Periodic solution of second-order impulsive delay differential system via generalized mountain pass theorem
© Chen and Dai; licensee Springer. 2014
Received: 23 July 2014
Accepted: 16 October 2014
Published: 6 November 2014
The Erratum to this article has been published in Boundary Value Problems 2015 2015:80
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.
- (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 , . Impulses only in the velocity occur also in impulsive mechanics . Such impulsive problems with impulses in the derivative only have been considered in many literatures; see, for instance –.
In recent years, impulsive and periodic boundary value problems have been studied by numerous mathematicians; see, for instance, , – 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 , – and the references therein.
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.
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.
for all .
A function is said to be a classic periodic solution of (IP), if u satisfies equation in (1a) for all and (1b), (1c) hold.
L is selfadjoint on.
The proof is completed. □
Similarly to , we introduce the following concept for the solution of problem (IP).
holds for any .
Since we have the following result, Definition 2.2 is suitable.
Ifis a weak solution of (IP), then u is a classical solution of (IP).
for any , since v is arbitrary in . By (10), . Therefore u is a classical solution of (IP). The proof is completed. □
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.
where is compact.
Let , and Q be boundary. We call S and ∂Q link if whenever and for all t, then for all t.
Then  Theorem 5.29 can be stated as follows.
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
Q is bounded and ,
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.
for all .
for all .
for all and with .
(H4): as and as uniformly for all t.
(H5): for all .
Assume that (H1)-(H5) hold. Then problem (IP) has at least one periodic solution.
and , for .
for L being large enough. This implies (H3).
We will use Theorem A to prove Theorem 3.1.
Set and .
which implies that , that is, .
Then a simple calculation shows that and and . Then . Combining with , one has and . □
Lemma 3.1 is a new orthogonal decomposition different from the one in . We will show that it is a useful result.
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.
Under the assumptions of Theorem 3.1, ϕ satisfies (PS) condition.
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 .
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.
Thus ϕ satisfies (i) of (I3) with and .
- (i)If , one has . By (H2) and (H5) there exists , for any ,
- (ii)If , one has , which implies that(20)
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 , 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. □
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.
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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