Existence and Multiplicity of Weak Quasi-periodic Solutions for Second Order Hamiltonian System with a Forcing Term

In this paper, we first obtain three inequalities and two of them, in some sense, generalize Sobolev's inequality and Wirtinger's inequality from periodic case to quasi-periodic case, respectively. Then by using the least action principle and the saddle point theorem, under subquadratic case, we obtain two existence results of weak quasi-periodic solutions for the second order Hamiltonian system: d[P(t) ˙ u(t)] dt = ∇F(t, u(t)) + e(t), which generalize and improve the corresponding results in recent literature [J. Kuang, F(t, −x) and e(t) ≡ 0 are also made, we obtain two results on existence of infinitely many weak quasi-periodic solutions for the second order Hamiltonian system under the subquadratic case.


Introduction and main results
In this paper, we are concerned with the existence and multiplicity of weak-quasi periodic solutions for the second order Hamiltonian system: where u(t) = (u 1 (t), . . ., u N (t)) τ , N > 1 is an integer, F ∈ C 1 (R × R N , R), ∇F(t, x) = (∂F/∂x 1 , . . ., ∂F/∂x N ) τ , P(t) = (p ij (t)) N×N is a symmetric and continuous N × N matrixvalue functions on R, e : R → R N , (•) τ stands for the transpose of a vector or a matrix.
Definition 1.1 ([8]).A function f (t) is said to be Bohr almost periodic, if for any ε > 0, there is a constant l ε > 0, such that in any interval of length l ε , there exists τ such that the inequality | f (t + τ) − f (t)| < ε is satisfied for all t ∈ R.

Definition 1.2 ([9]
).A function f ∈ C 0 (R × R m , R N ) is called almost periodic in t uniformly for x ∈ R m when, for each compact subset K in R m , for each ε > 0, there exists l > 0, and for each α ∈ R, there exists τ ∈ [α, α + l] such that Let p > 1 be a positive integer and {T j } p j=1 be rationally independent positive real constants.Define where Λ j = 2mπ T j m ∈ Z .To be precise, in [13], Kuang obtained the following results.

Theorem 1.3 ([13, Theorem 2.3]
). Suppose F satisfies the following conditions: Then (1.1) with P(t) ≡ I N×N and e(t) ≡ 0 has at least a quasi periodic solution, where the definition of V can be seen in Section 2 below.Theorem 1.4 ([13, Theorem 2.4]).Suppose that F satisfies ( f 1 )-( f 4 ) and Weak quasi-periodic solutions for Hamiltonian system Then (1.1) with P(t) ≡ I N×N and e(t) ≡ 0 has at least one quasi-periodic solution by saddle point theorem.
Obviously, ( f 4 ) implies that |∇F| is bounded, which makes lots of functions eliminated.For example, a simple function which does not satisfy ( f 4 ).However, in this paper, we obtain that system (1.1) still has quasi-periodic solution for such potential F like (1.3).To be precise, in this paper, inspired by [10,13,15,24,28,32], we obtain the following results.
(I) Existence of weak quasi-periodic solution By using the least action principle and the saddle point theorem, we obtain that system (1.1) has at least one weak quasi-periodic solution.
Theorem 1.6.Suppose that (P ), (E ), (W ), ( f 1 )-( f 3 ) and ( f 4 ) hold.If  [15] because of the presence of (W) and ( f 4 ) .(W) and ( f 4 ) were given by Wang and Zhang in [28], which present some advantages compared to the well known condition: there exist g, h Finally, one can also compare Theorem 1.5 and Theorem 1.6 with the corresponding results in [32], in which, Zhang and Tang investigated the existence of T-periodic solution under (W) and the following condition: there exist g ∈ where g ∈ L 2 ([0, T]; R + ) is demanded from proofs of their theorems.In our Theorem 1.5 and Theorem 1.6, when P(t) ≡ I N×N , e(t) ≡ 0, V only contains a frequency 2π/T and F(t, x) is periodic in t with period T, we only demand that g ∈ L 1 ([0, T]; R + ).Hence, our results are different from those in [32].
(II) Multiplicity of weak quasi-periodic solutions Moreover, by using a critical point theorem due to Ding in [6], we obtain the following multiplicity results.Theorem 1.8.Suppose that (P ), (W ), ( then system (1.1) has infinitely many weak quasi-periodic solutions.

Preliminaries
In this section, we need to make some preliminaries.Some knowledge and statements below come from [3,4,8,9,13] .
Then the mean value of f is the limit (when it exists) lim A fundamental property of almost periodic functions is that such functions have convergent means, that is, the limit lim The elements of these spaces B p (R N ) are called Besicovitch almost periodic functions. For then we say that u, v belong to a class of equivalence.We will identify the equivalence class u with its continuous representant ) is a Hilbert space with its norm • 2 and the inner product endowed with the inner product and the corresponding norm Inspired by [13] and [16], we present the following two lemmas: Combining (1.2), we obtain that By Parseval's equality, we have (2.5) Weak quasi-periodic solutions for Hamiltonian system Then by [14, Theorem 3.5-2], we have and Hence, (2.11) holds.

Remark 2.3.
A version of Lemma 2.1 and (2.10) has been given in [13] (see [13, Lemma 3.1 and Lemma 3.3]), where the author obtained that there exists a constant C > 0 such that and when (2.9) holds, However, the value of C are not given.Our Lemma 2.1 and Lemma 2.2 present the value of C, which will play an important role in our main results and their proofs.Moreover, we also present the inequality (2.11).One can compare (2.10) and (2.11) with Sobolev's inequality and Wirtinger's inequality in [16] which investigate periodic functions u ∈ W 1,2 T .It is easy to see that when V only contains a frequency 2π/T, (2.11) reduces to Wirtinger's inequality.

Existence
In this section, we will use the least action principle (see [16,Theorem 1.1]) to prove Theorem 1.5 and use the saddle point theorem (see [19]) to prove Theorem 1. 6.
Then V = Ṽ ⊕ V.For u ∈ V, u can be written as u = ū + ũ, where It is easy to obtain that lim Then ũ ∈ Ṽ.For the sake of convenience, we denote Proof of Theorem 1.5.Since V is a Hilbert space, then V is reflexive.Note that P is positive definite.Then , (3.1) and (W)(iv) imply that Then by the least action principle (see [16, Theorem 1.1]), we know that ϕ has at least one critical point u * which minimizes ϕ.Thus we complete the proof.
Proof of Theorem 1.6.It follows from ( f 5 ) that there exists At first, we prove that ϕ satisfies (PS) condition.Assume that Similar to the argument of (3.2), we have lim
Proof.By (2.8) and (2.9), it is obvious that (2.10) holds.Moreover, by (2.5) and (2.9), we have [13]f.The proof with P(t) ≡ I N×N and e(t) ≡ 0 can be seen in[13, Theorem 2.1].With the aid of the conditions (P) and (E ), it is easy to see that the proof is the essentially same as Theorem 2.1 of[13].So we omit the details.we refer readers to Theorem 2.1 and its proof in[13].
Definition 2.6.When u satisfies (2.14), we say that u is a weak solution of system (1.1).