Abstract and Applied Analysis

Volume 2014 (2014), Article ID 250870, 7 pages

http://dx.doi.org/10.1155/2014/250870

Research Article

Periodic Solutions for Second Order Hamiltonian Systems with Impulses via the Local Linking Theorem

Longsheng Bao and Binxiang Dai

School of Mathematics and Statistics, Central South University, Changsha, Hunan 410075, China

Received 19 April 2014; Accepted 29 June 2014; Published 10 July 2014

Academic Editor: Leszek Gasinski

Copyright © 2014 Longsheng Bao and Binxiang Dai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

A class of second order impulsive Hamiltonian systems are considered. By applying a local linking theorem, we establish the new criterion to guarantee that this impulsive Hamiltonian system has at least one nontrivial T-periodic solution under local superquadratic condition. This result generalizes and improves some existing results in the known literature.

1. Introduction and Main Results

Consider the second order Hamiltonian systems with impulsive effects

where , , , , where and denote the right and left limits of at , respectively,    are continuous, and , . is a symmetric constant matrix.

Impulsive differential equations serve as basic models to study the dynamics of processes that are subject to sudden changes in their states. The theory of impulsive differential systems has been developed by numerous mathematicians (see [1–4]). These kinds of processes naturally occur in control theory, biology, optimization theory, medicine, and so on (see [5–9]).

In recent years, many existence results are obtained for impulsive differential systems by critical point theory, such as [10–23] and their references. In most superquadratic cases, there is so-called Ambrosetti-Rabinowitz condition (see [18–23]):

where is a constant, which implies that is of superquadratic growth as ; that is,

Moreover, Wu and Zhang [24] study the homoclinic solutions without any periodicity assumption under the local Ambrosetti-Rabinowitz type condition. Two key conditions of the main results of [24] are listed as follows.

(A1)There exist and such that

(A2)There exists such that , uniformly in .

In recent paper [25], Zhang and Tang had obtained some results of the nontrivial T-periodic solutions under much weaker assumptions instead of (A1) and (A2).

(B1)There exist constants , , and and a function such that

(B2)There exists a subset of with such that

Remark 1. Condition (B2) is weaker than (A2) because condition (A2) implies

Recently, applying the local linking theorem (see [26]), the works in [27–30] obtained the existence of periodic solutions or homoclinic solutions with (3) superquadratic condition under different systems. As shown in [25], condition (B2) is a local superquadratic condition; this situation has been considered only by a few authors.

Motivated by papers [24, 25, 31], in this paper, we aim to consider problem (1) under local superquadratic condition via a version of the local linking theorem (see [26]). In particular, the impulsive function satisfies a kind of new superquadratic condition which is different from that in the known literature. Our main results are the following theorems.

Theorem 2. Suppose that and ,  , satisfies (B2) and consider the following.

(H1)There exists a positive constant such that

(H2) uniformly for .(H3)There exist constants , and such that, for every and with ,

(H4)There exist constants , and such that, for every and with ,

(I1)There exist constants and such that

(I2)There are two constants and such that

(I3) satisfies , for all .

Then problem (1) has at least one nontrivial T-periodic solution.

Remark 3. Noting (3), obviously, conditions (B2) and (H4) are weaker than those of (2). From (B2), we only need to hold in a subset of . What is more, in (2) is asked to be positive globally. Here need not be nonnegative globally; we also generalized Theorems 1.3 and 1.4 in [25]. For example, let

where

Let ; then satisfies our Theorem 2 but does not satisfy (2) and (3) and does not satisfy the corresponding conditions in [25].

Theorem 4. Suppose that and ,   , satisfies (H1), (H2), (H3), (I1), (I2), and (I3) and the following condition holds.

(H5)There exist constants and such that

 where

Then problem (1) has at least one nontrivial T-periodic solution.

Theorem 5. Suppose that and ,   , satisfies (B2), (H1), (H2), (H3), (I1), (I2), and (I3) and the following condition holds.

(H6)There exist constants , , and and a function and such that, for every and with ,

Then problem (1) has at least one nontrivial T-periodic solution.

The remaining of this paper is organized as follows. In Section 2, some fundamental facts are given. In Section 3, the main results of this paper are presented.

2. Preliminaries

Let be a real Banach space with direct sum decomposition . Consider two sequences of subspaces , such that . For every multi-index , let  ; we define . A sequence is admissible if for every there is such that . For every , we define by the function restricted to .

Definition 6 (see [26]). Let . The function satisfies the condition if every sequence , such that is admissible and

possesses a subsequence which converges to a critical point of .

Definition 7 (see [26]). Let be a Banach space with direct sum decomposition . The function has local linking at with respect to , if there exists such that

Theorem 8 (see [26]). Suppose that satisfies the following assumptions:

(A1) has local linking at 0 and ,(A2) satisfies condition,(A3) maps bounded sets into bounded sets,(A4)for every and , as .

Then has at least three critical points.

Let us recall some basic notation. In the Sobolev space , consider the inner product

for any . The corresponding norm is defined by

for any . Moreover, it is well known that is compactly embedded in , which implies that

for some constant , where .

Define the corresponding functional on by

By the conditions of and , , we get that functional is a continuously Gáteaux differential functional whose Gáteaux derivative is the functional , given by

If , then is absolutely continuous and . In this case, may not hold for some ; this leads to impulsive effects.

Following the ideas of [11, 12], we can prove that the critical points of are the weak solutions of problem (1).

To prove our main results, we have the following facts (see [32]).

Letting

we see that

where is the liner self-adjoint operator defined and is the identity matrix. By the Riesz representation theorem, we have

The compact imbedding of into implies that is compact. Summing up the above discussion, can be rewritten as

By classical spectral theory, we can decompose into the orthogonal sum of invariant subspace for

where and and are such that, for some ,

Notice that is finite dimensional.

In this paper, we set .

3. Proof of Main Results

3.1. The Proof of Theorem 2

Let and ; then . Suppose is an orthogonal basis of . Correspondingly, let

then , . We divide our proof into four steps.

Step 1. has local linking at .

In view of (I1), we obtain

Combining this inequality, we have

Since , this implies

Applying (H2) and (33), for any , there exists such that

which implies that

On one hand, by (34) and (35), for all with . Choose ; then one has

On the other hand, since is finite, there exists a constant such that

For all with . Choose ; by (34), (35), and (37), we obtain

Let ; one has

Step 2. maps bounded sets into bounded sets.

By (H3) and , there exists such that

which implies that

Note that

It implies that maps bounded sets into bounded sets.

Step 3. satisfies the condition.

Consider a sequence such that is admissible. Then there exists a constant such that

By (41), for , one has

together with (H4), one has

for all .

It follows from (21), (43), (45), (H1), and (I2) that

Since , , and , (46) shows that is bounded in . By a method similar to that of [10], we can prove that has a convergent subsequence. Thus satisfies condition.

Step 4. For every and , as .

Since is finite, there exists such that

In fact, for , by (B2), there exists such that

Hence, it follows from (47), (48), and (I3) that

for and a.e. , which implies that

Therefore, all the assumptions of Theorem 8 are verified. Then, the proof of Theorem 2 is completed.

3.2. The Proof of Theorem 4

Following the same procedures in the proof of Theorem 2, we can prove that satisfies (A1), (A2), and (A3) in Theorem 8.

To prove (A4), set , ; then by (H5), we have

When , it follows from (51) that

this implies

Set ; then by (53), we have

; since   is finite, there exists such that

By (54), (55), and (I3), we have

Since and , (56) implies

Consequently, the conclusion follows from Theorem 8. This completes the proof.

3.3. The Proof of Theorem 5

Similar to the proof of Theorem 2, satisfies all conditions of Theorem 8. Thus, problem (1) has at least one nontrivial T-periodic solution.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is supported by the National Natural Science Foundation of China (no. 11271371).

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