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A Generalized Wirtinger's Inequality with Applications to a Class of Ordinary Differential Equations
Journal of Inequalities and Applicationsvolume 2009, Article number: 710475 (2009)
Abstract
We first prove a generalized Wirtinger's inequality. Then, applying the inequality, we study estimates for lower bounds of periods of periodic solutions for a class of delay differential equations , and , where , , and and , are two given constants. Under some suitable conditions on and , lower bounds of periods of periodic solutions for the equations aforementioned are obtained.
1. Introduction and Statement of Main Results
In the present paper, we are concerned with a generalized Wirtinger's inequality and estimates for lower bounds of periods of periodic solutions for the following autonomous delay differential equation:
and the following nonautonomous delay differential equation
where , , and , and are two given constants.
For the special case that and , various problems on the solutions of (1.1), such as the existence of periodic solutions, bifurcations of periodic solutions, and stability of solutions, have been studied by many authors since 1970s of the last century, and a lot of remarkable results have been achieved. We refer to [1–6] for reference.
The delay equation (1.1) with more than one delay and is also considered by a lot of researchers (see [7–13]). Most of the work contained in literature on (1.1) is the existence and multiplicity of periodic solutions. However, except the questions of the existence of periodic solutions with prescribed periods, little information was given on the periods of periodic solutions. Moreover, few work on the nonautonomous delay differential equation (1.2) has been done to the best of the author knowledge. Motivated by these cases, as a part of this paper, we study the estimates of periods of periodic solutions for the differential delay equation (1.1) and the nonautonomous equation (1.2). We first give a generalized Wirtinger's inequality. Then we turn to consider the problems on (1.1) and (1.2) by using the inequality.
In order to state our main results, we make the following definitions.
Definition 1.1.
For a positive constant , is called Lipschitz continuous, if for all ,
where denotes the norm in .
Definition 1.2.
For a positive constant , is called Lipschitz continuous uniformly in , if for all , and any ,
Then our main results read as follows.
Theorem 1.3.
Let be a nontrivial periodic solution of the autonomous delay differential equation (1.1) with the second derivative. Suppose that the function is Lipschitz continuous. Then one has .
Theorem 1.4.
Let be a nontrivial periodic solution of the nonautonomous delay differential equation (1.2) with the second derivative. Suppose that the function is periodic with respect to and Lipschitz continuous uniformly in . If the following limit
exists for all and and is uniformly bounded, then one has .
2. Proof of the Main Results
We will apply Wirtinger's inequality to prove the two theorems. Firstly, let us recall some notation concerning the Sobolev space. It is well known that is a Hilbert space consisting of the periodic functions on which together with weak derivatives belong to . For all , let and denote the inner product and the norm in , respectively, where is the inner product in . Then according to [14], we give Wirtinger's inequality and its proof.
Lemma 2.1.
If and , then
Proof.
By the assumptions, has the following Fourier expansion:
Then Parseval equality yields that
This completes the proof.
Now, we generalize Wirtinger's inequality to a more general form which includes (2.1) as a special case. We prove the following lemma.
Lemma 2.2.
Suppose that and with . Then
Proof.
Since , by Lemma 2.1, we have
that is,
Let denote the average of , that is, . This means that . Hence, Schwarz inequality, together with (2.6) and implies that
Then the proof is complete.
Corollary 2.3.
Under the conditions of Lemma 2.1, the inequality (2.4) implies Wirtinger's inequality (2.1).
Proof.
If and , then (2.1) follows (2.4) on taking .
We call (2.4) a generalized Wirtinger's inequality. For other study of Wirtinger's inequality, one may see [15] and the references therein. Now, we are ready to prove our main results. We first give the proof of Theorem 1.3.
Proof of Theorem 1.3.
From (1.1) and Definition 1.1, for all , one has
Hence, since has the second derivative,
Noting that is also periodic, = , for . Hence, by Hölder inequality, one has
that is,
From (2.1) and , we have
Combining (2.11) and (2.12), one has
Now, we prove Theorem 1.4.
Proof.
From (1.2), Definition 1.2 and the assumptions of Theorem 1.4, for all , one has
Since is nonnegative and uniformly bounded (for all and ), there is such that . Together with the fact that has the second derivative, our estimates imply that
As in the proof of Theorem 1.3, we get
that is,
Thus, (2.1) together with (2.16) yields that
By an argument of Viete theorem with respect to the quadratic function , we have that
Remark 2.4.
Roughly speaking, the period can reach the lower bound . Let us take an example for (1.1). Take and . For each , we define a function by
Then one can check easily that is Lipschitz continuous with . Let . One has
This means that is a periodic solution of (1.2) with period .
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Acknowledgments
The authors would like to thank the referee for careful reading of the paper and many valuable suggestions. Supported by the specialized Research Fund for the Doctoral Program of Higher Education for New Teachers, the National Natural Science Foundation of China (10826035) and the Science Research Foundation of Nanjing University of Information Science and Technology (20070049).
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Keywords
 Positive Constant
 Periodic Solution
 Sobolev Space
 Periodic Function
 Quadratic Function