Journal of Applied Mathematics

Volume 2013 (2013), Article ID 896168, 3 pages

http://dx.doi.org/10.1155/2013/896168

## Lower Bounds of Periods of Periodic Solutions for a Class of Differential Equations with Variable Delays

School of Mathematics and Statistics, Central South University, Changsha 410083, China

Received 2 June 2013; Accepted 25 July 2013

Academic Editor: Nazim Idrisoglu Mahmudov

Copyright © 2013 Xin-Ge Liu and Mei-Lan Tang. 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.

#### Abstract

Based on generalized Wirtinger's inequality, periods of periodic solutions of the nonautonomous differential equations with variable delays are investigated. Based on Hölder inequality, lower bounds of periods of periodic solutions for a class of functional differential equations with variable delays are obtained by a simple method.

#### 1. Introduction

The existence and multiplicity of periodic solutions, bifurcations of periodic solutions, and stability of solutions of functional differential equations have attracted the attention of many mathematicians [1–5]. A lot of remarkable results have been achieved [6–10]. However, only a few works on periods of periodic solutions have been done (see, e.g., [11–13]). Supposeis Lipschitz continuous in a Banach space with constantandis a-periodic nonconstant solution of. Lasota and Yorke [12] have showed that. Busenberg et al. [14] refined the earlier estimate ofin [12]; they [14] showed that. At the same time, they [14] also gave a simple proof of the better lower boundin spaces with the norm defined via an inner product. Mawhin and Walter [15] showed how some lower bounds on the period of the possible periodic solutions of autonomous ordinary differential equations due to Yorke [11] are easy consequences of the general principle. Zevin and Pinsky [16] investigated a class of Lipschitzian differential equations of even order; they obtained the minimal periods of periodic solutions. In 2012, Domoshnitsky et al. [17] investigated componentwise positivity of solutions to periodic boundary problem for linear functional differential system. Recently, Cheng and Zhang [18] proved a generalized Wirtingers inequality. Based on this inequality, they [18] studied estimates for lower bounds of periods of periodic solutions for the following autonomous delay differential equation:

where, andis a given constant. In their paper [18], delays are required to be constants with the form of. In this paper, we will replace the constant delaywith the generalized delay functionwith,. Furthermore, the method used in our paper is simpler than that in [18]. Lower bounds of periods of periodic solutions for a class of functional differential equations with variable delays are obtained.

Consider the lower bounds of periods of periodic solutions for the following delay differential equations:

where,,, andfor.

In order to estimate the lower bounds of periods of periodic solution of (2), we need the following definitions and lemmas.

*Definition 1. *For a positive constant,is called-Lipschitz continuous if, for all,

wheredenotes the Euclidean norm in.

Letbe the Hilbert space consisting of the-periodic functionson which together with weak derivatives belong to. For all, letand denote the inner product and the norm in, respectively, whereis the inner product in. Let, wherehas the second derivative,.

Lemma 2 (see [8]). *Supposeand. Then the functionhas an inversesatisfying with. *

Lemma 3 (see [18]). *Ifand, then
*

#### 2. Main Results

Since, by Lemma 2, the inverse ofexists. Letbe the inverse of.

Theorem 4. *Let be a nonconstant -periodic solution of the nonautonomous delay differential equation (2) and. Suppose that the functionis-Lipschitz continuous and,,. Then. *

*Proof. *Sinceis a nonconstant -periodic solution of the nonautonomous delay differential equation (2), for all, we have
We claim that if, then, for, there exists at least onesuch that
Otherwise, iffor, then
From (5), one has
Noting that, we obtain

Noting that, then. From Lemma 3, we have

Then.is a constant-periodic solution. This contradicts the assumption thatis a nonconstant-periodic solution.

For simplicity of proof, we suppose thatfor, (5) can be rewritten as
Let; one has
Applying Hölder inequality gives

Raising both sides of inequality (13) to powerand integrating both sides fromto, we have
Sinceis the inverse of , by using Lemma 2, we have
That is,
Since, obviously,. By Lemma 3, we have. So

*Remark 5. *When delay, from the second inequality of (13), we can easily obtain Theorem 1 in [18].

We can easily obtain the following result.

Corollary 6. *Letbe a nonconstant-periodic solution of the nonautonomous delay differential equation (2) and. Suppose that the functionis-Lipschitz continuous and,,. Then. *

#### Acknowledgments

The authors are grateful to the referees for their valuable comments. This study was partly supported by NSFC under Grant nos. 61271355 and 61070190, the ZNDXQYYJJH under Grant no. 2010QZZD015, and NFSS under Grant no. 10BJL020.

#### References

- M. Han, “Bifurcations of periodic solutions of delay differential equations,”
*Journal of Differential Equations*, vol. 189, no. 2, pp. 396–411, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Gopalsamy, X. Z. He, and L. Z. Wen, “On a periodic neutral logistic equation,”
*Glasgow Mathematical Journal*, vol. 33, no. 3, pp. 281–286, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. Dormayer, “The stability of special symmetric solutions of $\dot{x}(t)=\alpha f(x(t-1))$ with small amplitudes,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 14, no. 8, pp. 701–715, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Li, X.-Z. He, and Z. Liu, “Hamiltonian symmetric groups and multiple periodic solutions of differential delay equations,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 35, no. 4, pp. 457–474, 1999. View at Publisher · View at Google Scholar · View at MathSciNet - G. Fei, “Multiple periodic solutions of differential delay equations via Hamiltonian systems. I,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 65, no. 1, pp. 25–39, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Kuang and A. Feldstein, “Boundedness of solutions of a nonlinear nonautonomous neutral delay equation,”
*Journal of Mathematical Analysis and Applications*, vol. 156, no. 1, pp. 293–304, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Liu and Y. Mao, “Existence theorem for periodic solutions of higher order nonlinear differential equations,”
*Journal of Mathematical Analysis and Applications*, vol. 216, no. 2, pp. 481–490, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Lu and W. Ge, “Existence of positive periodic solutions for neutral logarithmic population model with multiple delays,”
*Journal of Computational and Applied Mathematics*, vol. 166, no. 2, pp. 371–383, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Luo and Z. Luo, “Existence of positive periodic solutions for neutral multi-delay logarithmic population model,”
*Applied Mathematics and Computation*, vol. 216, no. 4, pp. 1310–1315, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - M.-L. Tang, X.-G. Liu, and X.-B. Liu, “New results on periodic solutions for a kind of Rayleigh equation,”
*Applications of Mathematics*, vol. 54, no. 1, pp. 79–85, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. A. Yorke, “Periods of periodic solutions and the Lipschitz constant,”
*Proceedings of the American Mathematical Society*, vol. 22, pp. 509–512, 1969. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Lasota and J. A. Yorke, “Bounds for periodic solutions of differential equations in Banach spaces,”
*Journal of Differential Equations*, vol. 10, pp. 83–91, 1971. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Y. Li, “Bounds for the periods of periodic solutions of differential delay equations,”
*Journal of Mathematical Analysis and Applications*, vol. 49, pp. 124–129, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. N. Busenberg, D. C. Fisher, and M. Martelli, “Better bounds for periodic solutions of differential equations in Banach spaces,”
*Proceedings of the American Mathematical Society*, vol. 98, no. 2, pp. 376–378, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Mawhin and W. Walter, “A general symmetry principle and some implications,”
*Journal of Mathematical Analysis and Applications*, vol. 186, no. 3, pp. 778–798, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. A. Zevin and M. A. Pinsky, “Minimal periods of periodic solutions of some Lipschitzian differential equations,”
*Applied Mathematics Letters*, vol. 22, no. 10, pp. 1562–1566, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Domoshnitsky, R. Hakl, and J. Sremr, “Component-wise positivity of solutions to periodic boundary value problem for linear functional differential systems,”
*Journal of Inequalities and Applications*, vol. 2012, article 112, 2012. View at Publisher · View at Google Scholar - R. Cheng and D. Zhang, “A generalized Wirtinger's inequality with applications to a class of ordinary differential equations,”
*Journal of Inequalities and Applications*, vol. 2009, Article ID 710475, 7 pages, 2009. View at Publisher · View at Google Scholar · View at Scopus