Existence of periodic solutions for first-order delay differential equations via critical point theory
© Zhang and Chen; licensee Springer. 2013
Received: 11 September 2013
Accepted: 28 October 2013
Published: 22 November 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
where , and is a given constant.
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 ;
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 ()
Definition 1.2 ()
The following lemma is Example 1 in .
and take , . Then A links B weakly.
by making the change of variable . Thus, a 4r-periodic solution of (1.1) corresponds to a 2π-periodic solution of (1.7).
where denotes the weak derivative of u.
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:
for all .
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 all .
2 Main results
Theorem 2.1 Assume that f satisfies (), () and (). Then (1.1) possesses at least one 4r-periodic solution.
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.
According to (2.5) and (2.6), we get that is weak-to-weak continuous.
This implies (2.8).
for all .
Hence, by , we get that .
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.
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,
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 ,
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.
By Proposition B.37 in , () implies that is compact. Hence () holds.
Next, we show that I satisfies () of Lemma 2.2.
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 (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.
where are constants.
which implies that is bounded in E.
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 .
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.
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