Infinitely many solutions for a class of second-order damped

In this paper, by using the variational approach, we study the existence of infinitely many solutions for a class of second-order damped vibration systems under super- quadratic and sub-quadratic conditions. Some new results are established and some recent results in the literature are generalized and significantly improved.

In this paper, we will investigate system (1.1) which is the extension of system (1.2) and system (1.3) and under more general super-quadratic conditions than those in [1], we obtain that system (1.1) has infinitely many solutions.Moreover, we also obtain a new result under sub-quadratic conditions.Next, we state our results.
(I) For super-quadratic case Theorem 1.1.Assume the following conditions hold: (P) there exists a constant m > 1 2 such that the matrix P (t) satisfies (H4) F (t, x) is even in x and F (t, 0) ≡ 0.
Then system (1.1) has an unbounded sequence of solutions.
When the condition F (t, 0) ≡ 0 is deleted, we have the following result: Theorem 1.2.Assume that (P) and (H1)-(H3) hold and F (t, x) is even in x.Then system (1.1) has an infinite sequence of distinct solutions.
By Theorem 1.1 and Theorem 1.2, we can obtain the following corollaries.
Corollary 1.2.Assume that (P), (H1) and (AR)-condtion hold and F (t, x) is even in x.Then system (1.1) has an unbounded sequence of solutions.
Remark 1.1.It is remarkable that in [4], the following condition which is similar to (H3) has been presented: ( Ŝ2 ) there exist p > 2, c 1 , c 2 , c 3 > 0 and ν ∈ (0, 2) such that, for all |z| ≥ r 1 , which was used to consider the existence of homoclinic solutions for the first order Hamiltonian system ż = J∇H(t, z).In [5], the author and Tang investigated the existence of periodic and subharmonic solutions for the second order Hamiltonian system ü(t) + Au(t) + ∇F (t, u(t)) = 0, a.e.t ∈ R (1.4) and presented that the conditions like in ( Ŝ2 ) are not necessary when one considered the existence of periodic solutions for system (1.4) and where Then system (1.1) has infinitely many nontrivial solutions.

Preliminaries
In this section, we will present the variational structure of system (1.1), which is the slight modification of those in [1].Let ) is a Hilbert space.It follows from assumption (A) and Theorem 1.4 in [6] that the functional ϕ on H 1 T given by is continuously differentiable and is also a norm on H 1 T .Obviously, if the condition (P ) holds, u H 1 T and u are equivalent.Moreover, there exists C 0 > 0 such that [6]).Hence, there exists C * > 0 such that for all v ∈ H 1 T .By the Fundamental Lemma and Remarks in page 6-7 of [6], we have Then by (2.3), we obtain that e Q(t) P (t) u0 is completely continuous on [0, T ] and Note that T 0 q(t)dt = 0. Then by (2.3) and (2.4), it is easy to see that P (0) u(0) = P (T ) u0 (T ).Therefore, u 0 is a solution of system (1.1).This completes the proof.
By the Riesz theorem, define the operator K : for all u, v ∈ H 1 T .Then K is a bounded self-adjoint linear operator (see [1]).By the definition of K, the functional ϕ can be written as By the classical spectral theory, we have the decomposition: , where H 0 = ker(I − K) and H 0 , H − are finite dimensional.Moreover, by the spectral theory, there is a δ > 0 such that (see [1]).

The super-quadratic case
Similar to the proofs in [1], we will also use symmetric mountain pass theorem (see Theorem 9.12 in [2]) to prove Theorem 1.1 and use an abstract critical point theorem due to Bartsch and Ding (see [3]) to prove Theorem 1.2.
Remark 3.1.As shown in [7], a deformation lemma can be proved with replacing the usual (PS)-condition with (C)-condition, and it turns out that symmetric mountain pass theorem in EJQTDE, 2013 No. 15, p. 7 [2] are true under (C)-condition.We say that ϕ satisfies (C)-condition, i.e. for every sequence Proof of Theorem 1.1.
Then, obviously, {u n } is also unbounded.
Proof of Theorem 1.2.Similar to the proofs of Theorem 3.2 in [1], by combining the proofs of Theorem 1.1 and the abstract critical point theorem due to Bartsch and Ding (see [3]), the proof is easy to be completed and so we omit the details.
Proofs of Corollary 1.1 and Corollary 1.2.It is easy to see that (AR)-condition implies that (H2) and (H3).So by Theorem 1.1 and Theorem 1.2, the proofs are easy to be completed.we know that assumption (A), (H2) and (H3) ′ imply (H3).Then by Theorem 1.1 and Theorem 1.2, the proofs are easy to be completed.

The sub-quadratic case
In this section, we will investigate the subquadratic case.The following abstract critical point theorem will be used to prove Theorem 1.3.Lemma 4.1.(seeLemma 2.4 in [8]) Let E be an infinite dimensional Banach space and let f ∈ C 1 (E, R) be even, satisfy (PS), and f (0  Proof of Theorem 1.3.We will consider the functional Then it is easy to see that the critical point of φ is still the solution of system (1.1).
In fact, since Ẽ is finite dimensional, all norms on Ẽ are equivalent.Hence there exist