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Homoclinic solutions for a class of neutral Duffing differential systems
Advances in Difference Equations volume 2014, Article number: 121 (2014)
By using an extension of Mawhin’s continuation theorem and some analysis methods, the existence of a set with -periodic for a n-dimensional neutral Duffing differential systems, , is studied. Some new results on the existence of homoclinic solutions is obtained as a limit of a certain subsequence of the above set. Meanwhile, is a constant symmetrical matrix and is allowed to change sign.
The aim of this paper is to consider a kind of neutral Duffing differential systems as follows:
where with , , , and is a continuous T-periodic function with ; and τ are given constants; is a constant symmetrical matrix and is allowed to change sign.
As is well known, a solution of Eq. (1.1) is called homoclinic (to O) if and as . In addition, if , then u is called a nontrivial homoclinic solution.
Under the condition of , system (1.1) transforms into a classic second-order Duffing equation
which has been studied by Li et al.  and some new results on the existence and uniqueness of periodic solutions for (1.2) are obtained. Very recently, by using Mawhin’s continuation theorem, Du  studied the following neutral differential equations:
where ; ; ; , () and τ are given constants, obtaining the existence of homoclinic solutions for (1.3).
In this paper, like in the work of Rabinowitz in , Izydorek and Janczewska in  and Tan and Xiao in , the existence of a homoclinic solution for (1.1) is obtained as a limit of a certain sequence of -periodic solutions for the following equation:
where , is a -periodic function such that
is a constant independent of k. However, the approaches to show as are different from the corresponding ones used in the past and the existence of -periodic solutions to Eq. (1.4) is obtained by using an extension of Mawhin’s continuation theorem, which is quite different from the approach of [3–5]. Furthermore, is a constant symmetrical matrix and is allowed to change sign, different from the corresponding ones of .
Throughout this paper, denotes the standard inner product, and denotes the absolute value and the Euclidean norm on . For each , let , and . If the norms of and are defined by and , respectively, then and are all Banach spaces. Furthermore, for , , .
Define the linear operator
Lemma 2.1 
Suppose that Ω is an open bounded set in X such that the following conditions are satisfied:
[A1] For each , the equation
has no solution on ∂ Ω.
[A2] The equation
has no solution on .
[A3] The Brouwer degree
Equation (1.4) has a -periodic solution in .
Lemma 2.2 
If set and , , where is a constant with , then operator has continuous inverse on , satisfying
Lemma 2.3 
If is continuously differentiable on R, , , and are constants, then for every , the following inequality holds:
This lemma is a special case of Lemma 2.2 in .
Lemma 2.4 
Suppose that are eigenvalues of matrix C. If (), then A has a continuous bounded inverse with the following relationships:
, , , where
q is a constant with .
Lemma 2.5 
Let with and , . Suppose , and with . Then
Throughout this paper, we suppose in addition that , , where are eigenvalues of matrix C with and let , , .
For convenience, we list the following assumptions which will be used to study the existence of homoclinic solutions to Eq. (1.1) in Section 3.
[H1] There are constants and such that
[H2] is a bounded function with and
Remark 2.1 
From (1.5), we see that . So if assumption [H2] holds, for each , .
3 Main results
In order to investigate the existence of -periodic solutions to system (1.4), we need to study some properties of all possible -periodic solutions to the following system:
For each , let represent the set of all the -periodic solutions to system (3.1).
Theorem 3.1 Suppose assumptions [H1]-[H2] hold, , and
then for each , if , then there are positive constants , , , and which are independent of k and λ, such that
Proof For each , if , then u must satisfy
Multiplying both sides of Eq. (3.2) by and integrating on the interval , we have
Clearly, , then we have
and from (3.4) and [H1] that
By using [H1] and Lemma 2.5, we get
In a similar way as in the proof of (3.6), we have
By using [H2], we get
By applying (3.6)-(3.9), we see that
Thus, from (3.10)
By using Lemma 2.4, we have , and from (3.10)-(3.11)
there is a constant such that
and by (3.11)
Obviously, and are constants independent of k and λ. Thus by using Lemma 2.2, for all , we get
From (3.14) and (3.15), we obtain
where is a constant independent of k and λ.
For , from the continuity of , one can find that there is a such that
and it follows from (3.14) that for , ,
By Lemma 2.4 and (3.17), we get
Clearly, is a constant independent of k and λ. Hence the conclusion of Theorem 3.1 holds. □
Theorem 3.2 Assume that the conditions of Theorem 3.1 are satisfied. Then for each , Eq. (3.2) has at least one -periodic solution such that
where , , , and are constants defined by Theorem 3.1.
Proof In order to use Lemma 2.1, for each , we consider the following equation:
Let represent the set of all the -periodic of system (3.18), since , then , where Σ is defined by Theorem 3.1. If , by using Theorem 3.1, we have
Let , where
If , then (constant vector) and by [H1], we see that
Now, if we set , then . So condition [A1] and condition [A2] of Lemma 2.1 are satisfied. What remains is verifying condition [A3] of Lemma 2.1. In order to do this, let
where is determined by Lemma 2.1. From assumption [H1], we have
So condition [A3] of Lemma 2.1 is satisfied. Therefore, by using Lemma 2.1, we see that Eq. (1.2) has a -periodic solution Evidently, is a -periodic solution to Eq. (3.1) for the case of , so . Thus, by using Theorem 3.1, we get
Theorem 3.3 Suppose that the conditions in Theorem 3.1 hold, then Eq. (1.1) has a nontrivial homoclinic solution.
Proof From Theorem 3.2, we see that for each , there exists a -periodic solution to Eq. (1.2). So for every , satisfies
Let for . By (3.17),
and, by (3.20),
Obviously, is a constant independent of k. Similar to the proof of Lemma 2.4 in , we see that there exists a such that for each interval , there is a subsequence of with R, and uniformly on .
For all with , there must be a positive integer such that for , . So for , from (1.5) and (3.20) we see that
uniformly on .
By the fact that is a continuous differential on , for , uniformly . We have , , in view of being arbitrary, that is, is a solution to system (1.1).
Now, we will prove and for . We have
Clearly, for every if , by (3.14) and (3.15), we get
Let and ; we have
From (3.13), in a similar way we get
So, by using Lemma 2.3,
Finally, in order to obtain
we show that
From (3.16), we have and by (1.1), we get
If (3.25) does not hold, then there exist and a sequence such that
From this, we have, for ,
It follows that
which contradicts (3.24), so (3.25) holds.
Since C is symmetrical, it is easy to see that there is an orthogonal matrix T such that .
Let , then we get as . By (3.25), we have
By using (3.19), we see that , which implies
For all , there exists such that (), for . Similarly, by (3.26), we see that there is a constant such that , for .
Then, by using Lemma 2.2 and (3.27), when , we get
Now, by (3.27) and (3.28), we conclude that , there exists such that for ,
Thus, we get , as .
In the similar way, when , we can proof , as .
Therefore, , as ; i.e.,
we know T is an orthogonal matrix, then as .
Thus, we have
Clearly, ; otherwise, , which contradicts the assumption [H2].
As an application, we consider the following equation:
where , , and . Clearly, . Also, and , which implies that assumption [H1] is satisfied with , . is a bounded function and , which implies that assumption [H2] holds. Furthermore, we can choose , , , and , then
By applying Theorem 3.3, we see that Eq. (3.29) has a nontrivial homoclinic solution. □
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The author would like to express the sincere gratitude to Editor for handling the process of reviewing the paper, as well as to the reviewers who carefully reviewed the manuscript.
The authors declare that they have no competing interests.
The author drafted the manuscript, read and approved the final manuscript.
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Cite this article
Chen, W. Homoclinic solutions for a class of neutral Duffing differential systems. Adv Differ Equ 2014, 121 (2014). https://doi.org/10.1186/1687-1847-2014-121
- homoclinic solution
- continuation theorem
- periodic solution