- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Abstract and Applied Analysis

Volume 2013 (2013), Article ID 583107, 8 pages

http://dx.doi.org/10.1155/2013/583107

## Homoclinic Solutions for a Class of the Second-Order Impulsive Hamiltonian Systems

^{1}College of Mathematics and Statistics, Jishou University, Jishou, Hunan 416000, China^{2}Department 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.

#### Abstract

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 .

When , , and , system (1) and (2) reduces to Hamiltonian system Rabinowitz [1] studied the existence of nontrivial homoclinic solutions of it.

When and satisfied (V1), system (1), (2) reduces to Hamiltonian system Izydorek and Janczewska [2] studied the existence of homoclinic solutions of it.

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. [27] used dual variational methods, and Lions [28] and Hofer and Wysocki [29] employed concentration compactness method, Ekeland's variational principle, that they established the existence of homoclinic solutions of the first-order Hamiltonian systems. Rabinowitz [1] and Izydorek and Janczewska [2] obtained homoclinic solutions of a class of second order Hamiltonian systems as a limit of its periodic solutions.

In recent paper [18], 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 [18]). *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. *

The rest of this paper is organized as follows. In Section 2 we present some preliminary results. Our main result's proofs are given in Section 3.

#### 2. Preliminaries

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 [2]). *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 [30]). *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 [2]). *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.

Lemma 6. *Assume that (V1), (K1), (K2), (W1), (W2), (W3), (W4), (G1), (G2), (G3), and (F1) are satisfied; system (2), (7) possesses a -periodic solution. *

*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

Define

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 .

Lemma 8. *The function determined by Lemma 7 is a non-zero homoclinic solution of the system (1), (2). *

*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 [13] 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.

#### Acknowledgments

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).

#### References

- P. H. Rabinowitz, “Homoclinic orbits for a class of Hamiltonian systems,”
*Proceedings of the Royal Society of Edinburgh. Section A*, vol. 114, no. 1-2, pp. 33–38, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Izydorek and J. Janczewska, “Homoclinic solutions for a class of the second order Hamiltonian systems,”
*Journal of Differential Equations*, vol. 219, no. 2, pp. 375–389, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. P. Agarwal and D. O'Regan, “A multiplicity result for second order impulsive differential equations via the Leggett Williams fixed point theorem,”
*Applied Mathematics and Computation*, vol. 161, no. 2, pp. 433–439, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. P. Agarwal, D. Franco, and D. O'Regan, “Singular boundary value problems for first and second order impulsive differential equations,”
*Aequationes Mathematicae*, vol. 69, no. 1-2, pp. 83–96, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Chu and J. J. Nieto, “Impulsive periodic solutions of first-order singular differential equations,”
*Bulletin of the London Mathematical Society*, vol. 40, no. 1, pp. 143–150, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. He and X. He, “Monotone iterative technique for impulsive integro-differential equations with periodic boundary conditions,”
*Computers & Mathematics with Applications*, vol. 48, no. 1-2, pp. 73–84, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. J. Nieto and D. O'Regan, “Variational approach to impulsive differential equations,”
*Nonlinear Analysis: Real World Applications*, vol. 10, no. 2, pp. 680–690, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. Qian and X. Li, “Periodic solutions for ordinary differential equations with sublinear impulsive effects,”
*Journal of Mathematical Analysis and Applications*, vol. 303, no. 1, pp. 288–303, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. H. Rabinowitz,
*Minimax Methods in Critical Point Theory with Applications to Differential Equations*, vol. 65 of*CBMS Regional Conference Series in Mathematics*, American Mathematical Society, Washington, DC, USA, 1986. View at MathSciNet - I. Rachůnková and M. Tvrdý, “Non-ordered lower and upper functions in second order impulsive periodic problems,”
*Dynamics of Continuous, Discrete & Impulsive Systems. Series A*, vol. 12, no. 3-4, pp. 397–415, 2005. View at MathSciNet - J. Sun, H. Chen, and L. Yang, “Variational methods to fourth-order impulsive differential equations,”
*Journal of Applied Mathematics and Computing*, vol. 35, no. 1-2, pp. 323–340, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Shen and W. Wang, “Impulsive boundary value problems with nonlinear boundary conditions,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 69, no. 11, pp. 4055–4062, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Tian and W. Ge, “Applications of variational methods to boundary-value problem for impulsive differential equations,”
*Proceedings of the Edinburgh Mathematical Society*, vol. 51, no. 2, pp. 509–527, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Tian, J. Wang, and W. Ge, “Variational methods to mixed boundary value problem for impulsive differential equations with a parameter,”
*Taiwanese Journal of Mathematics*, vol. 13, no. 4, pp. 1353–1370, 2009. View at Zentralblatt MATH · View at MathSciNet - J. Xie and Z. Luo, “Multiple solutions for a second-order impulsive Sturm-Liouville equation,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 527082, 6 pages, 2013. View at Publisher · View at Google Scholar - J. Xie and Z. Luo, “Solutions to a boundary value problem of a fourth-order impulsive differential equation,”
*Boundary Value Problems*, vol. 2013, 154 pages, 2013. View at Publisher · View at Google Scholar - W. Zuo, D. Jiang, D. O'Regan, and R. P. Agarwal, “Optimal existence conditions for the periodic delay
*ϕ*-Laplace equation with upper and lower solutions in the reverse order,”*Results in Mathematics*, vol. 44, no. 3-4, pp. 375–385, 2003. View at Publisher · View at Google Scholar · View at MathSciNet - H. Zhang and Z. Li, “Periodic and homoclinic solutions generated by impulses,”
*Nonlinear Analysis: Real World Applications*, vol. 12, no. 1, pp. 39–51, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Zhang and Z. Li, “Variational approach to impulsive differential equations with periodic boundary conditions,”
*Nonlinear Analysis: Real World Applications*, vol. 11, no. 1, pp. 67–78, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Zhang and R. Yuan, “An application of variational methods to Dirichlet boundary value problem with impulses,”
*Nonlinear Analysis: Real World Applications*, vol. 11, no. 1, pp. 155–162, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Lv and S. Lu, “Homoclinic orbits for a class of second-order Hamiltonian systems without a coercive potential,”
*Journal of Applied Mathematics and Computing*, vol. 39, no. 1-2, pp. 121–130, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - P. D. Makita, “Homoclinic orbits for second order Hamiltonian equations in ℝ,”
*Journal of Dynamics and Differential Equations*, vol. 24, no. 4, pp. 857–871, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. H. Tang and L. Xiao, “Homoclinic solutions for a class of second-order Hamiltonian systems,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 71, no. 3-4, pp. 1140–1152, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Winkler, “Spatially monotone homoclinic orbits in nonlinear parabolic equations of super-fast diffusion type,”
*Mathematische Annalen*, vol. 355, no. 2, pp. 519–549, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - X. Zhang and X. Tang, “Subharmonic solutions for a class of non-quadratic second order Hamiltonian systems,”
*Nonlinear Analysis: Real World Applications*, vol. 13, no. 1, pp. 113–130, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Zhou, J. Yu, and Y. Chen, “Homoclinic solutions in periodic difference equations with saturable nonlinearity,”
*Science China Mathematics*, vol. 54, no. 1, pp. 83–93, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. Coti-Zelati, I. Ekeland, and E. Séré, “A variational approach to homoclinic orbits in Hamiltonian systems,”
*Mathematische Annalen*, vol. 288, no. 1, pp. 133–160, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P.-L. Lions, “The concentration-compactness principle in the calculus of variations. The locally compact case. I,”
*Annales de l'Institut Henri Poincaré*, vol. 1, no. 2, pp. 109–145, 1984. View at Zentralblatt MATH · View at MathSciNet - H. Hofer and K. Wysocki, “First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems,”
*Mathematische Annalen*, vol. 288, no. 3, pp. 483–503, 1990. View at Publisher · View at Google Scholar · View at MathSciNet - A. Ambrosetti and P. H. Rabinowitz, “Dual variational methods in critical point theory and applications,”
*Journal of Functional Analysis*, vol. 14, no. 4, pp. 349–381, 1973. View at Zentralblatt MATH · View at MathSciNet