tailieunhanh - Báo cáo hóa học: " Research Article Lagrangian Stability of a Class of Second-Order Periodic Systems"

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 Lagrangian Stability of a Class of Second-Order Periodic Systems | Hindawi Publishing Corporation Boundary Value Problems Volume 2011 Article ID 845413 18 pages doi 2011 845413 Research Article Lagrangian Stability of a Class of Second-Order Periodic Systems Shunjun Jiang Junxiang Xu and Fubao Zhang Department of Mathematics Southeast University Nanjing 210096 China Correspondence should be addressed to Junxiang Xu xujun@ Received 24 November 2010 Accepted 5 January 2011 Academic Editor Claudianor O. Alves Copyright 2011 Shunjun Jiang et al. 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. We study the following second-order differential equation p x F x t x a p p x aỊxỊlx e x t 0 where p s s p-2 s p 1 a 0 and v 0 are positive constants and l satisfies -1 l p - 2. Under some assumptions on the parities of F x t and e x t by a small twist theorem of reversible mapping we obtain the existence of quasiperiodic solutions and boundedness of all the solutions. 1. Introduction and Main Result In the early 1960s Littlewood 1 asked whether or not the solutions of the Duffing-type equation x g x t 0 are bounded for all time that is whether there are resonances that might cause the amplitude of the oscillations to increase without bound. The first positive result of boundedness of solutions in the superlinear case . g x t x TO as x to was due to Morris 2 . By means of KAM theorem Morris proved that every solution of the differential equation is bounded if g x t 2x3 -pt where pt is piecewise continuous and periodic. This result relies on the fact that the nonlinearity 2x3 can guarantee the twist condition of KAM theorem. Later several authors see 3-5 improved Morris s result and obtained similar result for a large class of superlinear function g t t . 2 Boundary Value Problems When g x satisfies 0 k gx K x Vx e R x that is the differential equation .