- Open Access
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.
- second-order delay differential equation
- periodic solution
- critical point theorems
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 .
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.
- Ding Y: Variational Methods for Strongly Indefinite Problems. World Scientific, Singapore; 2007.MATHGoogle Scholar
- Guo Y: Nontrivial periodic solutions for asymptotically linear Hamiltonian systems with resonance. J. Differ. Equ. 2001, 175: 71-87. 10.1006/jdeq.2000.3966MATHView ArticleGoogle Scholar
- Mawhin J, Willem M: Critical Point Theory and Hamiltonian Systems. Springer, New York; 1989.MATHView ArticleGoogle Scholar
- Rabinowits PH American Mathematical Society 65. Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS 1986.View ArticleGoogle Scholar
- Su JB: Nontrivial periodic solutions for the asymptotically linear Hamiltonian systems with resonance at infinity. J. Differ. Equ. 1998, 145: 252-273. 10.1006/jdeq.1997.3360MATHView ArticleGoogle Scholar
- Zhao F, Zhao L, Ding Y: Existence and multiplicity of solutions for a non-periodic Schrödinger equation. Nonlinear Anal. TMA 2008, 69: 3671-3678. 10.1016/j.na.2007.10.024MATHMathSciNetView ArticleGoogle Scholar
- Li J, He X: Multiple periodic solutions of differential delay equations created by asymptotically linear Hamiltonian systems. Nonlinear Anal. 1998, 31(1/2):45-54.MATHMathSciNetView ArticleGoogle Scholar
- Guo Z, Yu J: Multiplicity results for periodic solutions to delay differential equations via critical point theory. J. Differ. Equ. 2005, 218: 15-35. 10.1016/j.jde.2005.08.007MATHMathSciNetView ArticleGoogle Scholar
- Fei G: Multiple periodic solutions of differential delay equations via Hamiltonian systems (I). Nonlinear Anal. 2006, 65: 25-39. 10.1016/j.na.2005.06.011MATHMathSciNetView ArticleGoogle Scholar
- Fei G: Multiple periodic solutions of differential delay equations via Hamiltonian systems (II). Nonlinear Anal. 2006, 65: 40-58. 10.1016/j.na.2005.06.012MATHMathSciNetView ArticleGoogle Scholar
- Guo Z, Xiaomin Z: Multiple results for periodic solutions to a class of second-order delay differential equations. Commun. Pure Appl. Anal. 2010, 9(6):1529-1542.MATHMathSciNetView ArticleGoogle Scholar
- Wu K, Wu X, Zhou F: Multiplicity results of periodic solutions for a class of second order delay differential equations. Nonlinear Anal. 2012, 75: 5836-5844. 10.1016/j.na.2012.05.026MATHMathSciNetView ArticleGoogle Scholar
- Yu J, Xiao H:Multiplicity periodic solutions with minimal period 4 of delay differential equations.J. Differ. Equ. 2013, 254: 2158-2172. 10.1016/j.jde.2012.11.022MATHMathSciNetView ArticleGoogle Scholar
- Zhang X, Meng Q: Nontrivial periodic solutions for delay differential systems via Morse theory. Nonlinear Anal. 2011, 74: 1960-1968. 10.1016/j.na.2010.11.003MATHMathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.