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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 960276, 12 pages
Existence and Multiplicity of Fast Homoclinic Solutions for a Class of Damped Vibration Problems with Impulsive Effects
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.
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.
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.  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  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 ). 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 , 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 , we have the following definition of fast homoclinic solutions.
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
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.
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.
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 ). 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 ). 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 , 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