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.
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
Then there exists a nontrivial weak homoclinic orbit of Eqs. (1.1) and (1.2).
2 Proof of main results
Lemma 2.1 (Mountain pass lemma )
We now prove some technical lemmas.
Lemma 2.2The space
is a Hilbert space with the inner product
and the corresponding norm
It is easy to see that
Furthermore, from the above argument, we have
Finally, we obtain that
The proof is complete. □
Therefore, we have
by the definition of the weak derivative, which implies
It follows from (2.8), (H4) and Lemma 2.3 that
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
Therefore, by Lemma 2.3, we have
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.
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
Hence, we have
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
it follows from Lemma 2.3 that
Similarly, we have
By the Cauchy-Schwarz inequality, we have
From (2.11)-(2.14), we have
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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