- Research Article
- Open Access

# The Existence of Periodic Solutions for Non-Autonomous Differential Delay Equations via Minimax Methods

- Rong Cheng
^{1, 2}Email author

**2009**:137084

https://doi.org/10.1155/2009/137084

© Rong Cheng. 2009

**Received:**9 April 2009**Accepted:**19 October 2009**Published:**26 October 2009

## Abstract

By using variational methods directly, we establish the existence of periodic solutions for a class of nonautonomous differential delay equations which are superlinear both at zero and at infinity.

## Keywords

- Periodic Solution
- Compact Operator
- Fourier Expansion
- Shift Operator
- Differential Delay Equation

## 1. Introduction and Main Result

where is odd and is parameter. Since Jone's work in [4], there has been a great deal of research on problems of existence, multiplicity, stability, bifurcation, uniqueness, density of periodic solutions to (1.1) by applying various approaches. See [2, 4–23]. But most of those results concern scalar equations (1.1) and generally slowly oscillating periodic solutions. A periodic solution of (1.1) is called a "slowly oscillating periodic solution" if there exist numbers and such that for , for and for all .

where and are symmetric constant matrices. Before Guo and Yu's work, many authors generally first use the reduction technique introduced by Kaplan and Yorke in [7] to reduce the search for periodic solutions of (1.2) with and its similar ones to the problem of finding periodic solutions for a related system of ordinary differential equations. Then variational method was applied to study the related systems and the existence of periodic solutions of the equations is obtained.

where is odd with respect to and satisfies the following superlinear conditions both at zero and at infinity

When (1.2) satisfies (1.3), we can apply the twist condition between the zero and at infinity for to establish the existence of periodic solutions of (1.2). Under the superlinear conditions (1.5), there is no twist condition for , which brings difficulty to the study of the existence of periodic solutions of (1.4). But we can use minimax methods to consider the problem without twist condition for .

Throughout this paper, we assume that the following conditions hold.

- (H1)
- (H2)
- (H3)

Then our main result can be read as follows.

Theorem 1.1.

Suppose that satisfies (1.5) and the conditions hold. Then (1.4) possesses a nontrivial -periodic solution.

Remark 1.2.

We shall use a minimax theorem in critical point theory in [25] to prove our main result. The ideas come from [25–27]. Theorem 1.1 will be proved in Section 2.

## 2. Proof of the Main Result

First of all in this section, we introduce a minimax theorem which will be used in our discussion. Let be a Hilbert space with . Let be the projections of onto and , respectively.

Definition 2.1.

Let and be boundary. One calls and link if whenever and for all , then .

Definition 2.2.

A functional satisfies condition, if every sequence that , and being bounded, possesses a convergent subsequence.

Then [25, Theorem ] can be stated as follows.

Theorem 2 A.

Let be a real Hilbert space with , and inner product . Suppose satisfies condition,

, where and is bounded and selfadjoint, ,

there exists a subspace and sets , and constants such that

Then possesses a critical value .

Then and , where denotes the gradient of with respect to . We have the following lemma.

Lemma 2.3.

Under the conditions of Theorem 1.1, the function satisfies the following.

Proof.

The definition of implies (i) directly. We prove case (ii) and case (iii).

*Case* (ii). Let

Then and or is equivalent to or , respectively.

From (1.5) and L'Hospital rules, we have (2.3) by a direct computation.

*Case* (iii). By (H2), we have a constant
such that
for
with
.

Firstly, it follows from that .

By reducing method, we have

Thus, the inequality for holds.

Take and . Then (2.5) and (2.6) hold with and .

Below we will construct a variational functional of (1.4) defined on a suitable Hilbert space such that finding -periodic solutions of (1.4) is equivalent to seeking critical points of the functional.

where is -periodic with respect to . Therefore we only seek -periodic solution of (2.12) which corresponds to the -periodic solution of (1.4).

It is not difficult to see that is a bounded linear operator on and .

It is easy to compute that is bounded and linear. Moreover is isometric, that is, and , where denotes the identity mapping on .

Lemma 2.4.

Critical points of over are critical points of on , where is the restriction of over .

Proof.

Note that any is -periodic and is odd with respect to . It is enough for us to prove for any and being a critical point of in .

The proof is complete.

Remark 2.5.

By Lemma 2.4, we only need to find critical points of over . Therefore in the following will be assumed on .

Let and denote the positive definite and negative definite subspace of in , respectively. Then . Letting , , we see that of Theorem A holds. Since is compact, of Theorem A holds. Now we establish of Theorem A by the following three lemmas.

Lemma 2.6.

Under the assumptions of Theorem 1.1, of holds for .

Proof.

Lemma 2.7.

Under the assumptions of Theorem 1.1, satisfies of .

Proof.

Henceforth, for any and , that is, . Then of holds.

Lemma 2.8.

Proof.

Now it remains to verify that satisfies -condition.

Lemma 2.9.

Under the assumptions of Theorem 1.1, satisfies -condition.

Proof.

We first show that is bounded. If is not bounded, then by passing to a subsequence if necessary, let as .

By (2.4), there exists a constant such that as . By (2.5), one has

where is a constant independent of .

By the above inequality, one has

which is a contradiction. Hence is bounded.

Below we show that has a convergent subsequence. Notice that and is compact. Since is bounded, we may suppose that

Henceforth has a convergent subsequence.

Now we are ready to prove Theorem 1.1.

Proof of Theorem 1.1.

It is obviously that Theorem 1.1 holds from Lemmas 2.3, 2.4, 2.6, 2.7, 2.8, and 2.9 and Theorem A.

## Declarations

### Acknowledgments

This work is supported by the Specialized Research Fund for the Doctoral Program of Higher Education for New Teachers and the Science Research Foundation of Nanjing University of Information Science and Technology (20070049).

## Authors’ Affiliations

## References

- Cunningham WJ:
**A nonlinear differential-difference equation of growth.***Proceedings of the National Academy of Sciences of the United States of America*1954,**40**(4):708-713.MATHMathSciNetView ArticleGoogle Scholar - Furumochi T:
**Existence of periodic solutions of one-dimensional differential-delay equations.***The Tôhoku Mathematical Journal*1978,**30**(1):13-35.MATHMathSciNetView ArticleGoogle Scholar - May R:
*Stablity and Complexity in Model Ecosystems*. Princeton University Press, Princeton, NJ, USA; 1973.Google Scholar - Jones GS:
**The existence of periodic solutions of**.*Journal of Mathematical Analysis and Applications*1962,**5**(3):435-450. 10.1016/0022-247X(62)90017-3MATHMathSciNetView ArticleGoogle Scholar - Chow S-N, Walther H-O:
**Characteristic multipliers and stability of symmetric periodic solutions of**.*Transactions of the American Mathematical Society*1988,**307**(1):127-142.MATHMathSciNetGoogle Scholar - Herz AV:
**Solutions of**approach the Kaplan-Yorke orbits for odd sigmoid .*Journal of Differential Equations*1995,**118**(1):36-53. 10.1006/jdeq.1995.1066MATHMathSciNetView ArticleGoogle Scholar - Kaplan J, Yorke J:
**Ordinary differential equations which yield periodic solutions of differential delay equations.***Journal of Mathematical Analysis and Applications*1974,**48**(2):317-324. 10.1016/0022-247X(74)90162-0MATHMathSciNetView ArticleGoogle Scholar - Kaplan J, Yorke J:
**On the stability of a periodic solution of a differential delay equation.***SIAM Journal on Mathematical Analysis*1975,**6**(2):268-282. 10.1137/0506028MATHMathSciNetView ArticleGoogle Scholar - Yorke J:
**Asymptotic stability for one dimensional differential-delay equations.***Journal of Differential Equations*1970,**7**(1):189-202. 10.1016/0022-0396(70)90132-4MATHMathSciNetView ArticleGoogle Scholar - Chapin S:
**Periodic solutions of differential-delay equations with more than one delay.***The Rocky Mountain Journal of Mathematics*1987,**17**(3):555-572. 10.1216/RMJ-1987-17-3-555MATHMathSciNetView ArticleGoogle Scholar - Walther H-O:
**Density of slowly oscillating solutions of**.*Journal of Mathematical Analysis and Applications*1981,**79**(1):127-140. 10.1016/0022-247X(81)90014-7MATHMathSciNetView ArticleGoogle Scholar - Nussbaum RD:
**Periodic solutions of special differential equations: an example in nonlinear functional analysis.***Proceedings of the Royal Society of Edinburgh. Section A*1978,**81**(1-2):131-151. 10.1017/S0308210500010490MATHMathSciNetView ArticleGoogle Scholar - Nussbaum RD:
**A Hopf global bifurcation theorem for retarded functional differential equations.***Transactions of the American Mathematical Society*1978,**238**(1):139-164.MATHMathSciNetView ArticleGoogle Scholar - Nussbaum RD:
**Uniqueness and nonuniqueness for periodic solutions of**.*Journal of Differential Equations*1979,**34**(1):25-54. 10.1016/0022-0396(79)90016-0MATHMathSciNetView ArticleGoogle Scholar - Li J, He X-Z, Liu Z:
**Hamiltonian symmetric groups and multiple periodic solutions of differential delay equations.***Nonlinear Analysis: Theory, Methods & Applications*1999,**35**(4):457-474. 10.1016/S0362-546X(97)00623-8MATHMathSciNetView ArticleGoogle Scholar - Li J, He X-Z:
**Multiple periodic solutions of differential delay equations created by asymptotically linear Hamiltonian systems.***Nonlinear Analysis: Theory, Methods & Applications*1998,**31**(1-2):45-54. 10.1016/S0362-546X(96)00058-2MATHView ArticleGoogle Scholar - Guo Z, Yu J:
**Multiplicity results for periodic solutions to delay differential equations via critical point theory.***Journal of Differential Equations*2005,**218**(1):15-35. 10.1016/j.jde.2005.08.007MATHMathSciNetView ArticleGoogle Scholar - Llibre J, Tarţa A-A:
**Periodic solutions of delay equations with three delays via bi-Hamiltonian systems.***Nonlinear Analysis: Theory, Methods & Applications*2006,**64**(11):2433-2441. 10.1016/j.na.2005.08.023MATHMathSciNetView ArticleGoogle Scholar - Jekel S, Johnston C:
**A Hamiltonian with periodic orbits having several delays.***Journal of Differential Equations*2006,**222**(2):425-438. 10.1016/j.jde.2005.08.013MATHMathSciNetView ArticleGoogle Scholar - Fei G:
**Multiple periodic solutions of differential delay equations via Hamiltonian systems. I.***Nonlinear Analysis: Theory, Methods & Applications*2006,**65**(1):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 Analysis: Theory, Methods & Applications*2006,**65**(1):40-58. 10.1016/j.na.2005.06.012MATHMathSciNetView ArticleGoogle Scholar - Han MA:
**Bifurcations of periodic solutions of delay differential equations.***Journal of Differential Equations*2003,**189**(2):396-411. 10.1016/S0022-0396(02)00106-7MATHMathSciNetView ArticleGoogle Scholar - Dormayer P:
**The stability of special symmetric solutions of****with small amplitudes.***Nonlinear Analysis: Theory, Methods & Applications*1990,**14**(8):701-715. 10.1016/0362-546X(90)90045-IMATHMathSciNetView ArticleGoogle Scholar - Benci V:
**On critical point theory for indefinite functionals in the presence of symmetries.***Transactions of American Mathematical Society*1982,**247**(2):533-572.MathSciNetView ArticleGoogle Scholar - Rabinowitz PH:
*Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics*.*Volume 65*. American Mathematical Society, Providence, RI, USA; 1986.Google Scholar - Fei G:
**On periodic solutions of superquadratic Hamiltonian systems.***Electronic Journal of Differential Equations*2002,**8:**1-12.View ArticleGoogle Scholar - Chang KC:
*Critical Point Theory and Applications*. Shanghai Science & Technology Press, Beijing, China; 1986.MATHGoogle Scholar - Long Y, Zehnder E:
**Morse-theory for forced oscillations of asymptotically linear Hamiltonian systems.**In*Stochastic Processes in Physics and Geometry*. World Scientific, Singapore; 1990:528-563.Google Scholar - Mawhin J, Willem M:
**Critical point theory and Hamiltonian systems.**In*Applied Mathematical Sciences*.*Volume 74*. Springer, Berlin, Germany; 1989.Google Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.