Variational Approach to Impulsive Differential Equations with Dirichlet Boundary Conditions

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.

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.

Motivated by the above facts, in this paper, our aim is to study the existence of distinct pairs of nontrivial solutions to the Dirichlet boundary problem for the second-order impulsive differential equations

(1.1)

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.

To begin with, we introduce some notation. Denote by the Sobolev space , and consider the inner product

(2.1)

and the norm

(2.2)

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).

Consider the functional

(2.3)

defined by

(2.4)

where . Clearly, is a Fréchet differentiable functional, whose Fréchet derivative at the point is the functional given by

(2.5)

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.

For , then . Hence, for , by Hölder's inequality, we have

(2.6)

which completes the proof.

3. Main Results

Theorem 3.1.

Suppose that the following conditions hold.

(i) There exist and such that

(3.1)

(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 .

Next, we will verify that is bounded from below. In view of (i), (iii), and Lemma 2.4, we have

(3.2)

for any . That is, is bounded from below.

In the following we will show that satisfies the Palais-Smale condition. Let , such that is a bounded sequence and . Then, there exists such that

(3.3)

In view of (3.2), we have

(3.4)

So is bounded in . From the reflexivity of , we may extract a weakly convergent subsequence that, for simplicity, we call , in . Next, we will verify that strongly converges to in . By (2.5), we have

(3.5)

By in , we see that uniformly converges to in . So,

(3.6)

By and , we have

(3.7)

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

Let , , then

(3.8)

Define

(3.9)

Then, for any , there exists an odd homeomorphism . Let , then for any . By (ii), we have

(3.10)

then for any .

Let , , then , . Let , then when , for any , we have

(3.11)

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.

(i) There exists and such that

(3.12)

(ii) There exists and such that

(3.13)

(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 .

Next, we will verify that is bounded from below. Let , . In view of (i), (ii), and Lemma 2.4, we have

(3.14)

for any . That is, is bounded from below.

In the following, we will show that satisfies the Palais-Smale condition. As in the proof of Theorem 3.1, by (3.3) and (3.14), we have

(3.15)

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.

Take the same as in Theorem 3.1, then for any , there exists an odd homeomorphism . Let , then for any . By (iii), we have

(3.16)

Then, for any .

Let , , then . Let , then when , for any , we have

(3.17)

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 .

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

Proof.

Let

(3.18)

then is continuous, bounded, and odd. Consider boundary value problem

(3.19)

Next, we will verify that the solutions of problem (3.19) are solutions of problem (1.1). In fact, let be the solution of problem (3.19). If , then there exists an interval such that

(3.20)

When , by (i), we have

(3.21)

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

Case 1.

, then is nondecreasing in . By and , we have

(3.22)

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 every , we consider the functional

(3.23)

defined by

(3.24)

where .

It is clear that is Fréchet differentiable at any and

(3.25)

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 .

In the following, we will show that is bounded from below. since for , thus

(3.26)

By (iii), we have

(3.27)

for any . That is, is bounded from below.

In the following we will show that satisfies the Palais-Smale condition. Let such that is a bounded sequence and . Then, there exists such that

(3.28)

By (3.27), we have

(3.29)

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.

Take the same as in Theorem 3.1, then, for any , there exists an odd homeomorphism . Let , then for any . By (i) and (ii), we have

(3.30)

Then, for any .

Let , , then , . Let , then when , for any , we have

(3.31)

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 .

(ii) There exist , and such that

(3.32)

(iii) and are odd about .

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

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 .

(ii) There exist , and such that

(3.33)

(iii) and are odd about and uniformly for .

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

Proof.

Let

(3.34)

then is continuous, bounded, and odd. Consider boundary value problem

(3.35)

Next, we will verify that the solutions of problem (3.35) are solutions of problem (1.1). In fact, let be the solution of problem (3.35). If , then there exists an interval such that

(3.36)

When , by (i), we have

(3.37)

Thus, is nondecreasing in . By and , we have

(3.38)

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 every , we consider the functional

(3.39)

defined by

(3.40)

where .

It is clear that is Fréchet differentiable at any and

(3.41)

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 .

Next, we will show that is bounded from below. Let , . since for , thus

(3.42)

By (ii) and Lemma 2.4, we have

(3.43)

for any . That is, is bounded from below.

In the following we will show that satisfies the Palais-Smale condition. Let such that is a bounded sequence and . Then, there exists such that

(3.44)

By (3.43), we have

(3.45)

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.

Take the same as in Theorem 3.1, then for any , there exists an odd homeomorphism . By (iii), for any , there exists , when , we have

(3.46)

Let , then for any . Then, for any .

Let , , then . Let , then when , for any , we have

(3.47)

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 .

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

Proof.

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

4. Some Examples

Example 4.1.

Consider boundary value problem

(4.1)

It is easy to see that conditions (i), (ii), and (iii) of Theorem 3.1 hold. Let

(4.2)

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

Example 4.2.

Consider boundary value problem

(4.3)

It is easy to see that conditions (i), (ii), and (iii) of Theorem 3.3 hold. Let ,

(4.4)

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

Example 4.3.

Consider boundary value problem

(4.5)

Let , it is easy to see that conditions (i), (ii), and (iii) of Theorem 3.5 hold. Let

(4.6)

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

Example 4.4.

Consider boundary value problem

(4.7)

Let , it is easy to see that conditions (i), (ii), and (iii) of Theorem 3.7 hold. Let ,

(4.8)

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