Abstract and Applied Analysis

Volume 2014 (2014), Article ID 960276, 12 pages

http://dx.doi.org/10.1155/2014/960276

Research Article

Existence and Multiplicity of Fast Homoclinic Solutions for a Class of Damped Vibration Problems with Impulsive Effects

Qiongfen Zhang

College of Science, Guilin University of Technology, Guilin, Guangxi 541004, China

Received 11 April 2014; Accepted 26 May 2014; Published 15 June 2014

Academic Editor: Jianqing Chen

Copyright © 2014 Qiongfen Zhang. 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.

This paper is concerned with the existence and multiplicity of fast homoclinic solutions for a class of damped vibration problems with impulsive effects. Some new results are obtained under more relaxed conditions by using Mountain Pass Theorem and Symmetric Mountain Pass Theorem in critical point theory. The results obtained in this paper generalize and improve some existing works in the literature.

1. Introduction

Consider fast homoclinic solutions of the following problem:

where is of class , , , , as , denotes the sets of integers, and are impulsive points. Moreover, there exist a positive integer and a positive constant such that , , , , and represent the right and left limits of at , respectively, is a continuous function, and with

When , problem (1) becomes the following damped vibration problem:

Chen et al. [1] investigated problem (3) and obtained some results of fast homoclinic solutions by critical point theory.

When and , problem (1) becomes the following problem:

Fang and Duan [2] obtained the following result of homoclinic solutions for (4) by employing Mountain Pass Theorem, a weak convergence argument, and a weak version of Lieb’s methods.

Theorem A (see [2]). Assume that the following conditions hold:

(V1)there exists a positive number such that

(V2) uniformly for ;(V3)there exists a constant such that

(V4)there exist constants and such that

(I)there exists a constant with such that

Then, problem (4) possesses a nontrivial weak homoclinic orbit.

For , problem (1) involves impulsive effects. Impulsive differential equations are suitable for the mathematical simulation of evolutionary processes in which the parameters undergo relatively long periods of smooth variation followed by a short-term rapid change (that is jumps) in their values. Impulsive differential equations are often investigated in various fields of science and technology, for example, many biological phenomena involving thresholds, bursting rhythm models in medicine and biology, optimal control models in economics, pharmacokinetics, and frequency modulated systems, and so on. For more details of impulsive differential equations, we refer the readers to the books [3, 4].

In recent years, some researchers have paid attention to the existence and multiplicity of solutions for impulsive differential equations via variational methods. See, for example, [1, 5–17] and references therein. However, there are few papers [2, 18, 19] concerning homoclinic solutions of impulsive differential equations by variational methods. So, it is a novel method to employ variational methods to investigate the existence of homoclinic solutions for impulsive differential equations.

Motivated by the above papers, we will establish some new results for (1). In order to introduce the concept of fast homoclinic solutions for (1), we first state some properties of the weighted Sobolev space on which certain variational functional associated with (1) is defined and the fast homoclinic solutions are the critical points of the certain functional.

Let

where is defined in (2) and denotes the space of sequences whose second powers are summable on ; that is,

The space is equipped with the following norm:

For , let

Similar to [2], it is easy to check that is a Hilbert space with the norm given by

It is obvious that

with the embedding being continuous. Here, denotes the Banach spaces of functions on with values in under the norm

Similar to [1], we have the following definition of fast homoclinic solutions.

Definition 1. If (2) holds, a solution of (1) is called a fast homoclinic solution if .

Here and in subsequence, and denote the inner product and norm in , respectively. denote different positive constants. Now, we state our main results.

Theorem 2. Suppose that , , , and satisfy (2), (V1), and the following conditions:

(A) and as ,(V2)′, , and there exists a constant such that

(V3)′there is a constant such that

(V5) and there exists a constant such that

(I)′ and there exists a constant with such that

 where .

Then, problem (1) has at least one nontrivial fast homoclinic solution.

Theorem 3. Suppose that , , , and satisfy (2), (V1), (A), (V3)′, (I)′, and the following conditions:

(V2)′′, , and

(V5)′ and there exists a constant such that

Then, problem (1) has at least one nontrivial fast homoclinic solution.

Theorem 4. Suppose that , , , and satisfy (2), (V1), (A), (V2)′, (V3)′, (V5), (I)′, and the following condition:

(V6).

Then, problem (1) has an unbounded sequence of fast homoclinic solutions.

Theorem 5. Suppose that , , , and satisfy (2), (V1), (A), (V2)′′, (V3)′, (V5)′, (I)′, and (V6). Then, problem (1) has an unbounded sequence of fast homoclinic solutions.

The rest of this paper is organized as follows. In Section 2, some preliminaries are presented. In Section 3, we give the proofs of our results. In Section 4, some examples are given to illustrate our results.

2. Preliminaries

Let and be given in Section 1. By a similar argument in [2, 20], we have the following important lemma.

Lemma 6. For any ,

where , , and is the same as that in assumption (I)′.

The following two lemmas are Mountain Pass Theorem and Symmetric Mountain Pass Theorem, which are useful in the proofs of our theorems.

Lemma 7 (see [21]). Let be a real Banach space and satisfying (PS)-condition. Suppose that and

(i)there exist constants , such that ;(ii)there exists an such that .

Then, possesses a critical value which can be characterized as , where and is an open ball in of radius centered at .

Lemma 8 (see [21]). Let be a real Banach space and with even. Assume that and satisfies (PS)-condition, assumption (i) of Lemma 7, and the following condition:

(iii)for each finite dimensional subspace , there is such that , for , and is an open ball in of radius centered at .

Then, possesses an unbounded sequence of critical values.

Remark 9. Since it is very difficult to check condition (iii) of Lemma 8, few results about infinitely many homoclinic solutions can be seen in the literature by using Lemma 8, let alone infinitely many fast homoclinic solutions obtained by this lemma. Motivated by the idea of [22], we will use Lemma 8 to prove that problem (1) has infinitely many homoclinic fast solutions.

Lemma 10. Assume that (V3)′ and (V5) or (V5)′ hold. Then, for every ,

(i) is nondecreasing on ;(ii) is nonincreasing on .

The proof of Lemma 10 is routine and we omit it.

The functional corresponding to (1) on is given by

We now show that and, for ,

Firstly, we show that . By (V2)′, for any given , there exists such that

where , is the same as that in assumption (I)′. Then, by and (28), we have

Therefore, from (29), we have

From (I)′ and Lemma 6, we have

It follows from (26), (30), and (31) that . Next, we prove that . Rewrite as the following:

where

It is easy to check that and

Next, we prove that and

Let in , without loss of generality, and we can assume that . Since , we have

By (29) and (36), we have

Since , for almost every , we have

Then, by (37), (38), and Lebesgue’s dominated convergence theorem, we have

Therefore, for any and for any function , from (39), we have

Finally, we prove that . From (40), in , and , we have

This shows that . Therefore, and (27) holds. Similarly, we can prove and (27) holds by (V1), (V2)′′, and (I)′. Furthermore, the critical points of in are classical solutions of (1) with .

3. Proofs of Theorems

Proof of Theorem 2. It is clear that . We first show that the functional satisfies the (PS)-condition. Let satisfying which is bounded and let as . Then, there exists a constant such that

From (I)′ and (25), we have

From (26), (27), (31), (42), (43), (V3)′, and (V5), we have

Since and , the above inequalities imply that there exists a constant such that

Now, we prove that in . Passing to a subsequence if necessary, it can be assumed that in . For any given number , by (V2)′, we can choose such that

Since as , we can choose such that

From (A), we can choose such that

It follows from (23), (45), (47), and (48) that

where . Similarly, by (24), (45), (47), and (48), we have

Since in , it is easy to verify that converges to pointwise for all . Hence, it follows from (49) and (50) that

Since on , the operator defined by is a linear continuous map. So, in . Sobolev theorem implies that uniformly on , so there is such that

From (45), (46), (49), (50), and (51), we have

It follows from (52) and (53) that

From (27) and (I)′, as , we have

It follows from (54) and (55) that

Hence, in by (56). This shows that satisfies (PS)-condition.

We now show that there exist constants such that assumption (i) of Lemma 7 holds. From (V2)′, there exists such that

By and (57), we have

Let

Setting and , it follows from Lemma 6 that for . From Lemma 10 (i) and (59), we have

By (V5), (31), (58), and (60), we have

Therefore, we can choose a constant depending on such that for any with , which shows that satisfies assumption (i) of Lemma 7.

Finally, it remains to show that satisfies assumption (ii) of Lemma 7. From Lemma 10 (ii) and (22), we have for any

where , . Take such that

and for . For , from Lemma 10 (i) and (63), we get

where . From (26), (31), (62), (63), and (64), we get for

Since and , it follows from (65) that there exists such that and . Set , and then , , and . By Lemma 7, has a critical value given by

where

Hence, there exists such that

The function is a desired solution of problem (1). Since , is a nontrivial fast homoclinic solution. The proof is complete.

Proof of Theorem 3. In the proof of Theorem 2, the condition in (V5) is only used in the proofs of (45) and assumption (i) of Lemma 7. Therefore, we only need to prove that (45) and assumption (i) of Lemma 7 still hold if we use (V2)′′ and (V5)′ instead of (V2)′ and (V5), respectively. We first prove that (45) holds. From (V3)′, (V5)′, (26), (27), (31), (42), and (43), we have

which implies that there exists a constant such that (45) holds. Next, we prove that assumption (i) of Lemma 7 still holds. From (V2)′′, there exists such that

By and (70), we have

Let , and it follows from Lemma 6 that . It follows from (31) and (71) that

Therefore, we can choose a constant depending on such that for any with . The proof of Theorem 3 is complete.

Proof of Theorem 4. Condition (V6) shows that is even. In view of the proof of Theorem 2, we know that and satisfies (PS)-condition and assumption (i) of Lemma 7. Now, we prove that (iii) of Lemma 8 holds. Let be a finite dimensional subspace of . Since all norms of a finite dimensional space are equivalent, there exists such that

Assume that dim and is a base of such that

For any , there exist , such that

Let

It is easy to see that is a norm of . Hence, there exists a constant such that . Since , by Lemma 6, we can choose such that

where is given in (71). Let

Hence, for , let such that

Then, by (73)–(76), (78), and (79), we have

This shows that and there exists such that , which together with (77) implies that . Let and

Since for all and and , it follows that . For any , from Lemmas 6 and 10 (i), we have

where and . Since , , it follows that there exists such that

Then, for with and , it follows from (75), (78), (79), (80), and (83) that

On the other hand, since for , then

Therefore, from (81), (84), and (85), we have

By (77) and (78), we have

By (26), (31), (58), (82), (86), (87), and Lemma 10, we have for and

Since , there exists such that

It follows that

which shows that (iii) of Lemma 8 holds. By Lemma 8, possesses an unbounded sequence of critical values with , where is such that for  . If is bounded, then there exists such that

By a similar fashion for the proof of (49) and (50), for the given in (58), there exists such that

Hence, by (22), (26), (31), (58), (91), and (92), we have

It follows from (93) that

This contradicts the fact that is unbounded, and so is unbounded. The proof is complete.

Proof of Theorem 5. In view of the proofs of Theorems 3 and 4, the conclusion of Theorem 5 holds. The proof is complete.

4. Examples

Example 1. Consider the following system:

where , , , , and as . Let

where , , , , and . Let

Then, it is easy to check that all the conditions of Theorem 4 are satisfied with and . Hence, problem (95) has an unbounded sequence of fast homoclinic solutions.

Example 2. Consider the following system:

where , , , , and as . Let

where , , , , and . Let

Then, it is easy to check that all the conditions of Theorem 5 are satisfied with and . Hence, by Theorem 5, problem (98) has an unbounded sequence of fast homoclinic solutions.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the NNSF of China (no. 11301108), Guangxi Natural Science Foundation (no. 2013GXNSFBA019004), and the Scientific Research Foundation of Guangxi Education Office (no. 201203YB093).

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