norm ‖ ⋅ ‖ and  = ⨁ ∈   with dim   < +∞.For any  ∈ N, define Assume that ( 1 ) the compact group  acts isometrically on .The spaces   are invariant and there exists a finite dimensional space  such that, for every  ∈ N,   ≃  and the action of  on  is admissible.
Proof of Theorem 4. Let {  } ∈N be the completely orthogonal basis of  and   = R  .Then, dim   < +∞ and  = ⨁ ∈   .For any  ∈ N, define In the following, we will check that all conditions in Lemma 13 are satisfied and we divide our proof into several steps.
Proof of Theorem 6.Since the proof of Theorem 6 is exactly similar to the proof of Theorem 4, we omit it here.