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On homoclinic orbits for a class of damped vibration systems
Advances in Difference Equations volume 2012, Article number: 102 (2012)
In this article, we establish the new result on homoclinic orbits for a class of damped vibration systems. Some recent results in the literature are generalized and significantly improved.
Introduction and main results
Consider the following second-order damped vibration problems
where , B is an antisymmetric constant matrix, is a symmetric matrix valued function and . As usual we say that a solution u of (VS) is homoclinic (to 0) if , , , and as .
When B is a zero matrix, (VS) is just the following second-order Hamiltonian systems (HSs)
Inspired by the excellent monographs and works [1–3], by now, the existence and multiplicity of periodic and homoclinic solutions for HSs have extensively been investigated in many articles via variational methods, see [4–22]. Also second-order HSs with impulses via variational methods have recently been considered in [23–26]. More precisely, in 1990, Rabinowitz  established the existence result on homoclinic orbit for the periodic second-order HS. It is well known that the periodicity is used to control the lack of compactness due to the fact that HS is set on all .
For the nonperiodic case, the problem is quite different from the one described in nature. Rabinowitz and Tanaka  introduced a type of coercive condition on the matrix L:
() , as .
They established a compactness lemma under the nonperiodic case and obtained the existence of homoclinic orbit for the nonperiodic system (HS) under the usual Ambrosetti-Rabinowitz (AR) growth condition
where is a constant. Later, Ding  strengthened condition () by
() there exists a constant such that
Under the condition () and some subquadratic conditions on , Ding proved the existence and multiplicity of homoclinic orbits for the system (HS). From then on, the condition () or () are extensively used in many articles.
Compared with the case where B is a zero matrix, the case where , i.e., the nonperiodic system (VS), has been considered only by a few authors, see [27–29]. Zhang and Yuan  studied the existence of homoclinic orbits for the nonperiodic system (VS) when W satisfies the subquadratic condition at infinity. Soon after, Wu and Zhang  obtained the existence and multiplicity of homoclinic orbits for the nonperiodic system (VS) when W satisfies the local (AR) growth condition
where and are two constants. It is worth noticing that the matrix L is required to satisfy the condition () in the above two articles.
() there exists a constant such that
() there exists a constant such that
which are first used in . By using a recent critical point theorem, we prove that the nonperiodic system (VS) has at least one homoclinic orbit when W satisfies weak superquadratic at the infinity, which improve and extend the results of [27, 28].
Remark 1 In fact, there are some matrix-valued functions satisfying () and (), but not satisfying () or (). For example,
We consider the following conditions:
() , and there exist positive constants and such that
() as uniformly in t.
() if , and
for any .
() There exist and such that if .
Theorem 2 Assume that ()-() and ()-() hold. Then the system (VS) has at least one homoclinic orbit.
Remark 3 To see that our result generalizes  we present the following examples. These functions satisfy the weak superquadratic conditions ()-(), but not verify the growth condition (1).
where , and , .
In fact it is easy to verify that ()-() are satisfied. However, similar to the discussion of Remark 1.2 in , let , where . Then for any , one has
That is, the condition (1) is not satisfied for any .
This article is organized as follows. In the following section, we formulate the variational setting and recall a critical point theorem required. In section ‘Linking structure’, we discuss linking structure of the functional. In section ‘The -sequence’, we study the Cerami condition of the functional and give the proof of Theorem 2.
Notation Throughout the article, we shall denote by various positive constants which may vary from line to line and are not essential to the problem.
In this section, we establish a variational setting for the system (VS). Let H be which is a Hilbert space with the inner product and norm given by
for , where denotes the inner product in . It is well known that H is continuously embedded in for . Define an operator by
for all . Since B is an antisymmetric constant matrix, J is self-adjoint on H. Moreover, we denote by A the self-adjoint extension of the operator with the domain . Let be the usual -norm, and the usual -inner product. Set , the domain of . Define on E the inner product
and the norm
Then E is a Hilbert space and it is easy to verify that E is continuously embedded in . Using a similar proof of Lemma 3.1 in , we can prove the following lemma.
Lemma 4 Suppose thatsatisfies () and (), then E is compactly embedded intofor.
By Lemma 4, it is easy to prove that the spectrum has a sequence of eigenvalues (counted with their multiplicities)
with as , and corresponding eigenfunctions , , form an orthogonal basis in . Assume , and let , , and . Then
is an orthogonal decomposition of E. We introduce on E the following product
and the norm
where , . Then and are equivalent (see ). So by Lemma 4, we see that there exists a constant such that
Define the functional Φ on E by
where . Furthermore, define
From the assumptions it follows that Φ is defined on the Banach space E and belongs to . A standard argument shows that critical points of Φ are solutions of the system (VS). Moreover, it is easy to verify that if is a solution of (VS), then and , as (see Lemma 3.1 in ).
In order to study the critical points of Φ, we now recall a critical point theorem, see .
Let E be a Banach space. A sequence is said to be a -sequence if
Φ is said to satisfy the -condition if any -sequence has a convergent subsequence.
Theorem 5 ()
Suppose, , where, there exist, andsuch thatand, whereis the sphere of radius ρ and center 0, and
Moreover, if Φ satisfies the-condition for all, then Φ has a critical value in.
First we discuss the linking structure of Φ. By condition (), one has
for all . Observe that if () holds, and together with (4), then if , one has
Let . Then we have
Remark that and (5) imply that, for any , there is such that
for all .
Lemma 6 Let ()-() be satisfied, and assume further thatholds. Then there existssuch that, where.
Proof By (7) we have
for all , the lemma follows from the form of Φ (see (3)). □
Then is a finite subspace.
Lemma 7 Under the assumptions of Theorem 2, there existssuch thatfor allwith.
Proof It suffices to show that in as . For any , let , where , , . Since , then
where . Thus , and together with (4), we obtain
which shows that as . □
As a special case we have
Lemma 8 Assume that the assumptions of Theorem 2 are satisfied. Then, lettingwith, there issuch thatwhereand κ is given by Lemma 6.
In this section, we discuss the -sequence of Φ.
Lemma 9 Let ()-() and ()-() hold. Then any-sequence is bounded.
Proof Let be such that
Then, for ,
Suppose to the contrary that is unbounded. Setting , then , for all . Passing to subsequence, in E, and in for .
From (10), we obtain
Set for ,
By () and (), for all , and as .
For , let
Then by () one has and
It follows from (8) and (12) that
Using (13) we obtain
as uniformly in j, and for any fixed ,
as . It follows from (14) that, for any ,
as uniformly in j.
Let . By () there is such that
for all . Consequently,
for all j.
Set . By (), (16) and Hölder inequality, we can take large enough such that
for all j. Note that there is independent of j such that for . By (15) there is such that
for all . By (17)-(19), one has
which contradicts with (10). The proof is complete. □
Lemma 10 Under the assumptions of Theorem 2, Ψ is nonnegative, weakly sequentially lower semi-continuous, andis weakly sequentially continuous. Moreover, is compact.
Proof We follow the idea of . Clearly, by assumptions, . Let in E. By Lemma 10, in for , and a.e. . Hence for a.e. . Thus, it follows from Fatou’s lemma that
which shows that the function Ψ is weakly sequentially lower semi-continuous.
Now we show that is compact. It is clear that, for any ,
Since is dense in E, for any , we take such that
By (6), one has
For any , fix n so that . By (21) there exists such that
Then for all , which proves the weakly sequentially continuity. Therefore, is compact by the weakly continuity of since E is a Hilbert space. □
Lemma 10 implies that is weakly sequentially continuous, i.e., if in E, then . Let be an arbitrary -sequence, by Lemma 9, it is bounded, up to a subsequence, we may assume in E. Plainly, u is a critical point of Φ.
Lemma 11 Under the assumptions of Lemma 9, Φ satisfies-condition.
Proof Let be any -sequence. By Lemmas 4, 9, and 10, one has
So as . Since , we have , and therefore as in E. □
Proof of the theorem
Proof of Theorem 2 Lemma 8 shows that Φ possesses the linking structure of Theorem 5, and Lemma 11 implies that Φ satisfies the -condition. Therefore, by Theorem 5 Φ has at least one critical point u. □
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The research of J. J. Nieto was partially supported by the Ministerio de Economía y Competitividad and FEDER, project MTM2010-15314. The research of J. Sun was supported by the National Natural Science Foundation of China (Grant No. 11201270, 11271372), Shandong Natural Science Foundation (Grant No. ZR2012AQ010).
The authors declare that they have no competing interests.
Each of the authors, JS, JN and MO contributed to each part of this study equally and read and approved the final version of the manuscript.