Abstract
A sufficient condition is obtained for the existence of nontrivial weak homoclinic orbits of secondorder 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.[36] 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 [712]). However, few papers have been published on the existence of homoclinic solutions for secondorder impulsive differential equations via variational methods.
In this paper, we consider the following impulsive differential equations:
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:
(H_{1}) There exists a positive numberTsuch that
(H_{3}) There exists a constantsuch that
(H_{4}) There exist constantsandsuch that
(H_{5}) 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 (H_{2}) implies that is an equilibrium of (1.1)(1.2).
Remark 1.2 Set , . It is easy to see that , satisfy (H_{1})(H_{5}).
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
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
is a Hilbert space with the inner product
and the corresponding norm
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
which implies that , that is, , . Therefore, q is continuous. Thus, and .
which implies . On the other hand, since
Therefore,
Consequently, and converges to q in H. The proof is complete. □
Lemma 2.3For any, the following inequalities hold:
Proof For any , there exists an integer k such that . Then it follows from CauchySchwarz inequality that
Furthermore, from the above argument, we have
Since
Finally, we obtain that
The proof is complete. □
Define the functional as follows:
Lemma 2.4If (H_{1})(H_{5}) hold, thenand
Proof From the continuity of V, and (H_{2})(H_{3}), we see that, for each , there exists , such that
Since as , there exists such that
Therefore, we have
It follows from (H_{5}) that, ,
and
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
Hence, the critical point of the functional φ satisfies (1.1). The proof is complete. □
Lemma 2.6Under the assumptions (H_{1})(H_{5}), there existsandsuch thatand
Proof If and , then, by Lemma 2.3, . Hence, by (H_{5}) and Lemma 2.3, we have
and
It follows from (2.8), (H_{4}) 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), (H_{4}) and Lemma 2.3, we have
Since , the righthand member is negative of λ sufficiently large, and there exists such that , . The proof is complete. □
Lemma 2.7Under the assumptions (H_{1})(H_{5}), 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 (H_{3}), (H_{5}), (2.4), (2.5), (2.8) and (2.9) that
Let . By (H_{2}) and (H_{3}), we have
which implies
For any , there exists such that, for , we have
Therefore, by Lemma 2.3, we have
If converges to 0 in measure on R, then it follows from (H_{5}) 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
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
Hence, we have
which implies
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
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 CauchySchwarz inequality, we have
Therefore,
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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