- Open Access
RETRACTED ARTICLE: Landesman-Lazer type condition for second-order differential equations at resonance with impulsive effects
Advances in Difference Equations volume 2014, Article number: 235 (2014)
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.
We are concerned with periodic boundary value problem of second-order impulsive differential equations at resonance
where , is a Carathéodory function, , , and is continuous for every j.
When , problem (1.1) becomes to the well-known periodic boundary value problem at resonance
There are many existence results for problem (1.2) in the literature. Let us mention some pioneering works by Lazer , Lazer and Leach , and Landesman and Lazer . In , a key sufficient condition for the existence of solutions of problem (1.2) is the so-called Landesman-Lazer condition,
It is well known that the theory of impulsive differential equations has been recognized to not only be richer than that of differential equations without impulses, but also to provide a more adequate mathematical model for numerous processes and phenomena studied in physics, biology, engineering, etc. We refer the reader to the book . Recently, the Dirichlet and periodic boundary conditions problems for second-order differential equations with impulses in the derivative and without impulses are studied by some authors via variational method [5–11]. In this paper, we will investigate problem (1.1) under a more general Landesman-Lazer type condition. Define
and for ,
Throughout this paper, we give the following fundamental assumptions.
() There exists such that , for a.e. and for all .
() There exist positive constants such that for all ,
() 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.
To demonstrate the impulsive effects clearly, we can take
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.
() For all ,
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.
() For all ,
holds. Then problem (1.2) has at least one 2π-periodic solution.
Remark 1.4 By a simple calculation, one can easily derive
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.
Remark 1.5 It is remarkable that Landesman-Lazer condition () is an ‘almost’ necessary and sufficient condition when and are replaced by and , where , , and (see [, p.70]). If the condition (1.5) is not satisfied, i.e., ,
problem (1.2) cannot be guaranteed to have periodic solution. For example, we consider resonant differential equation
Obviously, , , and , . Taking , we have
Then () is not satisfied. From now on, we prove that (1.6) has not 2π-periodic solution by contradiction. Assume that (1.6) has 2π-periodic solution. Multiplying both sides of (1.6) by and integrating over , we get
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.
with the norm
Consider the functional defined on H by
Similarly as in , is continuously differentiable on H, and
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.
To prove the main result, we will use the following saddle point theorem due to Rabinowitz  (or see ).
Theorem 2.2 Let and , , . We suppose that:
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).
Proof Let be a constant and be a sequence satisfying
We first prove that is bounded in H by contradiction. Assume that is unbounded. Let be an arbitrary sequence bounded in H. It follows from (3.2) that, for any ,
By () and (), we have
From (3.3) and (3.4), we obtain
Then we have
Replacing in (3.6) by , we get
Due to the compact embedding , going to a subsequence,
Furthermore, we have
which implies is Cauchy sequence in H. Thus, in H. It follows from (3.5) and the usual regularity argument for ordinary differential equations (see ) that
where (). (Different subsequences of correspond to different and .)
Write (3.7) as
where θ satisfies and .
Taking , we get, for any ,
Thus, it follows from (3.3) and (3.8) that
By () and (), we obtain
It follows from (3.9) and (3.10) that
Hence, replacing in (3.3) by , we have
Now, dividing (3.1) by , we get
Note that in H. Due to the compact embedding and , we have in . Furthermore,
Hence, from (3.11) and (3.12), we have
Using Fatou’s lemma, we get
Thus, by a simple computation, we have
Hence, it follows from (3.13) and (3.14) that
This contradicts (). It implies that the sequence is bounded. Thus, there exists such that weakly in H. Due to the compact embedding and , going to a subsequence,
From (3.3), we obtain
Replacing by in the above equality, we get
By () and (), we have
Thus, it follows from (3.15), (3.16), and (3.17) that
which implies in H. It shows that φ satisfies (PS). □
Now, we can give the proof of Theorem 1.1.
Proof of Theorem 1.1 Denote
We first prove that
by contradiction. Assume that there exists a sequence such that (as ) and there exists a constant satisfying
By (), we have
By (), we get
From (3.19) and the definition of φ, we obtain
For , we have
The equality in (3.23) holds only for
Set . Since , going to a subsequence, there exists such that in H and in . Then (3.20), (3.21), (3.22), and (3.23) imply that
By (3.19), we have, for n large enough,
It follows from that
From (3.24) and (3.25), we get, for n large enough,
Using an argument similar to the proof of Lemma 3.1, we get
which is a contradiction to ().
Then (3.18) holds.
Next, we prove that
and φ is bounded on bounded sets.
Because of the compact embedding of and , there exists constants , such that
Then by () and (), one has
Hence, φ is bounded on bounded sets of H.
Since , we have
Thus, from (3.26) and (3.27), we obtain
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. □
Lazer AC: On Schauder’s fixed point theorem and forced second order nonlinear oscillations. J. Math. Anal. Appl. 1968, 21: 421-425. 10.1016/0022-247X(68)90225-4
Lazer AC, Leach DE: Bounded perturbations of forced harmonic oscillators at resonance. Ann. Mat. Pura Appl. 1969, 82(4):49-68.
Landesman EM, Lazer AC: Nonlinear perturbations of linear elliptic boundary value problems at resonance. J. Math. Mech. 1970, 19: 60-623.
Lakshmikantham V, Bainov DD, Simeonov PS: Theory of Impulsive Differential Equations. World Scientific, Singapore; 1989.
Bogun I: Existence of weak solutions for impulsive p -Laplacian problem with superlinear impulses. Nonlinear Anal., Real World Appl. 2012, 13(6):2701-2707. 10.1016/j.nonrwa.2012.03.014
Ding W, Qian D: Periodic solutions for sublinear systems via variational approach. Nonlinear Anal., Real World Appl. 2010, 11(4):2603-2609. 10.1016/j.nonrwa.2009.09.007
Zhang Z, Yuan R: An application of variational methods to Dirichlet boundary value problem with impulses. Nonlinear Anal., Real World Appl. 2010, 11(1):155-162. 10.1016/j.nonrwa.2008.10.044
Nieto JJ, O’Regan D: Variational approach to impulsive differential equations. Nonlinear Anal., Real World Appl. 2009, 10(2):680-690. 10.1016/j.nonrwa.2007.10.022
Zhang H, Li Z: Variational approach to impulsive differential equations with periodic boundary conditions. Nonlinear Anal., Real World Appl. 2010, 11(1):67-78. 10.1016/j.nonrwa.2008.10.016
Zhang X, Meng Q: Nontrivial periodic solutions for delay differential systems via Morse theory. Nonlinear Anal. 2011, 74(5):1960-1968. 10.1016/j.na.2010.11.003
Tomiczek P: The Duffing equation with the potential Landesman-Lazer condition. Nonlinear Anal. 2009, 70(2):735-740. 10.1016/j.na.2008.01.006
Mawhin J, Willem M: Critical Point Theory and Hamiltonian Systems. Springer, Berlin; 1989.
Rabinowitz PH CBMS Reg. Conf. Ser. in Math. 65. In Minmax Methods in Critical Point Theory with Applications to Differential Equations. Am. Math. Soc., Providence; 1986.
Fučík S: Solvability of Nonlinear Equations and Boundary Value Problems. D. Reidel Publ. Company, Holland; 1980.
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.
The authors declare that they have no competing interests.
The first author has contributed in obtaining new results and written the whole article. The second author has written the references with BibTeX and formatted the manuscript such that it conforms to the journal style. All authors have also read and approved the final manuscript.
This article has been retracted by Professor Ravi P Agarwal, Editor-in-Chief of Advances in Difference Equations. Following publication of this article, it was brought to the attention of the editorial and publishing staff that this article has substantial overlap with an article by Jin Li, Jianlin Luo and Zaihong Wang, published in November 2014 in Mathematical Modelling and Analysis. This is a violation of publication ethics which, in accordance with the Springer Policy on Publishing Integrity, warrants a retraction of the article and a notice to this effect to be published in the journal.
An erratum to this article can be found online at http://dx.doi.org/10.1186/s13662-015-0446-2.
A retraction note to this article can be found online at http://dx.doi.org/10.1186/s13662-015-0446-2.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Li, J., Zheng, M. RETRACTED ARTICLE: Landesman-Lazer type condition for second-order differential equations at resonance with impulsive effects. Adv Differ Equ 2014, 235 (2014). https://doi.org/10.1186/1687-1847-2014-235
- impulsive differential equations
- Landesman-Lazer type condition
- variational method