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Periodic solution of second-order impulsive delay differential system via generalized mountain pass theorem
Boundary Value Problems volume 2014, Article number: 234 (2014)
Abstract
In this paper we use variational methods and generalized mountain pass theorem to investigate the existence of periodic solutions for some second-order delay differential systems with impulsive effects. To the authors’ knowledge, there is no paper about periodic solution of impulses delay differential systems via critical point theory. Our results are completely new.
1 Introduction
In this paper, we study the following second-order delay differential systems with impulsive conditions:
where , , with . , for each ; there exists an such that , and for all ; is π-periodic in t and satisfies the following assumption:
-
(A)
is measurable in t for and continuously differentiable in x for a.e. , and there exist , such that
for all and a.e. . For convenience, we denote (1a)-(1c) as problem (IP).
Impulsive effects are important problems in the world due to the fact that some dynamics of processes will experience sudden changes depending on their states or at certain moments of time. For a second-order differential equation , one usually considers impulses in the position u and the velocity . However, for the motion of spacecraft one has to consider instantaneous impulses depending on the position, that result in jump discontinuities in velocity but with no change in position [1], [2]. Impulses only in the velocity occur also in impulsive mechanics [3]. Such impulsive problems with impulses in the derivative only have been considered in many literatures; see, for instance [4]–[11].
In recent years, impulsive and periodic boundary value problems have been studied by numerous mathematicians; see, for instance, [4], [12]–[15] and the references therein. Some classical tools such as fixed point theory, topological degree theory, the comparison method, the upper and lower solutions method and the monotone iterative method have been used to get the solutions of impulsive differential equations; we refer the reader to [5], [16]–[19] and the references therein.
Recently, some authors studied boundary value problems for second-order impulsive differential equations via variational methods (see [6]–[9], [20]–[26]).
On the other hand, in the past two decades, a wide variety of techniques, especially critical point theorem, have been developed to investigate the existence of the periodic solutions to the functional differential equations by several authors (see [10], [27], [28]). In 2009, by applying the critical theory and -index theory, Guo and Guo [28] obtained some results on the existence and multiplicity of periodic solutions for the delay differential equations
In [10], the non-autonomous second-order delay differential systems
were studied by a new critical point theorem.
Motivated by the above work, in this paper our main purpose is to apply the critical point directly to study problem (IP). To the best of our knowledge, there is no paper studying this delay differential systems under impulsive conditions via variational methods.
The rest of the paper is organized as follows: in Section 2, some preliminaries are given; in Section 3, the main result of this paper is stated, and finally we will give the proof of it.
2 Preliminaries
In this section, we recall some basic facts which will be used in the proofs of our main results. In order to apply the critical point theory, we construct a variational structure. With this variational structure, we can reduce the problem of finding solutions of (IP) to that of seeking the critical points of a corresponding functional.
Denote . Let
with the inner product
The corresponding norm is defined by
The space has some important properties: there are constants c such that
for all .
Let .
Definition 2.1
A function is said to be a classic periodic solution of (IP), if u satisfies equation in (1a) for all and (1b), (1c) hold.
Taking and multiplying the two sides of the equality
by v and integrating between 0 and 2π, we have
Thus consider a functional ϕ defined on , given by
Let be the space of square integrable 2π periodic vector-valued functions with dimension n, and be the space of 2π-periodic vector-valued functions with dimension n. For any , it has the following Fourier expansion in the sense that it is convergent in the space :
where . Moreover, we infer from the above decomposition of that the norm can be written as
It is well known that is compactly embedded in . Let L be an operator from to defined by the following form:
By the Riesz representation theorem, Lu can also be viewed as an element belonging to such that for any . It is easy to see that L is a bounded linear operator on . Set
then can be rewritten as
Lemma 2.1
L is selfadjoint on.
Proof
For any , we have
The proof is completed. □
Remark 2.1
It follows from assumption (A) and the continuity of , by a standard argument as in [29], that ϕ is continuously differentiable and weakly lower semi-continuous on . Moreover, we have
for and is weakly continuous. Moreover, is a compact operator defined by
Similarly to [8], we introduce the following concept for the solution of problem (IP).
Definition 2.2
We say that a function is a weak solution of problem (IP) if the identity
holds for any .
Since we have the following result, Definition 2.2 is suitable.
Lemma 2.2
Ifis a weak solution of (IP), then u is a classical solution of (IP).
Proof
If u is a weak solution of (IP), then for any
For any and such that if for , (5) implies
By the definition of weak derivative, the above equality implies
Since is π-periodic in t and , one has
Hence . A classical regularity argument shows that u is a classical solution of (6), which implies that is bounded for , and this implies that and exist. Thus we obtain
where . Since j is arbitrary in K and , (7) and (5) imply that
Therefore
for all with for . Since is dense in , (9) holds for all . Thus from (8) and (9), we have
which implies
for any , since v is arbitrary in . By (10), . Therefore u is a classical solution of (IP). The proof is completed. □
Definition 2.3
([29])
Let E be a real Banach space and . ϕ is said to satisfy the (PS) condition on E if any sequence for which is bounded and as possesses a convergent subsequence in E.
Let E be a Hilbert space with . Let , be the projections of E onto and , respectively. Set
where is compact.
Definition 2.4
Let , and Q be boundary. We call S and ∂Q link if whenever and for all t, then for all t.
Then [30] Theorem 5.29 can be stated as follows.
Theorem A
Let E be a real Hilbert space with, and inner product. Supposesatisfies (PS) condition, and
(I1): , whereandis bounded and selfadjoint (), where, be the projections of E ontoand, respectively,
(I2): is compact, and
(I3): there exist a subspace, sets, and constantssuch that
-
(i)
and ,
-
(ii)
Q is bounded and ,
-
(iii)
S and ∂Q link.
Then ϕ possesses a critical value.
3 Main results
In order to state our main results, we have to further assume the following hypotheses.
(H1): () satisfy
for all .
(H2): For any , there exist numbers and such that
for all .
(H3): There are constants , , , and such that
for all and with .
(H4): as and as uniformly for all t.
(H5): for all .
Theorem 3.1
Assume that (H1)-(H5) hold. Then problem (IP) has at least one periodic solution.
Example
There are many examples which satisfy (H1)-(H5). For example,
and , for .
Obviously, satisfy (H1)-(H2) and satisfies (H4)-(H5). Note that
for L being large enough. This implies (H3).
We will use Theorem A to prove Theorem 3.1.
Set and .
Lemma 3.1
and.
Proof
For any and , we have
which implies that , that is, .
For every , set
Then a simple calculation shows that and and . Then . Combining with , one has and . □
Remark 3.1
Lemma 3.1 is a new orthogonal decomposition different from the one in [10]. We will show that it is a useful result.
By (4) and Lemma 3.1, we have
for every , where , . Combining this with Remark 2.1 and Lemma 3.1, (I1) and (I2) of Theorem A hold for ϕ.
Now we prove that ϕ satisfies (PS) condition.
Lemma 3.2
Under the assumptions of Theorem 3.1, ϕ satisfies (PS) condition.
Proof
Suppose is such a sequence that is bounded and . We shall prove that has a convergent subsequence. We now prove that is bounded in . If is unbounded, we may assume that, going to a subsequence if necessary, as . In view of (H3), there exists such that
for all , and combing (H1), we have
This implies
Let , then
By (H3), there exists such that
for . Define . We have
where c, are constants independent of n. By (14) we have
Combining this inequality with (12) and (13) yields
as . Since , by (15), we have
as . This implies
Similarly, we have
Therefore, combining (16) and (17), we have
which gives a contradiction. Therefore, is bounded in and, going if necessary to a subsequence, we can assume that in and in . Write and , then in , and in .
By (4), we have
Since in , it is then easy to verify
Combining this with , as and (18), we have in . Similarly, in and hence in , that is, ϕ satisfies the (PS) condition. □
Proof of Theorem 3.1
We prove that ϕ satisfies the other conditions of Theorem A.
Step 1: By (H3) and (H4), we have
By (H4), for any , there exists such that
Therefore, there exists such that
Combining this with (2), we have
Consequently, by (H1), for ,
Choose and ρ such that . Then for any ,
Thus ϕ satisfies (i) of (I3) with and .
Step 2: Let with and . We denote
For , we write , where , .
-
(i)
If , one has . By (H2) and (H5) there exists , for any ,
-
(ii)
If , one has , which implies that
(20)
Denote .
Claim: There exists such that, ,
For otherwise, , such that
Write . Notice that and . In the sense of subsequence, we have
Then (20) implies that
Note that , in the sense of subsequence as . Thus in the sense of subsequences,
This means that in , i.e.,
By (23) we know that . Therefore, . Then there exist , such that
Otherwise, for all , we must have
i.e.,
We have
We get a contradiction. Thus (25) holds. Let , , and . By (22), we have
Let j be large enough such that and . Then we have
This implies that
This is a contradiction to (24). Therefore the claim is true and (21) holds. For , let . By (H4), for , there exists such that
Choose . For ,
By (H5), for ,
which implies that there exists such that for
Setting , we have proved that for any and
Let . By (26) we have , i.e., ϕ satisfies (ii) of (I3) in Theorem A.
Finally, by Lemma 3.2, ϕ satisfies the (PS) condition. Similar to the proof of [30], we prove that S and ∂Q link. By Theorem A, there exists a critical point of ϕ such that . Moreover, u is a classical solution of (IP) and u is nonconstant by (H5). The proof is completed. □
Remark 3.2
In order to seek 2T-periodic solutions of more general systems
where f and impulsive effects are T-periodic in t, we make the substitution: and . Thus the above systems transforms to
This implies that a 2π-periodic solution of the second systems corresponds to a 2T-periodic solution of the first one. Hence we will only look for the 2π-periodic solutions in the sequel.
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Acknowledgements
This research was supported by the National Natural Science Foundation of China (11271371, 51479215) and the Postgraduate research and innovation project of Hunan Province (CX2011B078).
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Chen, D., Dai, B. Periodic solution of second-order impulsive delay differential system via generalized mountain pass theorem. Bound Value Probl 2014, 234 (2014). https://doi.org/10.1186/s13661-014-0234-z
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DOI: https://doi.org/10.1186/s13661-014-0234-z