Abstract
In this paper, we study the existence of periodic solutions of second order impulsive differential equations at resonance with impulsive effects. We prove the existence of periodic solutions under a generalized Lazer-Leach type condition by using variational method. The impulses can generate a periodic solution.
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1 Introduction
We are concerned with the periodic boundary value problem of second order impulsive differential equations at resonance
where , is a continuous function, , , and is continuous for every j.
When , problem (1.1) becomes the well-known periodic boundary value problem at resonance
Assume that
exist and are finite. Lazer and Leach [1] proved that (1.2) has at least one 2π-periodic solution provided that the following condition holds:
From then on, a series of relevant resonant problems were studied (see [2]–[5] and the references cited therein) by some classical tools such as topological degree method, variational method, etc. Recently, the periodic problem of the second order differential equation with impulses has been widely studied because of its background in applied sciences (see [6]–[18] and the references cited therein). In this paper, we investigate problem (1.1) under a more general Lazer-Leach type condition. Define
and for ,
Throughout this paper, we give the following fundamental assumptions.
(H1): The limits
exist and are finite.
(H2): There exist continuous, 2π-periodic functions such that for ,
(H3): For all ,
For the sake of convenience, we decompose (H3) into the following two conditions.
(): For all ,
(): For all ,
We now can state the main theorems of this paper.
Theorem 1.1
Assume that conditions (H1), (H2) and () hold. Then problem (1.1) has at least one 2π-periodic solution.
Theorem 1.2
Assume that conditions (H1), (H2) and () hold. Then problem (1.1) has at least one 2π-periodic solution.
From Theorem 1.1 and Theorem 1.2, we obtain the following theorem.
Theorem 1.3
Assume that conditions (H1), (H2) and (H3) hold. Then problem (1.1) has at least one 2π-periodic solution.
Moreover, we have the following corollary.
Corollary 1.4
Assume that conditions (H1) and
(): for all,
hold. Then problem (1.2) has at least one 2π-periodic solution.
Remark 1.5
It is easy to find a function such that (g) is not satisfied and (H1) holds. For example, we can take . Hence, Corollary 1.4 improves the related results in the literature mentioned above. Moreover, since we consider the problem with impulses, Theorem 1.3 is also a complement of the pioneering works.
Remark 1.6
When condition () is not satisfied, i.e., there exists such that
problem (1.2) may have no solution. For example, we consider the resonant differential equation
Obviously, , and , . We have
We take such that . Then () is not satisfied. From now on, we prove that (1.5) has no 2π-periodic solution by contradiction. Assume that (1.5) has 2π-periodic solution. Multiplying both sides of (1.5) by and integrating over , we get
which is impossible. Hence, problem (1.2) may have no solution if condition () is not satisfied. Now, we give the following boundary value condition:
and the impulsive condition
Clearly, and . Then
Hence, (H1), (H2) and (H3) hold. Equivalently, Eq. (1.5) with conditions (1.6) and (1.7) has at least one 2π-periodic solution. Therefore, the impulses in problem (1.1) 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 and Theorem 1.2.
2 Preliminaries
In the following, we introduce some notations and some necessary definitions.
Define
with the norm
Consider the functional defined on H by
Similarly as in [18], is continuously differentiable on H, and
Now, we have the following lemma.
Lemma 2.1
Ifis a critical point of φ, then x is a 2π-periodic solution of Eq. (1.1).
The proof of Lemma 2.1 is similar to Lemma 2.1 in [9], 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 [19] (or see [20]).
Theorem 2.2
Letand, , . We suppose that:
-
(a)
there exist a bounded neighborhood D of 0 in and a constant α such that ;
-
(b)
there exists a constant such that ;
-
(c)
φ satisfies (PS).
Then the functional φ has a critical point in H.
3 The proof of the main results
In this section, we first show that the functional φ satisfies the Palais-Smale condition.
Lemma 3.1
Assume that conditions (H1), (H2) and (H3) hold. Then φ defined by (2.1) satisfies (PS).
Proof
Let be a constant and be a sequence satisfying
and
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 ,
Thus
Hence
By (H1) and (H2), we have
From (3.3) and (3.4), we obtain
Set
Then we have
and furthermore,
Replacing in (3.6) by , we get
Due to the compact imbedding , going to a subsequence,
Therefore,
Furthermore, we have
which implies that is a Cauchy sequence in H. Thus, in H. It follows from (3.5) and the usual regularity argument for ordinary differential equations (see [21]) 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 (H1) and (H2), 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
Passing to the limits, we have
Noting that in H as and
where , , we get from the Lebesgue domain convergence theorem that
i.e.,
which contradicts (H3). This implies that the sequence is bounded. Thus, there exists such that weakly in H. Due to the compact imbedding and , going to a subsequence,
From (3.3), we obtain
Replacing by in the above equality, we get
By (H1) and (H2), we have
and
Thus, it follows from (3.12), (3.13) and (3.14) that
Therefore,
which implies in H. It shows that φ satisfies (PS). □
Remark 3.2
If conditions (H1), (H2) and () (or ()), φ defined by (2.1) still satisfies (PS).
Now, we can give the proof of Theorem 1.1.
Proof of Theorem 1.1
Denote
and
We first prove that
by contradiction. Assume that there exists a sequence such that (as ) and there exists a constant satisfying
By (H1), we have
By (H2), we get
From (3.16) and the definition of φ, we obtain
For , we get that there exist constants , such that
Since for , we have, for ,
Hence, for ,
The equality in (3.20) holds only for
Set . Since , going to a subsequence, there exists such that in H and in . Then (3.17), (3.18), (3.19) and (3.20) imply that
By (3.16), we have, for n large enough,
It follows from that
From (3.21) and (3.22), we get, for n large enough,
Passing to the limits and using an argument similarly as in the proof of Lemma 3.1, we get
which is a contradiction to ().
Then (3.15) holds.
Next, we prove that
and φ is bounded on bounded sets.
Because of the compact imbedding of and , there exist constants , such that
Then by (H1) and (H2), one has that there exist positive constants such that
Hence, φ is bounded on the bounded sets of H.
For , using an argument similar to the case , we have
Thus, from (3.23) and (3.24), we obtain
which implies
Up to now, the conditions (a) and (b) of Theorem 2.2 are satisfied. According to Remark 3.2, (c) is also satisfied. Hence, by Theorem 2.2, problem (1.1) has at least one solution. This completes the proof. □
Next, we prove Theorem 1.2 slightly differently from Theorem 1.1.
Proof of Theorem 1.2
Denote
and
We first prove that
For , we get that there exist constants , such that
Since for , we have, for ,
Hence, for ,
The equality holds only for
If and , then
For , we have
By (H1) and (H2), we get that there exists a constant such that
Hence, for , we obtain
Therefore, (3.25) holds.
Next, we prove that
and φ is bounded on bounded sets.
Because of the compact imbedding of and , there exist constants , such that
Then, by (H1) and (H2), one has that there exist positive constants such that
Hence, φ is bounded on the bounded sets of H.
In what follows, we prove that
by contradiction. Assume that there exists a sequence such that (as ), and there exists a constant satisfying
By (H1), we have
By (H2), we get
From (3.26) and the definition of φ, we obtain
For , we get
Hence, for , we have
The equality in (3.30) holds only for
Set . There exists such that weakly in H. Due to the compact imbedding , going to a subsequence, in . Then (3.27), (3.28), (3.29) and (3.30) imply that
By (3.26), we have, for n large enough,
It follows from that
From (3.31) and (3.32), we get, for n large enough,
Passing to the limits and using an argument similarly as in the proof of Lemma 3.1, we get
which is a contradiction to (). This completes the proof. □
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Acknowledgements
The author would like to express their thanks to the editor of the journal and the referees for their careful reading of the first draft of the manuscript and making many helpful comments and suggestions which improved the presentation of the paper. This research was supported by the National Natural Science Foundation of China (11401274), Science and Technology Landing Project of colleges and universities in Jiangxi Province (KJLD14092) and Natural Science Foundation Project of Science and Technology Department of Jiangxi Province (20132BAB201012, 20142BDH80027).
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Li, J. Lazer-Leach type condition for second order differential equations at resonance with impulsive effects via variational method. Bound Value Probl 2014, 233 (2014). https://doi.org/10.1186/s13661-014-0233-0
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DOI: https://doi.org/10.1186/s13661-014-0233-0