- Open Access
On homoclinic orbits for a class of damped vibration systems
© Sun et al.; licensee Springer. 2012
- Received: 12 April 2012
- Accepted: 8 June 2012
- Published: 5 July 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.
- homoclinic orbits
- second-order systems
- damped vibration problems
- variational methods
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 .
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 .
where is a constant. Later, Ding  strengthened condition () by
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.
where and are two constants. It is worth noticing that the matrix L is required to satisfy the condition () in the above two articles.
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].
We consider the following conditions:
() as uniformly in t.
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 , .
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.
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.
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 .
Φ is said to satisfy the -condition if any -sequence has a convergent subsequence.
Theorem 5 ()
Moreover, if Φ satisfies the-condition for all, then Φ has a critical value in.
for all .
Lemma 6 Let ()-() be satisfied, and assume further thatholds. Then there existssuch that, where.
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.
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.
Suppose to the contrary that is unbounded. Setting , then , for all . Passing to subsequence, in E, and in for .
By () and (), for all , and as .
as uniformly in j.
for all j.
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.
which shows that the function Ψ is weakly sequentially lower semi-continuous.
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.
So as . Since , we have , and therefore as in E. □
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. □
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).
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