tailieunhanh - Báo cáo hóa học: " Research Article A Generalized Wirtinger’s Inequality with Applications to a Class of Ordinary Differential Equations"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article A Generalized Wirtinger’s Inequality with Applications to a Class of Ordinary Differential Equations | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 710475 7 pages doi 2009 710475 Research Article A Generalized Wirtinger s Inequality with Applications to a Class of Ordinary Differential Equations Rong Cheng1 2 and Dongfeng Zhang1 1 Department of Mathematics Southeast University Nanjing 210096 China 2 College of Mathematics and Physics Nanjing University of Information Science and Technology Nanjing 210044 China Correspondence should be addressed to Rong Cheng mathchr@ Received 5 January 2009 Revised 27 February 2009 Accepted 10 March 2009 Recommended by Ondrej Dosly 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 sn 1f x t - kr and x t sn 1g t x t - ks where x e RP f e C RP RP and g e C RxRp 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. Copyright 2009 R. Cheng and D. 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. 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 n x t - y f x t - kr C1-1 and the following nonautonomous delay differential equation x t -Ệ g t x t - ks where x e RP f e C RP RP and g e C R x RP RP and r 0 s 0 are two given constants. 2 Journal of Inequalities and Applications For the special case that n 1 and p 1 various problems on the solutions of such as the existence of periodic solutions bifurcations of periodic solutions and stability of .