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Multiple periodic solutions to a class of nonautonomous second-order delay differential equation
Boundary Value Problems volume 2013, Article number: 244 (2013)
The existence of the nontrivial periodic solutions for nonautonomous second-order delay differential equation
is investigated, where , , . Multiple periodic solutions are obtained by some recent critical point theorems.
MSC:34K13, 34K18, 58E50.
It is well known that the critical point theory is a powerful tool to deal with the multiplicity of periodic solutions to ordinary differential systems as well as partial differential equations (see [1–6]). In 1998, Li and He  first applied the critical point theory to study the multiplicity of periodic solutions for delay differential equations. Especially, in 2005, Guo and Yu  established a variational framework for delay differential autonomous systems. In the past several years, some results on the existence of periodic solutions for the functional differential equation have been obtained by the critical point theory (see [7–14]). However, most of these functional differential equations are autonomous, the results on the non-autonomous functional differential equations are relatively few (see [12, 13]).
where , , .
In this paper, we have the following conditions on f.
(f 1) , and
for all , ;
(f 2) there exist four τ-periodic and continuous functions , , and such that
uniformly for ;
(f 3±) is bounded for all , and
uniformly for , where
Theorem 1.1 Assume that f satisfies (f 1)-(f 3−) with , for , . Then (1.1) possesses at least 2m pairs 2τ-periodic solutions, where
is the set of all positive integers.
Theorem 1.2 Assume that f satisfies (f 1)-(f 3+) with , for , , . Then (1.1) possesses at least pairs 2τ-periodic solutions, where
In this paper, the main purpose is to study the multiplicity of periodic solutions for systems (1.1) via some recent critical point theorems for strongly indefinite functionals. In order to achieve this, some preliminaries are necessary. Let X and Y be Banach spaces with X being separable and reflexive, and set . Let be a dense subset. For each , there is a semi-norm on E defined by
We denote by the topology on E induced by the semi-norm family , and let ω and denote the weak-topology and weak*-topology, respectively. Clearly, the topology contains the product topology on produced by the weak topology on X and the strong topology on Y.
For a functional , we write . Recall that is weakly sequentially continuous if in E, one has for each , i.e., is sequentially continuous. For , we say that Φ satisfies the condition if any sequence , such that and as , contains a convergent subsequence.
() , is -closed for every and is continuous;
() there exists such that , where ;
() there exist a finite dimensional subspace and such that and , where , and .
Theorem A Assume that Φ is even and ()-() are satisfied. Then Φ has at least pairs of critical points with critical values less than or equal to provided Φ satisfies the condition for all .
In this section, we establish a variational structure which enables us to reduce the existence of 2τ periodic solutions of (1.1) to a classic Hamiltonian system. First, we have the following lemma by using similar arguments .
Lemma 2.1 Suppose that satisfies (f 1), then the function
Proof It is obvious that . By (f 1), . Then we have
The proof of Lemma 2.1 is complete. □
Suppose that x is a periodic solution of (1.1) with 2τ, let , then
We denote . Then
where is defined in (2.1) and denotes the gradient of with respect to the z variable. It is easy to obtain the following lemma.
Lemma 2.2 For any 2τ periodic solution of (1.1), let , then is a 2τ periodic solution of (2.2). Conversely, for any 2τ periodic solution of (2.2) satisfying , x is a 2τ periodic solution of (1.1).
For , let denote the space of 2τ periodic functions on R with values in . For any , it has the following Fourier expansion in the sense that it is convergent in the space ,
where , .
Let . If there exists a function such that, for every ,
then y is called a weak derivative of z denoted by .
For any , and can be explicitly expressed by
We define an operator by the Riesz representation theorem
By a direct computation, L is a bounded self-adjoint linear operator on .
By (f 2) and (3.6) in Section 3, there exist two continuous functions and such that
By using similar arguments as in , we have that
Thus the critical points of in are classical solutions of (2.2).
Denote . By a direct computation, we have
where , .
Note that for any , and . Due to the fact that , we know that whatever . Therefore, we have the following lemma.
Lemma 2.3 If is a critical point of φ in E, then is a critical point of φ in .
Moreover, we also denote by , and the positive definite, negative definite and null subspaces of the self-adjoint linear operator defining it, respectively.
Then E has an orthogonal decomposition
where . We have
There exists such that
where , , . Therefore, we get
Remark 2.1 The condition is to ensure .
3 Proofs of theorems
In this section, stand for different positive constants for .
By a direct computation, (f 1) and (f 2) imply that is even and satisfies
uniformly for .
(f 3±) implies that
uniformly for .
Lemma 3.1 Suppose that f satisfies (f 1)-(f 3±). Then the function φ satisfies the condition for any .
Proof First we define an operator
for any . By a direct computation, B is a bounded self-adjoint linear operator on E. Thus is also a self-adjoint linear operator on E. Then E has an orthogonal decomposition
Also there exists such that
for any , is defined in Section 2.
Let be any sequence such that
We first prove that is bounded. For any , (3.2) implies that there exists a constant such that
for all and . Since , we have
where , , . By (3.3±), there exists a constant such that
Therefore we have
Thus we have that is bounded. Using similar arguments, we can prove that is bounded. Consider . Arguing indirectly, we suppose is unbounded, then we have . According to the definition of , this implies that there are constants such that
By (3.7), we have
By (3.8) and (3.3±), (3.9) is a contradiction. Hence is bounded. Therefore there exists a constant such that
So, we get is bounded, and going if necessary to a subsequence, we can assume that in E and in . Write and , then in E, in E and in .
In view of (3.6) and in , it is easy to verify
But then as , and
This yields in E. Similarly, in E and hence in E, that is, Φ satisfies the condition. Thus φ satisfies the condition. The proof of Lemma 3.1 is complete. □
Proof of Theorem 1.1 For , let , , , and
where and are from Section 2.
In order to obtain this theorem, we apply Theorem A to the functional . The proof of this theorem is divided into the following three steps.
Step 1. Φ satisfies ().
We first check that is -closed for any . Let be any sequence -converging to some . Write and , then in E and hence is bounded in the norm topology. Note that for and for , then for any and small , we have by (3.10) and (3.6)
which implies that Ψ is bounded from below on E. Consequently, combining and shows that is bounded in E by (2.3) and hence
Moreover, since , we have
It follows from (3.11) and the above inequality that is also bounded in E since all norms are equivalent in a finite dimensional space. Then () is bounded, and hence we can assume that converges weakly to in E. Thus we have . Note that is an equivalent norm on . By the lower semi-continuity of the norm, we get
that is, and hence is -closed.
Next, we prove that is continuous. To achieve this, it is sufficient to demonstrate that has the same property. Suppose in E, then converges uniformly to z on . Hence, for every given , we see that converges to in measure on . Moreover, by (3.6), one has
for all k and , where and denotes the natural norm of . Thus, the Vitali theorem is applicable and
for any . So, Φ satisfies ().
Step 2. Φ satisfies ().
By (3.1), there exist , such that
By Proposition 1.1 in , there is a positive constant ξ such that . Set small , then for each with , one has , and hence by (3.12)
and hence () holds.
Step 3. Φ satisfies ().
Obviously, and . In order to obtain the desired conclusion, it is sufficient to prove that as on .
Let . By the definition of m, there exists a constant such that
for . Clearly, for any ,
Let . We claim that
Indeed, for , by (3.6), one has
which implies that (3.15) is true by the arbitrariness of ε. Then, for , by (2.3), (3.13) and (3.14), one has
Since and are finitely dimensional, (3.15) and the above estimate imply that as , . Hence () holds.
The proof of Theorem 1.1 is complete. □
Proof of Theorem 1.2 For , let , , , and
Then the conclusion is obtained by the same argument as in the proof of Theorem 1.1. The proof of Theorem 1.2. is complete. □
Example 3.1 Consider the following equation:
where . Let
By a straightforward computation, we have
uniformly for .
Take , , , . By Theorem 1.2, we have , , for , , , ,
Then we have . By Theorem 1.2, (3.16) possesses at least two pairs 2π-periodic solutions.
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The authors are grateful for the referees’ careful reviewing and their valuable suggestions. The work is partially supported by the Natural Science Foundation of Shanxi Province (No. 2012011004-1) of China.
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.