Abstract and Applied Analysis

Volume 2013 (2013), Article ID 908062, 10 pages

http://dx.doi.org/10.1155/2013/908062

Research Article

Existence of Solution for Impulsive Differential Equations with Nonlinear Derivative Dependence via Variational Methods

Lizhao Yan,1,2 Jian Liu,³ and Zhiguo Luo²

¹Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, China

²Hunan Normal University Press, Changsha, Hunan 410081, China

³School of Economics and Management, Changsha University of Science and Technology, Changsha, Hunan 410004, China

Received 30 May 2013; Revised 1 August 2013; Accepted 22 August 2013

Academic Editor: M. Victoria Otero-Espinar

Copyright © 2013 Lizhao Yan 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.

We use variational methods and iterative methods to investigate the solutions of impulsive differential equations with nonlinear derivative dependence. The conditions for the existence of solutions are established. The main results are also demonstrated with examples.

1. Introduction

Many dynamical systems have an impulsive dynamical behavior due to abrupt changes at certain instants during the evolution process. The mathematical description of these phenomena leads to impulsive differential equations. Recent development in this field has been motivated by many applied problems, such as control theory, population dynamics, and medicine [1–9].

We consider the following nonlinear Dirichlet boundary value problems for impulsive differential equations:

where , , , is continuous, and , , are continuous.

The characteristic of (1) is the presence of the first order derivative in the nonlinearity 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 [10–14]. 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. Motivated by [15, 16], in this paper we will fill the gap in this area.

When there is no derivative in the nonlinearity term, problem (1) has been extensively studied by [17–23], using variational methods. We know, contrary to these equations, (1) is not variational and the well-developed critical point theory is of no avail for, at least, a direct attack to problem (1). The technique used in this paper consists of, associating with problem (1), a family of the following Dirichlet boundary value problems with no dependence on the derivative of the solution. Namely, for each , we consider the problem

Now problem (2) is variational and we can treat it by variational methods.

In this paper, we need the following conditions.

) is measurable in for every and continuous in for a.e. .() as uniformly for and and for and .() There exist constants and such that

() There exist constants and such that

 where .() There exist constants such that

() The function satisfies the following conditions:

 where is a constant.() are odd and nondecreasing, and there exist constants , , and such that

() There exists such that

(), for all , .

The paper is organized as follows: Section 2 is the preliminaries of this paper, Section 3 is devoted to show the solvability of problem (2), and Section 4 will show the solvability of problem (1).

2. Preliminaries

Firstly, we recall some facts which will be used in the proof of our main result. It has been shown, for instance, in [16] that the set of all eigenvalues of the following problem

is given by the sequence of positive numbers

where

Each eigenvalue is simple with the associated eigenfunction

We recall the well-known characterization of as the best constant in the Poincaré inequality

Let be the Sobolev space endowed with the norm

Throughout the paper, it will be assumed that . We also consider the norm

We need the following Lemmas.

Lemma 1 (see [24, Lemma 2.4]). If , then the norms and are equivalent.

Lemma 2 (see [24, Lemma 2.5]). There exists such that if , then

Lemma 3 (see [24, Lemma 2.6]). Let ; then there exists such that

For , we have that and are both absolutely continuous. Hence for any . If , then is absolutely continuous. In this case, may not hold for some . It leads to the impulsive effects. As a consequence, we need to introduce a different concept of solution.

Definition 4. A function

is said to be a classical solution of problem (1) if satisfies the equation a.e. on and the limits , and , exist and satisfy the impulsive condition

and the boundary condition .

We have the following fact. Take and multiply the equation in problem (1) by and integrate from 0 to :

The first term is now

Hence

Definition 5. We say that a function is a weak solution of problem (1) if the identity

holds for any .

Proposition 6. Under the hypotheses and , the functional defined by

is continuous and differentiable and

for any . Moreover, the critical point of is a classical solutions of problem (2).

Proof. Using the assumptions , we can obtain the continuity and differentiability of and that is defined by

for any . It follows that the critical point of is the weak solution of (2). Moreover, it is a classical solution of problem (2).

Evidently, since . By the definition of weak solution, we have

Choose with for every ; then

This implies that

Hence, for every . The impulsive condition in (2) is satisfied. This completes the proof.

We will obtain the critical points of by using the Mountain Pass Theorem. Therefore, we state this theorem precisely.

Lemma 7 (see [25]). Let be a real Banach space and satisfy (PS)-condition. Suppose that satisfies the following conditions:

  ;

there exists constants such that ;

there exists such that .

Then possesses a critical value given by

where is an open ball in of radius centered at 0 and

3. The Solvability of (2)

Theorem 8. Suppose that and hold; then there exist positive constants and such that, for each , problem (2) has one solution such that .

Proof. (I) We show that satisfies the (PS)-condition.

Assume that is a sequence such that is bounded and as . We will prove that the sequence is bounded. Obviously, there exists a constant such that

We set

From (17), (25), (26), and (33), , and , we have

By , we obtain that is bounded in .

Since is a reflexive Banach space, passing to a subsequence if necessary, we can assume that

Hence

Notice that

From [26, Lemma 4.2], we see that there exists , for any :

Combining this inequality with (39), we have

It follows from (37)–(39) that in . Hence, satisfies (PS)-condition.

(II) We verify assumption (ii) of Lemma 7.

By (), there exists such that

Since , it follows that

Using , we have

Hence, from (18), (25), (35), , and , for , we have

Set , . Equation (45) shows that implies that ; that is, satisfies assumption (ii) of Lemma 7.

(III) We verify assumption (iii) of Lemma 7.

By and , we know that for

Take such that . Since , , and , (46) implies that there exists such that , and if we set . By Lemma 7, possesses a critical value given by

where . Obviously, , so according to Lemma 7, there exists and , such that

(IV) We prove that .

Since is the solution of problem (2), we have

So

It follows from and that, given , there exists a positive constant , independent of , such that

Hence, using , we have

By Sobolev embedding theorem, we obtain

Set and ; then

which implies

From the inf max characterization of in (III), we obtain

with chosen in (III). We estimate using and :

whose maximum is achieved at some and the value can be taken as . Clearly it is independent of . Which implies

This completes the proof.

Lemma 9. Suppose that hold and , , and hold only for positive ; then there exist positive constants , and such that, for each , problem (2) has a positive solution such that .

Proof. Set

Consider the function

Obviously, satisfies , and satisfies , so using Theorem 8, we obtained a solution of (60). Multiplying the equation by and integrating by parks, we conclude that . So is positive.

Theorem 10. Suppose that and hold; then there exist positive constants and such that, for each , problem (2) has one solution such that .

Proof. (I) We show that satisfies the (PS)-condition.

Assume that is a sequence such that is bounded and as . We will prove that the sequence is bounded.

It follows from , , and that

Hence, is bounded in . The following proof of (PS)-condition is similar to that in (I) of Theorem 8.

(II) We verify assumption (ii) of Lemma 7.

Using , we have

The proof is similar to that in (II) of Theorem 8.

(III) We verify assumption (iii) of Lemma 7.

Take such that . By and , using Sobolev embedding theorem, we have

Since , (63) implies that there exists such that and if we set . By Lemma 7, possess a critical value , given by

where . Obviously, , so according to Lemma 7, there exists and , such that

(IV) We prove that .

Like (IV) of Theorem 8, we can obtain that there exist such that

It follows from satisfying (65) that

Let

Since , then can achieve its maximum at some , which implies there exist such that

This completes the proof.

Lemma 11. Suppose that hold and , , and hold only for positive , then there exist positive constants , and such that, for each , problem (2) has a positive solution such that .

Proof. The proof is similar to that of Lemma 9, so we omit it.

Remark 12. By the same method, we can discuss the negative solution of (2).

4. The Solvability of (1)

Theorem 13. Assume that and and hold; then problem (1) has one nontrivial solution provided

satisfying .

Proof. We construct a sequence as solutions of the following problem:

obtained in Theorem 8, starting with an arbitrary . It follows from the Sobolev embedding theorem that . Using and , we obtain

Hence,

Using (17) and (18) and by Hölder inequality, we have

that is,

Since is less than 1, then it follows that strongly converges in , as it easily follows proving that is a Cauchy sequence in . By Theorem 8, we know that . In this way we obtain a nontrivial solution of (1).

Example 14. Let and ; consider the following nonlinear Dirichlet impulsive problem:

Compared with (1), and . We can take and . Then by simple computation, it is easy to verify that all conditions of Theorem 13 are satisfied. Hence, by Theorem 13, problem (75) has one nontrivial solution.

Theorem 15. Assume that and and hold; then problem (1) has one nontrivial solution provided

satisfying .

Proof. We construct a sequence as solutions of the problem obtained in Theorem 10, starting with an arbitrary . The rest of proof is similar to that of Theorem 13, so we omit it.

Example 16. Let and ; consider the following nonlinear Dirichlet impulsive problem:

Compared with (1), and . We can take and . Then by simple computation, it is easy to verify that all conditions of Theorem 15 are satisfied. Hence, by Theorem 15, problem (77) has one nontrivial solution.

Remark 17. In order to discuss the positive solutions of (1), we can argue as in Theorem 8, by replacing by . By the same method, we can discuss the negative solutions of (1).

Acknowledgments

The authors are grateful to the referees for their useful suggestions. This work is partially supported by the National Natural Science Foundation of China (no. 71201013), the Humanities and Social Sciences Project of the Ministry of Education of China (no. 12YJC630118), the Innovation Platform Open Funds for Universities in Hunan Province (no. 13K059), and the Provincial Natural Science Foundation of Hunan (no. 11JJ3012).

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