A Generalized Wirtinger's Inequality with Applications to a Class of Ordinary Differential Equations
© R. Cheng and D. Zhang. 2009
Received: 5 January 2009
Accepted: 10 March 2009
Published: 16 March 2009
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
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.
Then our main results read as follows.
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 .
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 , we give Wirtinger's inequality and its proof.
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.
Then the proof is complete.
Under the conditions of Lemma 2.1, the inequality (2.4) implies Wirtinger's inequality (2.1).
We call (2.4) a generalized Wirtinger's inequality. For other study of Wirtinger's inequality, one may see  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.
Now, we prove Theorem 1.4.
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).
- Han M: Bifurcations of periodic solutions of delay differential equations. Journal of Differential Equations 2003,189(2):396–411. 10.1016/S0022-0396(02)00106-7MathSciNetView ArticleMATHGoogle Scholar
- Nussbaum RD: A Hopf global bifurcation theorem for retarded functional differential equations. Transactions of the American Mathematical Society 1978, 238: 139–164.MathSciNetView ArticleMATHGoogle Scholar
- Kaplan JL, Yorke JA: 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)90162-0MathSciNetView ArticleMATHGoogle Scholar
- Nussbaum RD: Uniqueness and nonuniqueness for periodic solutions of . Journal of Differential Equations 1979,34(1):25–54. 10.1016/0022-0396(79)90016-0MathSciNetView ArticleMATHGoogle 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-IMathSciNetView ArticleMATHGoogle Scholar
- Furumochi T: Existence of periodic solutions of one-dimensional differential-delay equations. Tohoku Mathematical Journal 1978,30(1):13–35. 10.2748/tmj/1178230094MathSciNetView ArticleMATHGoogle 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-1987-17-3-555MathSciNetView ArticleMATHGoogle 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)00623-8MathSciNetView ArticleMATHGoogle 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)00058-2View ArticleMathSciNetMATHGoogle 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.2005.08.023MathSciNetView ArticleMATHGoogle 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.2005.08.013MathSciNetView ArticleMATHGoogle 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.2005.06.011MathSciNetView ArticleMATHGoogle 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.2005.06.012MathSciNetView ArticleMATHGoogle Scholar
- Mawhin J, Willem M: Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences. Volume 74. Springer, New York, NY, USA; 1989:xiv+277.View ArticleMATHGoogle Scholar
- Milovanović GV, Milovanović IŽ: Discrete inequalities of Wirtinger's type for higher differences. Journal of Inequalities and Applications 1997,1(4):301–310. 10.1155/S1025583497000209MathSciNetMATHGoogle Scholar
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.