Open Access

A Generalized Wirtinger's Inequality with Applications to a Class of Ordinary Differential Equations

Journal of Inequalities and Applications20092009:710475

DOI: 10.1155/2009/710475

Received: 5 January 2009

Accepted: 10 March 2009

Published: 16 March 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq1_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq2_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq3_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq4_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq5_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq6_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq7_HTML.gif are two given constants. Under some suitable conditions on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq8_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq9_HTML.gif , 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:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_Equ1_HTML.gif
(1.1)
and the following nonautonomous delay differential equation
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_Equ2_HTML.gif
(1.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq10_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq11_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq12_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq13_HTML.gif are two given constants.

For the special case that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq14_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq15_HTML.gif , 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 [16] for reference.

The delay equation (1.1) with more than one delay and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq16_HTML.gif is also considered by a lot of researchers (see [713]). 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq17_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq18_HTML.gif is called https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq19_HTML.gif -Lipschitz continuous, if for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq20_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_Equ3_HTML.gif
(1.3)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq21_HTML.gif denotes the norm in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq22_HTML.gif .

Definition 1.2.

For a positive constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq23_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq24_HTML.gif is called https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq25_HTML.gif -Lipschitz continuous uniformly in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq26_HTML.gif , if for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq27_HTML.gif , and any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq28_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_Equ4_HTML.gif
(1.4)

Then our main results read as follows.

Theorem 1.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq29_HTML.gif be a nontrivial https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq30_HTML.gif -periodic solution of the autonomous delay differential equation (1.1) with the second derivative. Suppose that the function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq31_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq32_HTML.gif -Lipschitz continuous. Then one has https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq33_HTML.gif .

Theorem 1.4.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq34_HTML.gif be a nontrivial https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq35_HTML.gif -periodic solution of the nonautonomous delay differential equation (1.2) with the second derivative. Suppose that the function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq36_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq37_HTML.gif -periodic with respect to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq38_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq39_HTML.gif -Lipschitz continuous uniformly in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq40_HTML.gif . If the following limit
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_Equ5_HTML.gif
(1.5)

exists for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq41_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq42_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq43_HTML.gif is uniformly bounded, then one has https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq44_HTML.gif .

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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq45_HTML.gif is a Hilbert space consisting of the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq46_HTML.gif -periodic functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq47_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq48_HTML.gif which together with weak derivatives belong to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq49_HTML.gif . For all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq50_HTML.gif , let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq51_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq52_HTML.gif denote the inner product and the norm in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq53_HTML.gif , respectively, where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq54_HTML.gif is the inner product in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq55_HTML.gif . Then according to [14], we give Wirtinger's inequality and its proof.

Lemma 2.1.

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq56_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq57_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_Equ6_HTML.gif
(2.1)

Proof.

By the assumptions, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq58_HTML.gif has the following Fourier expansion:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_Equ7_HTML.gif
(2.2)
Then Parseval equality yields that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_Equ8_HTML.gif
(2.3)

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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq59_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq60_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq61_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_Equ9_HTML.gif
(2.4)

Proof.

Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq62_HTML.gif , by Lemma 2.1, we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_Equ10_HTML.gif
(2.5)
that is,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_Equ11_HTML.gif
(2.6)
Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq63_HTML.gif denote the average of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq64_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq65_HTML.gif . This means that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq66_HTML.gif . Hence, Schwarz inequality, together with (2.6) and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq67_HTML.gif implies that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_Equ12_HTML.gif
(2.7)

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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq68_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq69_HTML.gif , then (2.1) follows (2.4) on taking https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq70_HTML.gif .

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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq71_HTML.gif , one has
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_Equ13_HTML.gif
(2.8)
Hence, since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq72_HTML.gif has the second derivative,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_Equ14_HTML.gif
(2.9)
Noting that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq73_HTML.gif is also https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq74_HTML.gif -periodic, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq75_HTML.gif = https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq76_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq77_HTML.gif . Hence, by Hölder inequality, one has
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_Equ15_HTML.gif
(2.10)
that is,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_Equ16_HTML.gif
(2.11)
From (2.1) and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq78_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_Equ17_HTML.gif
(2.12)

Combining (2.11) and (2.12), one has https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq79_HTML.gif

Now, we prove Theorem 1.4.

Proof.

From (1.2), Definition 1.2 and the assumptions of Theorem 1.4, for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq80_HTML.gif , one has
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_Equ18_HTML.gif
(2.13)
Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq81_HTML.gif is nonnegative and uniformly bounded (for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq82_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq83_HTML.gif ), there is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq84_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq85_HTML.gif . Together with the fact that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq86_HTML.gif has the second derivative, our estimates imply that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_Equ19_HTML.gif
(2.14)
As in the proof of Theorem 1.3, we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_Equ20_HTML.gif
(2.15)
that is,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_Equ21_HTML.gif
(2.16)
Thus, (2.1) together with (2.16) yields that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_Equ22_HTML.gif
(2.17)
By an argument of Viete theorem with respect to the quadratic function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq87_HTML.gif , we have that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_Equ23_HTML.gif
(2.18)

Remark 2.4.

Roughly speaking, the period https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq88_HTML.gif can reach the lower bound https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq89_HTML.gif . Let us take an example for (1.1). Take https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq90_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq91_HTML.gif . For each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq92_HTML.gif , we define a function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq93_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_Equ24_HTML.gif
(2.19)
Then one can check easily that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq94_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq95_HTML.gif -Lipschitz continuous with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq96_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq97_HTML.gif . One has
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_Equ25_HTML.gif
(2.20)

This means that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq98_HTML.gif is a periodic solution of (1.2) with period https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710475/MediaObjects/13660_2009_Article_1993_IEq99_HTML.gif .

Declarations

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).

Authors’ Affiliations

(1)
Department of Mathematics, Southeast University
(2)
College of Mathematics and Physics, Nanjing University of Information Science and Technology

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© R. Cheng and D. Zhang. 2009

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