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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 583107, 8 pages
Homoclinic Solutions for a Class of the Second-Order Impulsive Hamiltonian Systems
1College of Mathematics and Statistics, Jishou University, Jishou, Hunan 416000, China
2Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, China
Received 16 May 2013; Accepted 18 July 2013
Academic Editor: Juan J. Nieto
Copyright © 2013 Jingli Xie et al. 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 of homoclinic solutions for a class of the second order impulsive Hamiltonian systems. By employing the Mountain Pass Theorem, we demonstrate that the limit of a -periodic approximation solution is a homoclinic solution of our problem.
1. Introduction and Main Results
In this paper, we consider the second-order impulsive differential equation where , , , , for each , and the operator is defined as , where denotes the right-hand (left-hand) limit of at . There exist an and a such that , , and , . satisfies(V1), , and is -periodic in its first variable.
We are mainly concerned with the existence of homoclinic solutions of system (1) and (2). A function is said to be a (classical) solution of (1) and (2) if satisfies (1) and (2). A (classical) solution of (1) and (2) is a homoclinic solution and if as and as .
Some classical tools such as some fixed point theorems in cones, topological degree theory, the upper and lower solutions method combined with monotone iterative technique, and variational methods [3–20] have been widely used to get solutions of impulsive differential equations. However, the existence of homoclinic solutions for the impulsive systems is paid little attention. It is well known that the homoclinic orbit rupture phenomenon can lead to chaos, which has been interesting to the mathematicians in recent years [21–26]. In the literature, Coti-Zelati et al.  used dual variational methods, and Lions  and Hofer and Wysocki  employed concentration compactness method, Ekeland's variational principle, that they established the existence of homoclinic solutions of the first-order Hamiltonian systems. Rabinowitz  and Izydorek and Janczewska  obtained homoclinic solutions of a class of second order Hamiltonian systems as a limit of its periodic solutions.
In recent paper , Zhang and Li studied the existence of homoclinic solutions of an impulsive Hamiltonian system , , , , as a limit of its periodic solutions. In detail, they obtained the following theorem.
Theorem A (see ). Assume that , is continuous and -periodic in , and , satisfy the following conditions: (H1) is continuous differentiable -periodic, and there exist positive constants such that (H2) for all ; (H3) for ; (H4) there exists a such that then the Hamiltonian , , , , , possesses at least one nonzero homoclinic solution.
Motivated by papers [1, 2, 18], in this paper, we synthesize their methods to study the existence of homoclinic solutions of systems (1), (2). In detail, firstly, we introduce the following sequence equations: where for each , is a -periodic extension of the restriction of to the interval . Secondly, we study periodic solutions of (2) and (7) by converting the problem to the existence of critical points of some variational structure. finally, we find the homoclinic solutions of (1) and (2) as the limit of the periodic solutions of (2) and (7).
Part of the difficulty in treating (1) and (2) is subjected to the impulsive perturbation which destroys continuities of the velocity and when we apply the Mountain Theorem to prove our main result, we need the constant , appearing in the theorem to be independent of .
Our result is the following theorem.
Theorem 1. Assume that satisfies (V1), and , , and satisfy the following:(K1) there exist constants and such that for all (K2) there exists such that (W1), , as uniformly for ;(W2) there exist constants and such that (W3) there exist , , and such that and (W4) as uniformly in ;(G1), , as , ;(G2) there exists such that , , ;(G3), , ;(F1), , where , , and is a constant of (17). Then the system (1) and (2) possesses at least one nonzero homoclinic solution.
Let Then is a Hilbert space with the norm defined by For the norm in , which denotes a space of periodic essentially bounded measurable functions from into , . Next we set and define a functional as where Note that is Fréchet differentiable at any , and for any , we have It is clear that critical points of the functional are classical -periodic solutions of system (2) and (7).
Lemma 2 (see ). There is a positive constant such that for each and the following inequality holds:
Lemma 3. Set ; then for every and , we have
Proof. Set , , and , , . By (G3), we have So we have , . If is empty, we have , , which implies . Therefore, Without loss of generality, we can assume that is nonempty, and we have
Lemma 4 (see ). Let be a real Banach space and let be a -smooth functional satisfying the Palais-Smale condition and . If satisfies the following conditions:(i)there exist constants such that ,(ii)there exists , such that , then possesses a critical value given by where
Lemma 5 (see ). Let be a continuous mapping such that . For every the following inequality holds:
3. Proof of Theorem 1
We have divided the proof of Theorem 1 into a sequence of lemmas.
Proof. It is clear that . It is well known that Lemma 4 holds true with the (C) condition replacing the usual (PS) condition. We say the functional satisfies the (C) condition; that is, for every sequence , has a convergent subsequence if is bounded and .
Step 1. Pick such that is bounded and then there exists a constant such that where and . (24) implies is bounded; that is, there exists a constant such that From (W2), (G2), (17), and (25), we have On the other hand, it follows from (K1) and (17) that Combining (26) and (27), we obtain Since , it follows that is bounded. In a similar way to [21, Proposition B35], we can prove that has a convergent subsequence. So, the functional satisfies the (C) condition.
Step 2. We show that the functional satisfies the assumption (i) of Lemma 4. Set , , . By (G3), we have Hence when , we have From (K1), (W2), and (30), we have Set Let ; then and (31) gives .
Step 3. We show that the functional satisfies assumption (ii) of Lemma 4.
In order to verify (ii), we choose , such that and . Set . By (K2), we have . So we have
Take . By (W4), there exists such that
By (33), (34), (35), and Lemma 3, we have Clearly, as , so (ii) holds. By Lemma 4, possesses a critical value . Let denote the corresponding critical point of on ; that is, Hence the system (2), (7) possesses a nontrivial -periodic solution .
Lemma 7. Let be the sequence given by (37). Then there exist a subsequence of and a function such that converges to weakly in and strongly in .
Proof. We assert that there is a constant independent of such that
Let such that , for some , and . Define
We extend in the way of -periodic to . For simplicity, we also note it again by . It is clear that and . Define by for . Then, we have
independently of . The rest detailed argument is similar to the proof of Step 1 in Lemma 6 and we thus omit it here.
Hence, is a bounded sequence in and we may pick a subsequence such that converges weakly in and strongly in . Next is a bounded sequence in , so we may pick a subsequence such that converges weakly in and strongly in . We can repeat this process and obtain, for any positive integer , a sequence which converges weakly in and strongly in , and Therefore, for any positive integer , the sequence converges weakly in and strongly in . Hence there exists a function such that the sequence converges weakly in and strongly in .
Proof. The proof will be divided into four steps.
Firstly, we show that is a solution of the system (1), (2). Here, for simplicity, we denote by . For any given interval and any , define so for any , we have Therefore, one has The remained detailer argument is similar to the proof of Lemma 2.5 in  and we thus omit it here, so is a solution of system (1) and (2).
Secondly we show that , as . is weak continuity, so it is weak lower semicontinuity. One has and so By (23) and (46), we obtain , as .
Thirdly, we prove that as . We have proved is a solution of system (1) and (2), so we have By (V1), (K1), and (W1), one has . Hence as . By (23), one has Therefore one has as .
Finally, we show when . Since , , , we can let and . By Hölder inequality and , , , we have Let , which is a constant independent of .
It is clearly that , , is a periodic solution of (2), (7). So we can assume the maximum of occurs in . Now we assume , so there is as ; therefore there exists integer such that . Combining (G1), there exists an integer such that when , one has By (49) and (50), when , one has Define a function by It is clear that and is monotone nondecreasing, so we have Hence we have
Since , is a solution of the system (1) and (2), so when , we have Combining (51), (54), and (55) we have Hence, we have By the property of the function , there exists a constant such that . This is a contradiction. Hence the system (1), (2) has a nontrivial homoclinic solution even if .
Next, we give an example to illustrate our main result.
Example 9. Let where , , . It is easy to verify that , , , , and satisfy conditions (V1), (K1), (K2), (W1), (W2), (W3), (W4), (G1), (G2), (G3), and (F1). So, system (1), (2) with , , , , and as in (58) has a nontrivial homoclinic solution.
The authors are very grateful to the referees for their very helpful comments and suggestions, which greatly improved the presentation of this paper. This work is supported by the Hunan Provincial Natural Science Foundation of China (no. 11JJ3012) and Major Project of Science Research Fund of Education Department in Hunan (no. 11A095).
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