Existence of nontrivial weak homoclinic orbits for second-order impulsive differential equations

Hui Fang* and Hongbo Duan

Author Affiliations

Department of Mathematics, Kunming University of Science and Technology, Kunming, Yunnan, 650500, China

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Boundary Value Problems 2012, 2012:138  doi:10.1186/1687-2770-2012-138

 Received: 25 July 2012 Accepted: 7 November 2012 Published: 26 November 2012

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A sufficient condition is obtained for the existence of nontrivial weak homoclinic orbits of second-order impulsive differential equations by employing the mountain pass theorem, a weak convergence argument and a weak version of Lieb’s lemma.

1 Introduction

Fečkan [1], Battelli and Fečkan [2] studied the existence of homoclinic solutions for impulsive differential equations by using perturbation methods. Tang et al.[3-6] studied the existence of homoclinic solutions for Hamiltonian systems via variational methods. In recent years, many researchers have paid much attention to multiplicity and existence of solutions of impulsive differential equations via variational methods (for example, see [7-12]). However, few papers have been published on the existence of homoclinic solutions for second-order impulsive differential equations via variational methods.

In this paper, we consider the following impulsive differential equations:

(1.1)

(1.2)

where is of class , with , and with . ℤ denotes the set of all integers, and () are impulsive points. Moreover, there exist a positive integer p and a positive constant T such that , , , . and represent the right and left limits of at respectively.

We say that a function is a weak homoclinic orbit of Eqs. (1.1) and (1.2) if q satisfies (1.1) and

Motivated by the works of Nieto and Regan [7], Smets and Willem [13], in this paper we study the existence of nontrivial weak homoclinic orbits of (1.1)-(1.2) by using the mountain pass theorem, a weak version of Lieb’s lemma and a weak convergence argument. Our method is different from those of [8,9].

The main result is the following.

Theorem 1.1Assume that Eqs. (1.1) and (1.2) satisfy the following conditions:

(H1) There exists a positive numberTsuch that

(H2) uniformly for;

(H3) There exists a constantsuch that

(H4) There exist constantsandsuch that

(H5) There exists a constantb, with, such that

and

Then there exists a nontrivial weak homoclinic orbit of Eqs. (1.1) and (1.2).

Remark 1.1 (H2) implies that is an equilibrium of (1.1)-(1.2).

Remark 1.2 Set , . It is easy to see that , satisfy (H1)-(H5).

2 Proof of main results

Lemma 2.1 (Mountain pass lemma [14])

LetEbe a Banach space and, , be such thatand

Let

Then, for each, , there existssuch that

(V1) ;

(V2) ;

(V3) .

In what follows, denotes the space of sequences whose second powers are summable on ℤ (the set of all integers), that is,

The space is equipped with the following norm:

We now prove some technical lemmas.

Lemma 2.2The space

(2.1)

is a Hilbert space with the inner product

(2.2)

and the corresponding norm

(2.3)

Proof Let be a Cauchy sequence in H, then is a Cauchy sequence in and there exists such that converges to y in . Define the function as follows:

It is easy to see that

Since , , we have

which implies that , that is, , . Therefore, q is continuous. Thus, and .

Noticing that, for , we have

which implies . On the other hand, since

and (), we have

Therefore,

Consequently, and converges to q in H. The proof is complete. □

Lemma 2.3For any, the following inequalities hold:

Furthermore, and

Proof For any , there exists an integer k such that . Then it follows from Cauchy-Schwarz inequality that

which implies .

Furthermore, from the above argument, we have

that is, .

Since

Finally, we obtain that

The proof is complete. □

Define the functional as follows:

(2.4)

Lemma 2.4If (H1)-(H5) hold, thenand

(2.5)

Proof From the continuity of V, and (H2)-(H3), we see that, for each , there exists , such that

Since as , there exists such that

Therefore, we have

It follows from (H5) that, ,

and

(2.6)

Thus, φ and the right hand of (2.5) is well defined on H. By the definition of Fréchet derivative, it is easy to see that and (2.5) holds. □

Lemma 2.5Ifis a critical point of the functionalφ, thenqsatisfies (1.1).

Proof If is a critical point of the functional φ, then for any , we have

, take such that for any , and . Therefore, we have

by the definition of the weak derivative, which implies

(2.7)

Hence, the critical point of the functional φ satisfies (1.1). The proof is complete. □

Lemma 2.6Under the assumptions (H1)-(H5), there existsandsuch thatand

Proof If and , then, by Lemma 2.3, . Hence, by (H5) and Lemma 2.3, we have

(2.8)

and

(2.9)

It follows from (2.8), (H4) and Lemma 2.3 that

Therefore, as and , there exists such that .

Now, let and . Then there exists a subset of ℝ and λ large enough such that

Since , by (2.4), (H4) and Lemma 2.3, we have

Since , the right-hand member is negative of λ sufficiently large, and there exists such that , . The proof is complete. □

Lemma 2.7Under the assumptions (H1)-(H5), there exists a bounded sequenceinHsuch that

where, . Furthermore, does not converge to 0 in measure.

Proof All we have to prove is that any sequence obtained by taking and in Lemma 2.1 is bounded and does not converge to 0 in measure. For n sufficiently large, it follows from (H3), (H5), (2.4), (2.5), (2.8) and (2.9) that

Since , is bounded in H.

Let . By (H2) and (H3), we have

which implies

For any , there exists such that, for , we have

Therefore, by Lemma 2.3, we have

(2.10)

If converges to 0 in measure on R, then it follows from (H5) and (2.10) that

a contradiction. The proof is complete. □

The following lemma is similar to a weak version of Lieb’s lemma [15], which will play an important role in the proof of Theorem 1.1.

Lemma 2.8Ifis bounded inHanddoes not converge to 0 in measure, then there exist a sequenceand a subsequenceofsuch that

Proof If

then, for any , there exists such that, for , we have

Therefore, for all and , we have

which implies

a contradiction. Therefore, there exist a constant and a subsequence of such that

where ℕ denotes the set of all positive integers. So, for , there exists such that

Let , . Since is bounded in H, by Lemma 2.3, it is easy to see that is bounded in . Therefore, has a subsequence which weakly converges to u in . Without loss of generality, we assume that in . Thus, in . Therefore, uniformly converges to u in . Noticing that

we have

that is, . □

Proof of Theorem 1.1 By Lemma 2.7, there exists a bounded in H such that

and does not converge to 0 in measure on ℝ, where d is the mountain pass value. By Lemma 2.8, there exists a sequence in ℤ such that

For any fixed , set and . Then () are impulsive points and

For any with , we have

Hence, we have

which implies

(2.11)

Since , in H, therefore

(2.12)

As in , is bounded in and hence for some and all . Also, uniformly converges to ω on and, being uniformly continuous on , uniformly converges to on . By the Lebesgue dominated convergence theorem, this implies that

(2.13)

For any and , take sufficiently large such that

Since in , in , therefore uniformly converges to ω in . By the continuity of I, there exists such that, when , we have

Since

it follows from Lemma 2.3 that

Similarly, we have

By the Cauchy-Schwarz inequality, we have

Therefore,

(2.14)

From (2.11)-(2.14), we have

Thus, and ω is a nontrivial weak homoclinic orbit of (1.1)-(1.2). □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the manuscript and read and approved the final manuscript.

Acknowledgements

This research is supported by the National Natural Science Foundation of China (Grant No. 10971085). The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the manuscript.

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