Abstract and Applied Analysis

Volume 2014 (2014), Article ID 356745, 8 pages

http://dx.doi.org/10.1155/2014/356745

## Multiple Solutions of Second-Order Damped Impulsive Differential Equations with Mixed Boundary Conditions

^{1}School of Economics and Management, Changsha University of Science and Technology, Changsha, Hunan 410004, China^{2}Hunan Normal University Press, Hunan Normal University, Changsha, Hunan 410081, China

Received 14 February 2014; Accepted 27 March 2014; Published 23 April 2014

Academic Editor: Fenghua Wen

Copyright © 2014 Jian Liu and Lizhao Yan. 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 use variational methods to investigate the solutions of damped impulsive differential equations with mixed boundary conditions. The conditions for the multiplicity of solutions are established. The main results are also demonstrated with examples.

#### 1. Introduction

Impulsive effect exists widely in many evolution processes in which their states are changed abruptly at certain moments of time. The theory of impulsive differential systems has been developed by numerous mathematicians [1–6]. Applications of impulsive differential equations with or without delays occur in biology, medicine, mechanics, engineering, chaos theory, and so on [7–11].

In this paper, we consider the following second-order damped impulsive differential equations with mixed boundary conditions: where , , is continuous, , are continuous, and for .

The characteristic of (1) is the presence of the damped term . Most of the results concerning the existence of solutions of these equations are obtained using upper and lower solutions methods, coincidence degree theory, and fixed point theorems [12–15]. On the other hand, when there is no presence of the damped term, some researchers have used variational methods to study the existence of solutions for these problems [16–21]. However, to the best of our knowledge, there are few papers concerned with the existence of solutions for impulsive boundary value problems like problem (1) by using variational methods.

For this nonlinear damped mixed boundary problem (1), the variational structure due to the presence of the damped term is not apparent. However, inspired by the work [22, 23], we will be able to transform it into a variational formulation. In this paper, our aim is to study the existence of distinct pairs of nontrivial solutions of problem (1). Our main results extend the study made in [22, 23], in the sense that we deal with a class of problems that is not considered in those papers.

#### 2. Preliminaries and Statements

Let , , , . We transform (1) into the following equivalent form:

Obviously, the solutions of (2) are solutions of (1).

Define the space . It is easy to see that and is a closed subset of . So is a Hilbert space with the usual inner product in .

Consider the Hilbert spaces with the inner product inducing the norm We also consider the inner product inducing the norm

Consider the problem As is well known, (7) possesses a sequence of eigenvalues with The corresponding eigenfunctions are normalized so that ; here

Now multiply (2) by and integrate on the interval : Then, a weak solution of (2) is a critical point of the following functional: where .

We say that is a classical solution of IBVP (1) if it satisfies the following conditions: satisfies the first equation of (1) a.e. on ; the limits , , exist and impulsive condition of (1) holds; satisfies the boundary condition of (1).

Lemma 1. *If is a weak solution of (1), then is a classical solution of (1).*

*Proof. *If is a weak solution of (1), then is a weak solution of (2), so holds for all ; that is,
By integrating by part, we have
Thus
holds for all . Without loss of generality, for any and with , for every , then substituting into (14), we get
Hence satisfies the first equation of (2). Therefore, by (14) we have

Next we will show that satisfies the impulsive and the boundary condition in (2). If the impulsive condition in (2) does not hold, without loss of generality, we assume that there exists such that
Let ; then
which contradicts (16). So satisfies the impulsive condition in (2) and (16) implies
If , pick ; one has
which contradicts (19), so satisfies the boundary condition. Therefore, is a solution of (1).

Lemma 2. *Let . Then there exists a constant , such that
**
where .*

*Proof. *By Hölder inequality, for ,

Lemma 3 (see [24, Theorem 9.1]). *Let be a real Banach space, with even, bounded from below, and satisfying P.S. condition. Suppose ; there is a set such that is homeomorphic to by an odd map and . Then possesses at least distinct pairs of critical points.*

#### 3. Main Results

Theorem 4. *Suppose that the following conditions hold.*(H1)*There exist , which is the kth eigenvalue of (7) such that
*(H2)*There exist and such that
*(H3)* and are odd about .*(H4)*, , as , .**Then, for , problem (1) has at least distinct pairs of solutions.*

*Proof. *Set
Consider

Next, we will verify that the solutions of problem (26) are solutions of problem (1).

In fact, let be the solution of problem (26). If , then there exists an interval such that
When , by (H1), we have
That is, is nondecreasing in . By and , we have
That is, for any . Since , then . So, there exists a constant such that , which contradicts (27). Then . Similarly, we can prove that .

Therefore, any solution of (26) is a solution of (1). Hence to prove Theorem 4, it suffices to produce at least distinct pairs of critical points of
where .

We will apply Lemma 3 to finish the proof.

By (30) and (H3), is even and .

Next, we will show that is bounded from below.

Let , . By (H1) and (H3), we have for ; thus
So, we have
for any . Therefore, is bounded from below.

In the following we will show that satisfies the P.S. condition. Let such that is a bounded sequence and ; then there exists such that
By (32), we have
So is bounded in . From the reflexivity of , we may extract a weakly convergent subsequence that, for simplicity, we call in . In the following we will verify that strongly converges to :
By in , we see that uniformly converges to in . So
So we obtain , as . That is, strongly converges to in , which means that satisfies the P.S. condition.

Now set , where is defined in (9). It is clear that is homeomorphic to by an odd map for any . In the following we verify that if is sufficiently small.

For any . By (H4) and (30), we have
for small . Since , and the proof is complete.

Theorem 5. *Suppose that the following conditions hold.*(H1)*There exist , which is the kth eigenvalue of (7) such that
*(H2)* for any .*(H3)* and are odd about .*(H4)*, , as , .**Then, for , problem (1) has at least distinct pairs of solutions.*

*Proof. *The proof is similar to the proof of Theorem 4, and therefore we omit it.

Theorem 6. *Suppose that the following conditions hold.*(H1)*There exist , which is the kth eigenvalue of (7) such that
*(H2)* and are odd about .*(H3)*, , as , .**Then, for , problem (1) has at least distinct pairs of solutions.*

*Proof. *Set
Consider

Next, we will verify that the solutions of problem (41) are solutions of problem (1).

In fact, let . By the definitions of and , (41) is reduced to
The solution of (42) satisfies . So and .

Let . By the definitions of and , (41) is reduced to
The solution of (43) satisfies , . So and .

Therefore, the solutions of (41) are solutions of (1). Hence to prove Theorem 6, it suffices to produce at least distinct pairs of critical points of
where .

We will apply Lemma 3 to finish the proof.

By (44) and (H2), is even and .

Next, we will show that is bounded from below.

By (H1) and (H2), we have and for ; thus
So, we have
for any . Therefore, is bounded from below.

In the following we will show that satisfies the P.S. condition. Let such that is a bounded sequence and ; then there exists such that
By (46), we have
So is bounded in . From the reflexivity of , we may extract a weakly convergent subsequence that, for simplicity, we call in . In the following we will verify that strongly converges to :
By in , we see that uniformly converges to in . So
So we obtain , as . That is, strongly converges to in , which means satisfies the P.S. condition.

Now set , where is defined in (9). It is clear that is homeomorphic to by an odd map for any . In the following we verify that if is sufficiently small.

For any , . By (H3) and (44), we have
for small . Since , and the proof is complete.

#### 4. Example

To illustrate how our main results can be used in practice we present the following example.

*Example 1. *Let , and consider the following problem:

Compared with (1), , . Obviously (H2), (H3), and (H4) are satisfied. Let , ; then (H1) is satisfied. By Theorem 4, for , , problem (52) has at least distinct pairs of solutions.

*Example 2. *Let , and consider the following problem:

Compared with (1), , . Obviously (H2), (H3), and (H4) are satisfied. Let , ; then (H1) is satisfied. By Theorem 5, for , , problem (53) has at least distinct pairs of solutions.

*Example 3. *Let , and consider the following problem:

Compared with (1), , . Obviously (H2) and (H3) are satisfied. Let ; then (H1) is satisfied. By Theorem 6, for , , problem (54) has at least distinct pairs of solutions.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This work is partially supported by the National Natural Science Foundation of China (no. 71201013) and the Innovation Platform Open Funds for Universities in Hunan Province (no. 13K059).

#### References

- M. Benchohra, J. Henderson, and S. Ntouyas,
*Impulsive Differential Equations and Inclusions*, vol. 2 of*Contemporary Mathematics and Its Applications*, Hindawi Publishing Corporation, New York, NY, USA, 2006. View at Publisher · View at Google Scholar · View at MathSciNet - A. M. Samoĭlenko and N. A. Perestyuk,
*Impulsive Differential Equations*, vol. 14, World Scientific, River Edge, NJ, USA, 1995. View at Publisher · View at Google Scholar · 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. J. Nieto, “Impulsive resonance periodic problems of first order,”
*Applied Mathematics Letters*, vol. 15, no. 4, pp. 489–493, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Yu, S. Wang, F. Wen, K. K. Lai, and S. He, “Designing a hybrid intelligent mining system for credit risk evaluation,”
*Journal of Systems Science & Complexity*, vol. 21, no. 4, pp. 527–539, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Luo and J. Shen, “Stability and boundedness for impulsive functional differential equations with infinite delays,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 46, no. 4, pp. 475–493, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Choisy, J.-F. Guégan, and P. Rohani, “Dynamics of infectious diseases and pulse vaccination: teasing apart the embedded resonance effects,”
*Physica D*, vol. 223, no. 1, pp. 26–35, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Jiao, X. Yang, L. Chen, and S. Cai, “Effect of delayed response in growth on the dynamics of a chemostat model with impulsive input,”
*Chaos, Solitons and Fractals*, vol. 42, no. 4, pp. 2280–2287, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Huang, C. Peng, X. Chen, and F. Wen, “Dynamics analysis of a class of delayed economic model,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 962738, 12 pages, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Zeng, F. Wang, and J. J. Nieto, “Complexity of a delayed predator-prey model with impulsive harvest and Holling type II functional response,”
*Advances in Complex Systems*, vol. 11, no. 1, pp. 77–97, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Dai and F. Wen, “Another improved Wei-Yao-Liu nonlinear conjugate gradient method with sufficient descent property,”
*Applied Mathematics and Computation*, vol. 218, no. 14, pp. 7421–7430, 2012. 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 - E. K. Lee and Y.-H. Lee, “Multiple positive solutions of singular two point boundary value problems for second order impulsive differential equations,”
*Applied Mathematics and Computation*, vol. 158, no. 3, pp. 745–759, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Zhao and H. Chen, “Multiplicity of solutions to two-point boundary value problems for second-order impulsive differential equations,”
*Applied Mathematics and Computation*, vol. 206, no. 2, pp. 925–931, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z.-G. Wang, G.-W. Zhang, and F.-H. Wen, “Properties and characteristics of the Srivastava-Khairnar-More integral operator,”
*Applied Mathematics and Computation*, vol. 218, no. 15, pp. 7747–7758, 2012. 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 - L. Yan, J. Liu, and Z. Luo, “Existence of solution for impulsive differential equations with nonlinear derivative dependence via variational methods,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 908062, 10 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - L. Z. Yan, J. Liu, and Z. G. Luo, “Existence and multiplicity of solutions for second-order impulsive differential equations on the half-line,”
*Advances in Difference Equations*, vol. 2013, article 293, 2013. View at Publisher · View at Google Scholar - 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 - J. Xie and Z. Luo, “Existence of three distinct solutions to boundary value problems of nonlinear differential equations with a $p$-Laplacian operator,”
*Applied Mathematics Letters*, vol. 27, pp. 101–106, 2014. View at Publisher · View at Google Scholar · View at MathSciNet - J. Mawhin and M. Willem,
*Critical Point Theory and Hamiltonian Systems*, vol. 74, Springer, New York, NY, USA, 1989. View at MathSciNet - J. J. Nieto, “Variational formulation of a damped Dirichlet impulsive problem,”
*Applied Mathematics Letters*, vol. 23, no. 8, pp. 940–942, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Xiao and J. J. Nieto, “Variational approach to some damped Dirichlet nonlinear impulsive differential equations,”
*Journal of the Franklin Institute*, vol. 348, no. 2, pp. 369–377, 2011. 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, Providence, RI, USA, 1986. View at MathSciNet