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.
Fečkan , Battelli and Fečkan  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:
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 , Smets and Willem , 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 constant such that
(H4) There exist constants and such that
(H5) There exists a constantb, with , such that
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 )
LetEbe a Banach space and , , be such that and
Then, for each , , there exists such 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
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
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 Cauchy-Schwarz inequality that
which implies .
Furthermore, from the above argument, we have
that is, .
Finally, we obtain that
The proof is complete. □
Define the functional as follows:
Lemma 2.4If (H1)-(H5) hold, then and
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, ,
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 2.5If is 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 (H1)-(H5), there exists and such that and
Proof If and , then, by Lemma 2.3, . Hence, by (H5) and Lemma 2.3, we have
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 sequence inHsuch 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
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 (H5) and (2.10) that
a contradiction. The proof is complete. □
The following lemma is similar to a weak version of Lieb’s lemma , which will play an important role in the proof of Theorem 1.1.
Lemma 2.8If is bounded inHand does not converge to 0 in measure, then there exist a sequence and a subsequence of such that
then, for any , there exists such that, for , we have
Therefore, for all and , we have
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
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
Since , in H, therefore
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
it follows from Lemma 2.3 that
Similarly, we have
By the Cauchy-Schwarz inequality, we have
From (2.11)-(2.14), we have
Thus, and ω is a nontrivial weak homoclinic orbit of (1.1)-(1.2). □
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript and read and approved the final manuscript.
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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