- 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

We study the existence of distinct pairs of nontrivial solutions for impulsive differential equations with Dirichlet boundary conditions by using variational methods and critical point theory.

## 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.

Let , then .

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 .

(iii) are odd and for any .

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.

then for any .

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:

(i) is bounded,

(ii) is odd about u and for every ,

(iii) are odd and for any .

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.

Then, for any .

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:

(i) is bounded,

(ii) are bounded,

(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 .

(ii) is odd about .

(iii) are odd and for any .

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).

where .

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.

Then, for any .

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 .

(iii) and are odd about .

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).

where .

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 .

(ii) uniformly for .

(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

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