Journal of Inequalities and Applications
Volume 2009 (2009), Article ID 710475, 7 pages
doi:10.1155/2009/710475
Research Article

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

Rong Cheng1,2 and Dongfeng Zhang1

1Department of Mathematics, Southeast University, Nanjing 210096, China
2College of Mathematics and Physics, Nanjing University of Information Science and Technology, Nanjing 210044, China

Received 5 January 2009; Revised 27 February 2009; Accepted 10 March 2009

Academic Editor: Ondrej Dosly

Copyright © 2009 Rong Cheng and Dongfeng Zhang. 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

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 x˙(t)=−∑k=1nf(x(t−kr)), and x˙(t)=−∑k=1ng(t,x(t−ks)), where x∈ℝp, f∈C(ℝp,ℝp), and g∈C(ℝ×ℝp,ℝp) and r>0, s>0 are two given constants. Under some suitable conditions on f and g, lower bounds of periods of periodic solutions for the equations aforementioned are obtained.