Abstract and Applied Analysis

Volume 2012 (2012), Article ID 616427, 8 pages

doi:10.1155/2012/616427

Research Article

Periodic Solutions of Some Impulsive Hamiltonian Systems with Convexity Potentials

Dezhu Chen and Binxiang Dai

School of Mathematical Sciences and Computing Technology, Central South University, Hunan, Changsha 410083, China

Received 31 August 2012; Accepted 18 November 2012

Academic Editor: Juntao Sun

Copyright © 2012 Dezhu Chen and Binxiang Dai. 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.

We study the existence of periodic solutions of some second-order Hamiltonian systems with impulses. We obtain some new existence theorems by variational methods.

1. Introduction

Consider the following systems:

where with for each , there exists an such that , and we suppose that satisfies the following assumption.

  is measurable in for and continuously differentiable in for a.e. , and there exist such that

 for all and .

Many solvability conditions for problem (1.1) without impulsive effect are obtained, such as, the coercivity condition, the convexity conditions (see [1–4] and their references), the sublinear nonlinearity conditions, and the superlinear potential conditions. Recently, by using variational methods, many authors studied the existence of solutions of some second-order differential equations with impulses. More precisely, Nieto in [5, 6] considers linear conditions, [7–10] the sublinear conditions, and [11–16] the sublinear conditions and the other conditions. But to the best of our knowledge, except [7] there is no result about convexity conditions with impulsive effects. By using different techniques, we obtain different results from [7].

We recall some basic facts which will be used in the proofs of our main results. Let

with the inner product

where denotes the inner product in . The corresponding norm is defined by

The space has some important properties. For , let , and . Then one has Sobolev's inequality (see Proposition in [1]):

Consider the corresponding functional on given by

It follows from assumption and the continuity of one has that is continuously differentiable and weakly lower semicontinuous on . Moreover, we have

for and is weakly continuous and the weak solutions of problem (1.1) correspond to the critical points of (see [8]).

Theorem 1.1 ([2, Theorem  1.1]). Suppose that and are reflexive Banach spaces, , is weakly upper semi-continuous for all , and is convex for all and is weakly continuous. Assume that

as and for every ,

as uniformly for . Then has at least one critical point.

2. Main Results

Theorem 2.1. Assume that assumption holds. If further

is convex for a.e. , and there exist , such that , for all , then (1.1) possesses at least one solution in .

Remark 2.2. implies there exists a point for which

Proof of Theorem 2.1. It follows Remark 2.2, (1.6), and that

for all and some positive constant . As if and only if , we have as . By Theorem 1.1 and Corollary 1.1 in [1], has a minimum point in , which is a critical point of . Hence, problem (1.1) has at least one weak solution.

Theorem 2.3. Assume that assumption and hold. If further

there exist , and such that for all and there exist some and such that

for and , where is a constant, then (1.1) possesses at least one solution in .

Remark 2.4. We can find that our condition is very different from condition (vii) in [7] since we prove this by the saddle point theorem substituted for the least action principle.

Proof of Theorem 2.3. We prove satisfies the (PS) condition at first. Suppose is such an sequence that is bounded and . We will prove it has a convergent subsequence. By and (1.6), we have

for some positive constants , , . By Remark 2.2, (1.6), and (2.4), we have

for some positive constants , , which implies that

for some positive constants , . By (1.6), the above inequality implies that

for the positive constant . The one has

If is unbounded, we may assume that, going to a subsequence if necessary,

By (2.8) and (2.9), we have

for all large and every . By (2.10) and , one has for all large . It follows from , (2.4), (2.6), (2.7), and above inequality that

for large and the positive constant , which contradicts the boundedness of since . Hence is bounded. Furthermore, is bounded by (2.6). A similar calculation to Lemma 3.1 in [9] shows that satisfies the (PS) condition. We now prove that satisfies the other conditions of the saddle point theorem. Assume that , then . From above calculation, one has

for all , which implies that

as in . Moreover, by and we have

for , which implies that

as in since . Now Theorem 2.3 is proved by (2.13), (2.15), and the saddle point theorem.

Theorem 2.5. Assume that assumption holds. Suppose that are concave and satisfy

for some positive constant , then (1.1) possesses at least one solution in .

Proof of Theorem 2.5. Consider the corresponding functional on given by

which is continuously differentiable, bounded, and weakly upper semi-continuous on . Similar to the proof of Lemma 3.1 in [2], one has that is convex in for every . By the condition, we have . Similar to the proof of Theorem 3.1, we have

which means as , uniformly for with by and (1.6). On the other hand,

which implies that as by and (1.6). We complete our proof by Theorem 1.1.

Acknowledgment

The first author was supported by the Postgraduate Research and Innovation Project of Hunan Province (CX2011B078).

References

J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, vol. 74, Springer, Berlin, Germany, 1989.

C.-L. Tang and X.-P. Wu, “Some critical point theorems and their applications to periodic solution for second order Hamiltonian systems,” Journal of Differential Equations, vol. 248, no. 4, pp. 660–692, 2010.

J. Mawhin, “Semicoercive monotone variational problems,” Académie Royale de Belgique, Bulletin de la Classe des Sciences, vol. 73, no. 3-4, pp. 118–130, 1987.

C.-L. Tang, “An existence theorem of solutions of semilinear equations in reflexive Banach spaces and its applications,” Académie Royale de Belgique, Bulletin de la Classe des Sciences, vol. 4, no. 7–12, pp. 317–330, 1996.

J. J. Nieto, “Variational formulation of a damped Dirichlet impulsive problem,” Applied Mathematics Letters, vol. 23, no. 8, pp. 940–942, 2010.

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.

J. Zhou and Y. Li, “Existence of solutions for a class of second-order Hamiltonian systems with impulsive effects,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 3-4, pp. 1594–1603, 2010.

H. Zhang and Z. Li, “Periodic solutions of second-order nonautonomous impulsive di erential equations,” International Journal of Qualitative Theory of Differential Equations and Applications, vol. 2, no. 1, pp. 112–124, 2008.

J. Zhou and Y. Li, “Existence and multiplicity of solutions for some Dirichlet problems with impulsive effects,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 7-8, pp. 2856–2865, 2009.

W. Ding and D. Qian, “Periodic solutions for sublinear systems via variational approach,” Nonlinear Analysis: Real World Applications, vol. 11, no. 4, pp. 2603–2609, 2010.

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.

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.

D. Zhang and B. Dai, “Infinitely many solutions for a class of nonlinear impulsive differential equations with periodic boundary conditions,” Computers & Mathematics with Applications, vol. 61, no. 10, pp. 3153–3160, 2011.

L. Yang and H. Chen, “Existence and multiplicity of periodic solutions generated by impulses,” Abstract and Applied Analysis, vol. 2011, Article ID 310957, 15 pages, 2011.

J. Sun, H. Chen, and J. J. Nieto, “Infinitely many solutions for second-order Hamiltonian system with impulsive effects,” Mathematical and Computer Modelling, vol. 54, no. 1-2, pp. 544–555, 2011.

J. Sun, H. Chen, J. J. Nieto, and M. Otero-Novoa, “The multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 12, pp. 4575–4586, 2010.