RETRACTED ARTICLE: Landesman-Lazer type condition for second-order differential equations at resonance with impulsive effects
© Li and Zheng; licensee Springer. 2014
Received: 10 June 2014
Accepted: 5 August 2014
Published: 9 September 2014
The Retraction Note to this article has been published in Advances in Difference Equations 2015 2015:204
In this paper, we study the existence of periodic solutions of second-order impulsive differential equations at resonance. We prove the existence of periodic solutions under a generalized Landesman-Lazer type condition by using the variational method. The impulses can generate a periodic solution.
where , is a Carathéodory function, , , and is continuous for every j.
Throughout this paper, we give the following fundamental assumptions.
() There exists such that , for a.e. and for all .
We now can state the main theorem of this paper.
Theorem 1.1 Assume that the conditions (), (), and () hold. Then problem (1.1) has at least one 2π-periodic solution.
where are constants. Hence, .
From Theorem 1.1, we obtain the following result.
Corollary 1.2 Assume that we have the conditions (), (1.4), and the following.
hold. Then problem (1.1) has at least one 2π-periodic solution.
Moreover, we have the following corollary.
Corollary 1.3 Assume that we have the conditions () and the following.
holds. Then problem (1.2) has at least one 2π-periodic solution.
A simple example illustrates it. Thus condition () generalizes condition (1.3). Hence, our results improve the related results in the literature mentioned above. Moreover, since we consider the problem with impulses, Theorem 1.1 is also a complement of the pioneering works.
which is impossible. Hence, problem (1.2) may have no solution if the condition () is not satisfied. However, as long as () holds, problem (1.1) will have at least one periodic solution. Therefore, the impulses can generate a periodic solution.
The rest of the paper is organized as follows. In Section 2, we shall state some notations, some necessary definitions, and a saddle theorem due to Rabinowitz. In Section 3, we shall prove Theorem 1.1.
In the following, we introduce some notations and some necessary definitions.
Now, we have the following lemma.
Lemma 2.1 If is a critical point of φ, then x is a 2π-periodic solution of (1.1).
The proof of Lemma 2.1 is similar to Lemma 2.1 in , so we omit it.
We say that φ satisfies (PS) if every sequence for which is bounded in ℝ and (as ) possesses a convergent subsequence.
There exists a bounded neighborhood D of 0 in and a constant α such that ;
there exists a constant such that ;
φ satisfies (PS).
Then the functional φ has a critical point in H.
3 The proof of Theorem 1.1
In this section, we first show that the functional φ satisfies the Palais-Smale condition.
Lemma 3.1 Assume that the conditions (), (), and () hold. Then φ defined by (2.1) satisfies (PS).
where (). (Different subsequences of correspond to different and .)
where θ satisfies and .
which implies in H. It shows that φ satisfies (PS). □
Now, we can give the proof of Theorem 1.1.
which is a contradiction to ().
Then (3.18) holds.
and φ is bounded on bounded sets.
Hence, φ is bounded on bounded sets of H.
Up to now, the conditions (a) and (b) of Theorem 2.2 are satisfied. According to Lemma 3.1, (c) is also satisfied. Hence, by Theorem 2.2, (1.1) has at least one solution. This completes the proof. □
The authors would like to express their thanks to the editor of the journal and the referees for their carefully reading of the first draft of the manuscript and making many helpful comments and suggestions which improved the presentation of the paper.
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