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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 896168, 3 pages
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.
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.
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  have showed that. Busenberg et al.  refined the earlier estimate ofin ; they  showed that. At the same time, they  also gave a simple proof of the better lower boundin spaces with the norm defined via an inner product. Mawhin and Walter  showed how some lower bounds on the period of the possible periodic solutions of autonomous ordinary differential equations due to Yorke  are easy consequences of the general principle. Zevin and Pinsky  investigated a class of Lipschitzian differential equations of even order; they obtained the minimal periods of periodic solutions. In 2012, Domoshnitsky et al.  investigated componentwise positivity of solutions to periodic boundary problem for linear functional differential system. Recently, Cheng and Zhang  proved a generalized Wirtingers inequality. Based on this inequality, they  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 , 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 . 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:
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 ). Supposeand. Then the functionhas an inversesatisfying with.
Lemma 3 (see ). 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
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.
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.
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