Multiple periodic solutions to a class of nonautonomous second-order delay differential equation
© Meng and Zhang; licensee Springer. 2013
Received: 18 May 2013
Accepted: 12 September 2013
Published: 19 November 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.
for all , ;
uniformly for ;
is the set of all positive integers.
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 .
The proof of Lemma 2.1 is complete. □
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).
where , .
then y is called a weak derivative of z denoted by .
By a direct computation, L is a bounded self-adjoint linear operator on .
Thus the critical points of in are classical solutions of (2.2).
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.
Remark 2.1 The condition is to ensure .
3 Proofs of theorems
In this section, stand for different positive constants for .
uniformly for .
uniformly for .
Lemma 3.1 Suppose that f satisfies (f 1)-(f 3±). Then the function φ satisfies the condition for any .
for any , is defined in Section 2.
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 .
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. □
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 ().
that is, and hence is -closed.
for any . So, Φ satisfies ().
Step 2. Φ satisfies ().
and hence () holds.
Step 3. Φ satisfies ().
Obviously, and . In order to obtain the desired conclusion, it is sufficient to prove that as on .
Since and are finitely dimensional, (3.15) and the above estimate imply that as , . Hence () holds.
The proof of Theorem 1.1 is complete. □
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. □
uniformly for .
Then we have . By Theorem 1.2, (3.16) possesses at least two pairs 2π-periodic solutions.
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.
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