Existence of solutions for a class of second-order sublinear and linear Hamiltonian systems with impulsive effects ∗

Existence of solutions for a class of second-order sublinear and linear Hamiltonian systems with impulsive effects ∗ Xiaofei He †1,2 and Peng Chen 3 1 Department of Mathematics and Computer Science, Jishou University, Jishou, Hunan 416000, P.R. China 2 Zhangjiajie College of Jishou University, Zhangjiajie 427000, P.R. China 3 School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan 410083, P.R.China

When α = 1, condition (1.2) reduces to the linearly bounded gradient condition, in this case, Zhao and Wu [21,22] also proved the existence of solutions for problem (1.1) under the condition and (1.3) or (1.4) with α = 1. For ) is an impulsive differential problem.Impulsive differential equations arising from the real world describe the dynamics of processes in which sudden, discontinuous jumps occur.For the background, theory and applications of impulsive differential equations, we refer the readers to the monographs and some recent contributions as [1,3,4,13,20].Some classical tools such as fixed point theorems in cones [1,5,19], the method of lower and upper solutions [3,23] have been widely used to study impulsive differential equations.
Recently, the Dirichlet and periodic boundary conditions problems with impulses in the derivative are studied by variational method.For some general and recent works on the theory of critical point theory and variational methods, we refer the readers to [10,14,19,27,28].It is a novel approach to apply variational methods to the impulsive boundary value problem (IBVP for short).
In the present paper, motivated by the above papers [15,21,22,28], we study the existence of solutions for problem (1.1) under the condition (1.2).We will use the saddle point theorem in critical theory to generalize some results in [28].In fact, we will establish some new existence criteria to guarantee that system (1.1) has at least one solutions under more relaxed assumptions on F (t, x), which are independent from (1.3) and more general than (1.4) in [17] and [28], to our best knowledge, it seems not to have been considered in the literature.

Preliminaries
In this section, we recall some basic facts which will be used in the proofs of our main results.In order to apply the critical point theory, we construct a variational structure.With this variational structure, we can reduce the problem of finding solutions of (1.1) to that of seeking the critical points of a corresponding functional.
Let H 1 T be the Sobolev space it is a Hilbert space with the inner product the corresponding norm is defined by Definition 2.1. [28]We say that a function u ∈ H 1 T is a weak solution of problem (1.1) if the identity holds for any v ∈ H 1 T .The corresponding functional ϕ on H 1 T given by where By Definition 2.1, the weak solutions of problem (1.1) correspond to the critical points of ϕ.
To prove our main results, we need the following definition and lemma.
Definition 2.2. [8]Let X be a real Banach space and I ∈ C 1 (X, R).I is said to satisfy the (PS) condition on X if any sequence {x n } ⊆ X for which I(x n ) is bounded and Lemma 2.1. [8]For and

Main results and Proofs
Theorem 3.1.Suppose that (A) and (1.2) hold, and the following conditions are satisfied: Then problem (1.1) has at least one weak solution in H 1 T .
Throughout this paper, for the sake of convenience, we denote 3 denote the positive number and fix Let , when δ 1 , δ 2 , δ 3 are small enough, it is easy to see that G(δ 1 , δ 2 , δ 3 ) is monotone increasing for every variable.
Furthermore, we have lim , 3 ) is monotone increasing for every variable.Furthermore, we have lim Now, we can prove our results.
Proof of Theorem 3.1.First, we prove that ϕ satisfies the (PS) condition.Suppose that {u n } ⊂ H 1 T is a (PS) sequence of ϕ, that is {ϕ(u n )} is bounded and ϕ ′ (u n ) → 0 as n → ∞ .By (F1), we can choose an which means that where M 3 is a positive constant dependent of the arbitrary positive number δ 1 which satisfies (3.9).
By (I1) and Lemma 2.1, we have for all u n , where M 4 is a positive constant dependent of the arbitrary positive number δ 2 which satisfies (3.9).
Since lim n→∞ ϕ ′ (x n ) = 0, we have by (3.13) and (3.14) On the other hand, by (2.4), we have where M 5 is a positive constant dependent of the arbitrary positive number δ 3 which satisfies (3.9).

Let H1
In order to use the saddle point theorem ( [12], Theorem 4.6), we only need to verify the following conditions: In fact, by (F1), we get From (I2) and (3.22), we have Thus, (A 1 ) is verified.
Next, for all u ∈ H1 T , by (1.2) and Sobolev's inequality, we have T from (3.25), i.e. (A 2 ) is verified.The proof of Theorem 3.1 is complete.
Similar to the proof of Theorem 3.1, we only need to verify (A 1 ) and (A 2 ).It is easy to verify (A 1 ) by (3.6).In what follows, we verify that (A 2 ) also holds .For all u ∈ H1 T , by (3.4) and Sobolev's inequality, we have