- Research Article
- Open Access

# Variational Approach to Impulsive Differential Equations with Dirichlet Boundary Conditions

- Huiwen Chen
^{1}and - Jianli Li
^{1}Email author

**2010**:325415

https://doi.org/10.1155/2010/325415

© Huiwen Chen and Jianli Li. 2010

**Received:**18 September 2010**Accepted:**9 November 2010**Published:**24 November 2010

## Abstract

## Keywords

- Variational Method
- Dirichlet Boundary
- Dirichlet Boundary Condition
- Fixed Point Theorem
- Bounded Sequence

## 1. Introduction

Impulsive effects exist widely in many evolution processes in which their states are changed abruptly at certain moments of time. Such processes are naturally seen in control theory [1, 2], population dynamics [3], and medicine [4, 5]. Due to its significance, a great deal of work has been done in the theory of impulsive differential equations. In recent years, many researchers have used some fixed point theorems [6, 7], topological degree theory [8], and the method of lower and upper solutions with monotone iterative technique [9] to study the existence of solutions for impulsive differential equations.

On the other hand, in the last few years, some researchers have used variational methods to study the existence of solutions for boundary value problems [10–16], especially, in [14–16], the authors have studied the existence of infinitely many solutions by using variational methods.

However, as far as we know, few researchers have studied the existence of distinct pairs of nontrivial solutions for impulsive boundary value problems by using variational methods.

where , , , , , , and denote the right and the left limits, respectively, of at , .

## 2. Preliminaries

Definition 2.1.

Suppose that is a Banach space and . If any sequence for which is bounded and as possesses a convergent subsequence in , we say that satisfies the Palais-Smale condition.

Let be a real Banach space. Define the set as symmetric closed set}.

Theorem 2.2 (see [17, Theorem 3.5.3]).

Let be a real Banach space, and let be an even functional which satisfies the Palais-Smale condition, is bounded from below and ; suppose that there exists a set and an odd homeomorphism and , then has at least n distinct pairs of nontrivial critical points.

Hence, is reflexive. We define the norm in as .

For
, we have that
and
are absolutely continuous and
. Hence,
for every
. If
, then
is absolutely continuous and
. In this case, the one-sided derivatives
,
may not exist. As a consequence, we need to introduce a different concept of solution. Suppose that
such that for every
,
satisfies
, and it satisfies the equation in problem (1.1) for
, a.e.
, the limits
, and
exist, and impulsive conditions and boundary conditions in problem (1.1) hold, we say it is *a classical solution* of problem (1.1).

for any . Obviously, is continuous.

Lemma 2.3.

If is a critical point of the functional , then is a classical solution of problem (1.1).

Proof.

The proof is similar to the proof of [16, Lemma 2.4], and we omit it here.

Lemma 2.4.

Proof.

which completes the proof.

## 3. Main Results

Theorem 3.1.

Suppose that the following conditions hold.

(ii) is odd about u and for every .

Then for any , there exists such that , and problem (1.1) has at least distinct pairs of nontrivial classical solutions.

Proof.

By (2.4), (ii), and (iii), is an even functional and .

for any . That is, is bounded from below.

In view of (3.5), (3.6), and (3.7), we obtain . Then, satisfies the Palais-Smale condition.

By Theorem 2.2, possesses at least distinct pairs of nontrivial critical points. That is, problem (1.1) has at least distinct pairs of nontrivial classical solutions.

Corollary 3.2.

Let the following conditions hold:

(ii) is odd about u and for every ,

Then, for any , there exists such that , and problem (1.1) has at least distinct pairs of nontrivial classical solutions.

Proof.

Let in Theorem 3.1, then Corollary 3.2 holds.

Theorem 3.3.

Suppose that the following conditions hold.

(iii) and are odd about u and for every .

Then, for any , there exists such that , and problem (1.1) has at least distinct pairs of nontrivial classical solutions.

Proof.

By (2.4) and (iii), is an even functional and .

for any . That is, is bounded from below.

It follows that is bounded in . In the following, the proof of the Palais-Smale condition is the same as that in Theorem 3.1, and we omit it here.

By Theorem 2.2, possesses at least distinct pairs of nontrivial critical points. That is, problem (1.1) has at least distinct pairs of nontrivial classical solutions.

Corollary 3.4.

Let the following conditions hold:

(iii) and are odd about u and for every .

Then, for any , there exists such that , and problem (1.1) has at least distinct pairs of nontrivial classical solutions.

Proof.

Let and in Theorem 3.3, then Corollary 3.4 holds.

Theorem 3.5.

Suppose that the following conditions hold.

(i) There exist constants such that for every .

Proof.

Thus, there exist constants such that for any . We consider the following two possible cases.

Case 1.

That is, for any . So, there exists a constant such that , which contradicts (3.20). Then, . Similarly, we can prove that .

Case 2.

, the arguments are analogous, then is solution of problem (1.1).

for any . Obviously, is continuous. By Lemma 2.3, we have the critical points of as solutions of problem (3.19). By (3.24), (ii), and (iii), is an even functional and .

for any . That is, is bounded from below.

It follows that is bounded in . In the following, the proof of the Palais-Smale condition is the same as that in Theorem 3.1, and we omit it here.

By Theorem 2.2, possesses at least distinct pairs of nontrivial critical points. Then, problem (3.19) has at least distinct pairs of nontrivial classical solutions, that is, problem (1.1) has at least distinct pairs of nontrivial classical solutions

Theorem 3.6.

Let the following conditions hold.

(i) There exist constants such that for every .

Proof.

The proof is similar to the proof of Theorem 3.5, and we omit it here.

Theorem 3.7.

Let the following conditions hold.

(i) There exist constants such that .

(iii) and are odd about and uniformly for .

Proof.

That is, for any . So, there exists a constant such that , which contradicts (3.36). Then . Similarly, we can prove that . Then, is solution of problem (1.1).

for any . Obviously, is continuous. By Lemma 2.3, we have the critical points of as solutions of problem (3.35). By (3.40) and (iii), is an even functional and .

for any . That is, is bounded from below.

It follows that is bounded in . In the following, the proof of the Palais-Smale condition is the same as that in Theorem 3.1, and we omit it here.

Let , then for any . Then, for any .

By Theorem 2.2, possesses at least distinct pairs of nontrivial critical points. Then, problem (3.35) has at least distinct pairs of nontrivial classical solutions, that is, problem (1.1) has at least distinct pairs of nontrivial classical solutions.

Theorem 3.8.

Let the following conditions hold.

(i) There exist constants such that .

(iii) and are odd about and for any .

Proof.

The proof is similar to the proof of Theorem 3.7, and we omit it here.

## 4. Some Examples

Example 4.1.

then . Applying Theorem 3.1, then for any , when , problem (4.1) has at least distinct pairs of nontrivial classical solutions.

Example 4.2.

then . Applying Theorem 3.3, then for any , when , problem (4.3) has at least distinct pairs of nontrivial classical solutions.

Example 4.3.

then . Applying Theorem 3.5, then for any , when , problem (4.5) has at least distinct pairs of nontrivial classical solutions.

Example 4.4.

then . Applying Theorem 3.7, then for any , when , problem (4.7) has at least distinct pairs of nontrivial classical solutions.

## Declarations

### Acknowledgments

This work was supported by the NNSF of China (no. 10871062) and a project supported by Hunan Provincial Natural Science Foundation of China (no. 10JJ6002).

## Authors’ Affiliations

## References

- George RK, Nandakumaran AK, Arapostathis A:
**A note on controllability of impulsive systems.***Journal of Mathematical Analysis and Applications*2000,**241**(2):276-283. 10.1006/jmaa.1999.6632MathSciNetView ArticleMATHGoogle Scholar - Jiang G, Lu Q:
**Impulsive state feedback control of a predator-prey model.***Journal of Computational and Applied Mathematics*2007,**200**(1):193-207. 10.1016/j.cam.2005.12.013MathSciNetView ArticleMATHGoogle Scholar - Nenov SI:
**Impulsive controllability and optimization problems in population dynamics.***Nonlinear Analysis. Theory, Methods & Applications*1999,**36:**881-890. 10.1016/S0362-546X(97)00627-5MathSciNetView ArticleMATHGoogle Scholar - Choisy M, Guégan J-F, Rohani P:
**Dynamics of infectious diseases and pulse vaccination: teasing apart the embedded resonance effects.***Physica D*2006,**223**(1):26-35. 10.1016/j.physd.2006.08.006MathSciNetView ArticleMATHGoogle Scholar - Gao S, Chen L, Nieto JJ, Torres A:
**Analysis of a delayed epidemic model with pulse vaccination and saturation incidence.***Vaccine*2006,**24**(35-36):6037-6045. 10.1016/j.vaccine.2006.05.018View ArticleGoogle Scholar - Li J, Nieto JJ:
**Existence of positive solutions for multipoint boundary value problem on the half-line with impulses.***Boundary Value Problems*2009,**2009:**-12.Google Scholar - Chu J, Nieto JJ:
**Impulsive periodic solutions of first-order singular differential equations.***Bulletin of the London Mathematical Society*2008,**40**(1):143-150. 10.1112/blms/bdm110MathSciNetView ArticleMATHGoogle Scholar - Qian D, Li X:
**Periodic solutions for ordinary differential equations with sublinear impulsive effects.***Journal of Mathematical Analysis and Applications*2005,**303**(1):288-303. 10.1016/j.jmaa.2004.08.034MathSciNetView ArticleMATHGoogle Scholar - Chen L, Sun J:
**Nonlinear boundary value problem of first order impulsive functional differential equations.***Journal of Mathematical Analysis and Applications*2006,**318**(2):726-741. 10.1016/j.jmaa.2005.08.012MathSciNetView ArticleMATHGoogle Scholar - Nieto JJ, O'Regan D:
**Variational approach to impulsive differential equations.***Nonlinear Analysis. Real World Applications*2009,**10**(2):680-690. 10.1016/j.nonrwa.2007.10.022MathSciNetView ArticleMATHGoogle Scholar - Tian Y, Ge W:
**Applications of variational methods to boundary-value problem for impulsive differential equations.***Proceedings of the Edinburgh Mathematical Society. Series II*2008,**51**(2):509-527.MathSciNetView ArticleMATHGoogle Scholar - Tian Y, Ge W, Yang D:
**Existence results for second-order system with impulse effects via variational methods.***Journal of Applied Mathematics and Computing*2009,**31**(1-2):255-265. 10.1007/s12190-008-0208-3MathSciNetView ArticleMATHGoogle Scholar - Zhang H, Li Z:
**Variational approach to impulsive differential equations with periodic boundary conditions.***Nonlinear Analysis. Real World Applications*2010,**11**(1):67-78. 10.1016/j.nonrwa.2008.10.016MathSciNetView ArticleMATHGoogle Scholar - Zhang Z, Yuan R:
**An application of variational methods to Dirichlet boundary value problem with impulses.***Nonlinear Analysis. Real World Applications*2010,**11**(1):155-162. 10.1016/j.nonrwa.2008.10.044MathSciNetView ArticleMATHGoogle Scholar - Zhou J, Li Y:
**Existence and multiplicity of solutions for some Dirichlet problems with impulsive effects.***Nonlinear Analysis. Theory, Methods & Applicationss*2009,**71**(7-8):2856-2865. 10.1016/j.na.2009.01.140View ArticleMATHMathSciNetGoogle Scholar - Sun J, Chen H:
**Variational method to the impulsive equation with Neumann boundary conditions.***Boundary Value Problems*2009,**2009:**-17.Google Scholar - Guo D, Sun J, Liu Z:
*Functional Methods of Nonlinear Ordinary Differential Equations*. Shandong Science and Technology Press; 2005.Google Scholar

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