Journal of Applied Mathematics

VolumeΒ 2012Β (2012), Article IDΒ 434938, 18 pages

http://dx.doi.org/10.1155/2012/434938

## Existence of Subharmonic Solutions for a Class of Second-Order -Laplacian Systems with Impulsive Effects

^{1}Department of Mathematics and Computation Science, Yulin Normal University, Yulin, Guangxi 537000, China^{2}College of Science, Guilin University of Technology, Guilin, Guangxi 541004, China^{3}School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan 410083, China

Received 14 April 2011; Accepted 1 July 2011

Academic Editor: MengΒ Fan

Copyright Β© 2012 Wen-Zhen Gong 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

By using minimax methods in critical point theory, a new existence theorem of infinitely many periodic solutions is obtained for a class of second-order -Laplacian systems with impulsive effects. Our result generalizes many known works in the literature.

#### 1. Introduction

Consider the following -Laplacian system with impulsive effects: where , , , and are continuous and is -periodic in for all , is the gradient of with respect to . is a -periodic positive definite symmetric matrix.

Throughout this paper, we always assume the following condition holds.(A) is measurable in for all and continuously differentiable in for a.e. , and there exist , such that for all and a.e. .

For the sake of convenience, in the sequel, we define .

When , , , problem (1.1) becomes the following second-order Hamiltonian system:

There are many papers concerning the existence of periodic solutions or homoclinic solutions for problem (1.3) by minimax methods. Here for identifying a few, we only mention [1β8].

For , , problem (1.1) 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. Since these processes are subject to short-term perturbations whose duration is negligible in comparison with the duration of the processes, it is natural to suppose that these perturbations act instantaneously, that is, in the form of impulse. Processes of this type 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 [9, 10].

There are many methods for finding periodic solutions of impulsive differential equations, such as the monotone-iterative technique, a numerical-analytical method, the method of upper and lower solutions, and the method of bilateral approximations. For more information about periodic solutions of impulsive differential equations, one can refer to the papers [11β18]. However, there are few papers [19β25] concerning periodic solutions of impulsive differential equations by variational methods. So it is a novel method to employ variational methods to investigate the existence of periodic solutions for impulsive differential equations.

Motivated by the above papers, we study the existence of subharmonic solutions for problem (1.1) by applying minimax methods in critical point theory. Our result is new, which seems not to be found in the literature.

Throughout this paper, let satisfy .

#### 2. Preliminaries

In this section, we recall some basic facts which will be used in the proofs of our main results. In order to apply the critical point theory, we construct a variational structure. With this variational structure, we can reduce the problem of finding solutions of problem (1.1) to that of seeking the critical points of the corresponding functional.

Let be a positive integer and the Sobolev space defined by with the norm

Take and multiply the two sides of the equality by and integrate from 0 to ; we have Moreover, by , one has Together with (2.4), we get

*Definition 2.1. *We say that a function is a weak solution of problem (1.1) if the identity
holds for any .

Define the functional on by where

It follows from assumption (A) that the functional is continuously differentiable on and for . By the continuity of , , one has that . Hence, . For any , we have

By Definition 2.1, the weak solutions of problem (1.1) correspond to the critical points of the functional .

For , let and ; then it follows from Propositionββ1.1 in [26] that where , and if , then where . Let ; then . We will use the following lemma to prove our main results.

Lemma 2.2 (see [27]). *Let be a real Banach space with , where is finite dimensional. Suppose that satisfies the (PS) condition, and *(a)*there exist constants , such that , where , and denotes the boundary of ; *(b)*there exists an and such that if , then .**
Then possesses a critical value which can be characterized as , where .*

It is well known that a deformation lemma can be proved with the weaker condition (C) replacing the usual (PS) condition. So Lemma 2.2 holds true under condition (C).

#### 3. Main Result and Proof

Theorem 3.1. *Assume that (A) holds and satisfy the following conditions: *(I1)*there exists such that
*(I2)*for any ,
*(H1)*, for all ;*(H2)* uniformly for a.e. ;*(H3)* uniformly for a.e. ;*(H4)*there exists a positive constant such that uniformly for a.e. ;*(H5)*there exists such that uniformly for a.e. ,**
where and . Then problem (1.1) has a sequence of distinct periodic solutions with period satisfying and as .*

*Remark 3.2. *As far as we know, there is no paper considering subharmonic solutions of impulsive differential equations. Our result is new.

*Proof. *The proof is divided into three steps. In the following, denote different positive constants.*Step 1. *The functional satisfies condition (C). Let satisfying as and is bounded; then, there exists a constant such that
From (H4), there exists such that
By assumption (A), for , there exists such that
which together with (3.4) implies that
By (3.3) and (3.6), we have
Since is continuous -periodic positive definite symmetric matrix on , there exist constants such that
It follows from (3.8) and (I1) that
By (3.7) and (3.9), we get
From (H5), there exists such that
By assumption (A), for , there exists such that
Thus, from (3.11) and (3.12), we have
which together with (3.3) and (I2) implies that
Hence, is bounded. If , we have
which together with (3.10) implies that is bounded. If , then from (2.12), we get
Since , it follows from (3.10) that is bounded too. Therefore, is bounded in . Hence, there exists a subsequence, still denoted by , such that
From (2.11), we have
From (3.3) and (3.18), we have
By (3.8), we know that , which together with the boundedness of and (3.19) implies that
From the boundedness of , the continuity of , and (3.18), we have
It follows from (A), (3.18) and the boundedness of that
which together with (3.20), (3.21), (3.22), and (3.23) implies that
It is easy to see from the boundedness of and (3.18) that
Let . Then, we have
It follows from (3.25) and (3.26) that
From (3.17), we get
By (3.27), (3.28), and HΓΆlder's inequality, we have
Hence, from (3.29) and (3.30), we obtain
That is, as . Since has the Kadec-Klee property, we have in . Therefore, the functional satisfies condition (C). *Step 2. *From (H2), for any small , there exists small enough such that
For and , it follows from (2.13) that
which implies that . Then from (I1), (3.8), and (3.33), we have
Let ; then from (3.24), we have
for all and . This implies that condition (a) of Lemma 2.2 holds. *Step 3. *Let , . Choose ; then from (H3), there exists such that
By assumption (A), for , there exists such that
which together with (3.37) implies that
Thus, from (H1), (I1), (3.8), and (3.39), we have
From (H3), we can choose suitable large such that
Let , where , . Since is finite dimensional, there exists a constant such that
By (I1), we have
From (3.39), (3.42), and (3.43), we obtain
From (H3), we can choose suitable such that
If , then we get
It follows from (3.46) that
where , . Notice that for any , we have
Hence, (3.47) holds for all whenever . Set
then , where
By (3.41) and (3.47), we have
Furthermore, for all , it follows from (H1), (3.8), and (3.43) that
Then by Lemma 2.2, has at least a critical point whose critical value satisfies
Similar to the proof of [28], let be a -periodic solution; we can prove that there exists a positive integer such that for all . Otherwise, as , which contradicts to (3.53). Repeating this process, we can obtain a sequence of distinct periodic solutions of problem (1.1). From (3.41), we know that is nonconstant. The proof is complete.

#### 4. Examples

In this section, we give an example to illustrate our result.

*Example 4.1. *Let , , , and consider the following -Laplacian system with impulsive effects
Let
where , . It is easy to check that satisfies (A), (H1), and (H2). By a direct computation, we have
which show that (H3), (H4), and (H5) hold. On the other hand,
where , . It is easy to see that satisfies (I1) and (I2). Hence, from Theorem 3.1, problem (4.1) has a sequence of distinct nonconstant periodic solutions with period satisfying and as .

#### Acknowledgments

This paper is partially supported by the NNSF (no. 10771215) of China. W.-Z. Gong is supported by Guangxi Natural Science Foundation (2010GXNSFA013125) and X. H. Tang is supported by the NNSF (no. 10771215) of China.

#### References

- Y. H. Ding, βExistence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems,β
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 25, no. 11, pp. 1095β1113, 1995. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Y. Ding and C. Lee, βHomoclinics for asymptotically quadratic and superquadratic Hamiltonian systems,β
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 71, no. 5-6, pp. 1395β1413, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - X. He and X. Wu, βPeriodic solutions for a class of nonautonomous second order Hamiltonian systems,β
*Journal of Mathematical Analysis and Applications*, vol. 341, no. 2, pp. 1354β1364, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Z.-Q. Ou and C.-L. Tang, βExistence of homoclinic solution for the second order Hamiltonian systems,β
*Journal of Mathematical Analysis and Applications*, vol. 291, no. 1, pp. 203β213, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - J. Wang, F. Zhang, and J. Xu, βExistence and multiplicity of homoclinic orbits for the second order Hamiltonian systems,β
*Journal of Mathematical Analysis and Applications*, vol. 366, no. 2, pp. 569β581, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Q. Zhang and X. H. Tang, βNew existence of periodic solutions for second order non-autonomous Hamiltonian systems,β
*Journal of Mathematical Analysis and Applications*, vol. 369, no. 1, pp. 357β367, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Q. Zhang and C. Liu, βInfinitely many homoclinic solutions for second order Hamiltonian systems,β
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 72, no. 2, pp. 894β903, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Z. Zhang and R. Yuan, βHomoclinic solutions for a class of non-autonomous subquadratic second-order Hamiltonian systems,β
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 71, no. 9, pp. 4125β4130, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - D. D. BaΔnov and P. Simeonov,
*Impulsive Differential Equations: Periodic Solutions and Applications*, vol. 66, Longman Scientific & Technical, Harlow, UK, 1993. View at Zentralblatt MATH - V. Lakshmikantham, D. D. BaΔnov, and P. S. Simeonov,
*Theory of Impulsive Differential Equations*, vol. 6, World Scientific Press, Singapore, 1989. - J. Chu and J. J. Nieto, βImpulsive periodic solutions of first-order singular differential equations,β
*Bulletin of the London Mathematical Society*, vol. 40, no. 1, pp. 143β150, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Y.-K. Chang, J. J. Nieto, and W.-S. Li, βOn impulsive hyperbolic differential inclusions with nonlocal initial conditions,β
*Journal of Optimization Theory and Applications*, vol. 140, no. 3, pp. 431β442, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - D. Franco and J. J. Nieto, βNonlinear boundary value problems for first order impulsive functional differential equations,β
*Journal of Computational and Applied Mathematics*, vol. 88, pp. 149β159, 1998. View at Google Scholar - Y. Li, βPositive periodic solutions of nonlinear differential systems with impulses,β
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 68, no. 8, pp. 2389β2405, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Z. Luo and J. J. Nieto, βNew results for the periodic boundary value problem for impulsive integro-differential equations,β
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 70, no. 6, pp. 2248β2260, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - E. K. Lee and Y.-H. Lee, βMultiple positive solutions of singular two point boundary value problems for second order impulsive differential equations,β
*Applied Mathematics and Computation*, vol. 158, no. 3, pp. 745β759, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - J. J. Nieto, βImpulsive resonance periodic problems of first order,β
*Applied Mathematics Letters*, vol. 15, no. 4, pp. 489β493, 2002. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - J. J. Nieto and R. RodrΓguez-LΓ³pez, βBoundary value problems for a class of impulsive functional equations,β
*Computers & Mathematics with Applications*, vol. 55, no. 12, pp. 2715β2731, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - P. Chen and X. H. Tang, βExistence of solutions for a class of p-Laplacian systems with impulsive effects,β
*Journal of Taiwanses Mathematics*. In press. - J. J. Nieto and D. O'Regan, βVariational approach to impulsive differential equations,β
*Nonlinear Analysis. Real World Applications*, vol. 10, no. 2, pp. 680β690, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - J. Sun, H. Chen, and L. Yang, βThe existence and multiplicity of solutions for an impulsive differential equation with two parameters via a variational method,β
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 73, no. 2, pp. 440β449, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - J. Sun, H. Chen, and L. Yang, βVariational methods to fourth-order impulsive differential equations,β
*Journal of Applied Mathematics and Computing*, vol. 35, no. 1-2, pp. 323β340, 2011. View at Publisher Β· View at Google Scholar - Y. Tian and W. Ge, βApplications of variational methods to boundary-value problem for impulsive differential equations,β
*Proceedings of the Edinburgh Mathematical Society. Series II*, vol. 51, no. 2, pp. 509β527, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - J. W. Zhou and Y. K. Li, βExistence and multiplicity of solutions for some Dirichlet problems with impulsive effects,β
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 71, no. 7-8, pp. 2856β2865, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - J. Zhou and Y. Li, βExistence of solutions for a class of second-order Hamiltonian systems with impulsive effects,β
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 72, no. 3-4, pp. 1594β1603, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - J. Mawhin and M. Willem,
*Critical Point Theory and Hamiltonian Systems*, vol. 74, Springer, New York, NY, USA, 1989. - 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, Providence, RI, USA, 1986. - S. W. Ma and Y. X. Zhang, βExistence of infinitely many periodic solutions for ordinary p-Laplacian systems,β
*Journal of Mathematical Analysis and Applications*, vol. 351, no. 1, pp. 469β479, 2009. View at Publisher Β· View at Google Scholar