#### Abstract

We study the existence of periodic solutions of some second-order Hamiltonian systems with impulses. We obtain some new existence theorems by variational methods.

#### 1. Introduction

Consider the following systems: where with for each , there exists an such that , and we suppose that satisfies the following assumption. is measurable in for and continuously differentiable in for a.e. , and there exist such that for all and .

Many solvability conditions for problem (1.1) without impulsive effect are obtained, such as, the coercivity condition, the convexity conditions (see [1–4] and their references), the sublinear nonlinearity conditions, and the superlinear potential conditions. Recently, by using variational methods, many authors studied the existence of solutions of some second-order differential equations with impulses. More precisely, Nieto in [5, 6] considers linear conditions, [7–10] the sublinear conditions, and [11–16] the sublinear conditions and the other conditions. But to the best of our knowledge, except [7] there is no result about convexity conditions with impulsive effects. By using different techniques, we obtain different results from [7].

We recall some basic facts which will be used in the proofs of our main results. Let with the inner product where denotes the inner product in . The corresponding norm is defined by

The space has some important properties. For , let , and . Then one has Sobolev's inequality (see Proposition in [1]):

Consider the corresponding functional on given by

It follows from assumption and the continuity of one has that is continuously differentiable and weakly lower semicontinuous on . Moreover, we have for and is weakly continuous and the weak solutions of problem (1.1) correspond to the critical points of (see [8]).

Theorem 1.1 ([2, Theorem 1.1]). * Suppose that and are reflexive Banach spaces, , is weakly upper semi-continuous for all , and is convex for all and is weakly continuous. Assume that
**
as and for every ,
**
as uniformly for . Then has at least one critical point. *

#### 2. Main Results

Theorem 2.1. *Assume that assumption holds. If further** is convex for a.e. , and** there exist , such that , for all , then (1.1) possesses at least one solution in .*

*Remark 2.2. * implies there exists a point for which

* Proof of Theorem 2.1. *It follows Remark 2.2, (1.6), and that
for all and some positive constant . As if and only if , we have as . By Theorem 1.1 and Corollary 1.1 in [1], has a minimum point in , which is a critical point of . Hence, problem (1.1) has at least one weak solution.

Theorem 2.3. *Assume that assumption and hold. If further** there exist , and such that for all and ** there exist some and such that
**for and , where is a constant, then (1.1) possesses at least one solution in . *

*Remark 2.4. *We can find that our condition is very different from condition (vii) in [7] since we prove this by the saddle point theorem substituted for the least action principle.

* Proof of Theorem 2.3. *We prove satisfies the (PS) condition at first. Suppose is such an sequence that is bounded and . We will prove it has a convergent subsequence. By and (1.6), we have
for some positive constants , , . By Remark 2.2, (1.6), and (2.4), we have
for some positive constants , , which implies that
for some positive constants , . By (1.6), the above inequality implies that
for the positive constant . The one has
If is unbounded, we may assume that, going to a subsequence if necessary,
By (2.8) and (2.9), we have
for all large and every . By (2.10) and , one has for all large . It follows from , (2.4), (2.6), (2.7), and above inequality that
for large and the positive constant , which contradicts the boundedness of since . Hence is bounded. Furthermore, is bounded by (2.6). A similar calculation to Lemma 3.1 in [9] shows that satisfies the (PS) condition. We now prove that satisfies the other conditions of the saddle point theorem. Assume that , then . From above calculation, one has
for all , which implies that
as in . Moreover, by and we have
for , which implies that
as in since . Now Theorem 2.3 is proved by (2.13), (2.15), and the saddle point theorem.

Theorem 2.5. *Assume that assumption holds. Suppose that are concave and satisfy ** for some positive constant , then (1.1) possesses at least one solution in . *

* Proof of Theorem 2.5. *Consider the corresponding functional on given by
which is continuously differentiable, bounded, and weakly upper semi-continuous on . Similar to the proof of Lemma 3.1 in [2], one has that is convex in for every . By the condition, we have . Similar to the proof of Theorem 3.1, we have
which means as , uniformly for with by and (1.6). On the other hand,
which implies that as by and (1.6). We complete our proof by Theorem 1.1.

#### Acknowledgment

The first author was supported by the Postgraduate Research and Innovation Project of Hunan Province (CX2011B078).