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Existence of periodic solutions for first-order delay differential equations via critical point theory
Boundary Value Problems volume 2013, Article number: 254 (2013)
By the weak linking theorem and the linking theorem, we study the existence of periodic solutions for the following system of delay differential equations:
where , and is a given constant. Two existence theorems of 4r-periodic solutions of (1) are obtained.
1 Introduction and preliminaries
Consider the following system of delay differential equations:
where , and is a given constant.
As , the existence of the periodic solutions for (1.1) has been extensively studied in the past years (for example, see [1–5]). However, their methods are not variational. Few results of the existence of periodic solutions for delay differential equations have been obtained by the variational method. In 2005, Guo and Yu  took the lead in using the variational approaches to study the existence of multiple periodic solutions for (1.1), and a multiplicity result was given. Recently, using the variational approaches, the multiplicity of the periodic solutions for the following system:
was studied by Wu and Wu in . In the present paper, our main purpose is to study the existence of the periodic orbits for system (1.1) via the linking and weak linking theory.
Throughout this paper, we always assume that
() f is odd, i.e., for any , ;
() there exists a continuously differentiable function F such that for all , and ;
() as , where is an matrix with
and (the set of all eigenvalues of A) for any , where and is the set of all positive integers.
In the following, we give some preliminaries.
Definition 1.1 ()
Let E be a Hilbert space and . The function is called weak-to-weak continuous if
Definition 1.2 ()
A subset A of a Banach space E links a subset B of E weakly if for every satisfying (1.2) and
there are a sequence and a constant c such that
The following lemma is Example 1 in .
Lemma 1.1 Let E be a separable Hilbert space, and let M, N be a closed subspace such that . Let
and take , . Then A links B weakly.
One can easily find that (1.1) can be changed to the equation
by making the change of variable . Thus, a 4r-periodic solution of (1.1) corresponds to a 2π-periodic solution of (1.7).
Similar to the treatment in Guo and Yu , we introduce the following spaces. Let denote the set of n-tuples of 2π periodic functions which are square integrable. Let 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 . Set is the closure of with respect to the Hilbert norm
More specifically, with the inner product
for any , where denotes the usual inner product in . In the sequel, we denote by H the Hilbert space . The norm on H is defined by
Now consider a functional I defined on H
where denotes the weak derivative of u.
We define an operator as follows: for any , Lu is defined by
where denotes the dual space of H. By the Riesz representation theorem, we can identify with H. Thus, Lu can also be viewed as a function belonging to H such that for any . Define
Then can be rewritten as
Define the bounded linear operator as follows: for any , . Next, we set . Then E is a closed subspace of H and is invariant with respect to L. It is easy to check that L is a bounded linear operator on H, is self-adjoint, and E is also invariant with respect to under condition () (see Guo and Yu ). By Proposition B.37 in  and Lemma 2.2 in , we have the following two lemmas.
Lemma 1.2 Assume that f satisfies () and the following condition:
() there are constants and such that
for all .
Then the functional I is continuously differentiable on H and is defined by
Moreover, is a compact mapping defined as follows:
By the Riesz theorem, we can view as an element of H for any . As usual, we identify and its continuous representative.
We have the following fact.
Lemma 1.3 Assume that f satisfies (), (), (). Then critical points of functional I restricted to E are 2π-periodic solutions of system (1.7).
Remark 1.1 It is pointed in  that a critical point u of I in H will be a weak solution of (1.7). However, a simple regularity argument shows that .
Remark 1.2 As usual, we should deal with (1.10) in the space H. But, according to Lemma 1.3, we only need to treat the functional I in the subspace E of H.
Lemma 1.4 ()
For each , is compactly embedded in . In particular there is such that
for all .
2 Main results
Theorem 2.1 Assume that f satisfies (), () and (). Then (1.1) possesses at least one 4r-periodic solution.
Proof Let denote the usual normal orthogonal bases in and set
where is the set of all positive integers. Then . For any , it has a Fourier expansion as follows:
where all ,
Consequently, we have
Then (2.2) and (2.3) show that and are two equivalent norms on E. Henceforth we use the norm as the norm for E. And the spaces M, N are mutually orthogonal with respect to the associated inner product.
First, we prove that satisfies (1.2) in E.
Let be any sequence which converges to some u weakly in E. By the compactness of the embedding , we assume that
Thus, a.e. for all . Since satisfies (), there exist positive constants and such that
Note that the right-hand side of (2.4) converges to in . Hence is uniformly absolutely continuous. Hence, by Vitali’s theorem,
Moreover, since L is a bounded self-adjoint linear operator on E,
According to (2.5) and (2.6), we get that is weak-to-weak continuous.
Next, we prove
Indeed, by (), we know that there exists a positive constant c such that
Thus, for , by (2.9), we have
where is a given constant. Since , (2.10) implies (2.7). The proof of (2.8) is similar. In fact, when , by (2.9), we get that
This implies (2.8).
Note that (2.10) implies
The combination of (2.8) and (2.11) implies that there is such that (1.3) holds with , . By Lemma 1.1 we know that there are a sequence and a constant c such that
Finally, we show that the sequence is bounded in E. To do this, assume that , and write . Then . From Lemma 1.4, there is a renamed subsequence such that
By (), for any , there exists a constant such that
Moreover, by the continuity of f, there is a constant such that
By (2.13) and (2.14), for any , we have
as and . This shows
By (2.12) and (2.17), we see that
for all , i.e.,
for all .
Then, by (2.18), one can obtain
where I is the unit matrix. For any j, take and , where . An easy computation shows that
Hence, by , we get that .
Let , where , . A proof similar to (2.16) shows that
where , , . Thus
On the other hand, (2.12) implies
which contradicts (2.23). Thus must be bounded. Consequently, there is a renamed subsequence of such that in E. Hence, by the weak-to-weak continuity of , we have
Now, the combination of (2.12) and (2.24) implies that . This completes the proof. □
Remark 2.1 Let , and . Then satisfies all the conditions of Theorem 2.1.
In order to give our another result, we still need the following preliminaries.
Let , be the projectors of E onto M, N associated with the given splitting of E. Set
Recall that K is continuous and maps bounded sets to relatively compact sets since K is compact. Let with , a given subspace of E. Then ∂Q will refer to the boundary of Q in .
Definition 2.1 We say S and ∂Q link if whenever and for all , then for all .
Lemma 2.1 ()
Let , , , , , , and . Then S and ∂Q link.
Lemma 2.2 ()
Suppose satisfies the (PS) condition and
() , where and , are bounded self-adjoint,
() is compact,
() there exist a subspace and sets , and constants such that
Q is bounded and ,
S and ∂Q link.
Then I possesses a critical value .
The following is our another main result.
Theorem 2.2 Assume that f satisfies (), () and the following conditions:
() for all ,
() as ,
() there exist constants , and such that
Then (1.1) possesses at least one nonconstant 4r-periodic solution.
Proof We will show that I satisfies the hypotheses of Lemma 2.2. This will lead to a nonconstant 4r-periodic solution of (1.1). We divide the proof of Theorem 2.2 into the following three parts.
First, we prove that I satisfies () and () of Lemma 2.2.
Note that , and L is bounded self-adjoint on E. We see that I satisfies () of Lemma 2.2 with , and
By Proposition B.37 in , () implies that is compact. Hence () holds.
Next, we show that I satisfies () of Lemma 2.2.
By (), for any , there is such that
whenever . By (), there is a constant such that
By (2.25) and Lemma 1.4, for any , we have
Choose . Since , there is small such that . Then, for , (2.26) implies that . Consequently, I satisfies ()(i) with .
where and are free constants for the moment. Define . Then and S and ∂Q link by Lemma 2.1.
By (), there are constants such that
for all . Thus, for , by the Hölder inequality (note that ) and orthogonality, we get that
where are constants. Now, choose large and such that
By (2.27), (2.29), (2.30), (2.31) and (), one can easily check that . Hence I satisfies ()(ii).
To sum up, I satisfies () of Lemma 2.2.
Finally, we check that I satisfies the (PS) condition. Let be a sequence such that and as . Then, for large k, by () and (2.28), we have
where are constants.
Let , where , . Then, for large k, by (), Lemma 1.4 and the Hölder inequality, we get that
This implies that
Similarly, one can easily see that
By (2.32), (2.34) and (2.35), there is a constant such that
which implies that is bounded in E.
By the compactness of , going if necessary to a subsequence, we can assume that
Let , where and . Then
as . Similarly, we have as . Hence in E. Hence I satisfies the (PS) condition.
Therefore, Theorem 2.2 follows from Lemma 2.2. □
Remark 2.2 Let and . Then satisfies all the conditions of Theorem 2.2 with and .
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Many thanks to the reviewers for carefully reading the manuscript and valuable comments on improving the paper. This work is supported partly by the Professor Program Foundation of Zhaotong University.
The authors declare that they have no competing interests.
All authors contributed equally in the paper. They read and approved the final manuscript.