Boundary Value Problems
Volume 2011 (2011), Article ID 845413, 18 pages
doi:10.1155/2011/845413
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

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.

Abstract

We study the following second-order differential equation: ( Φ 𝑝 ( 𝑥 ′ ) ) ′ + 𝐹 ( 𝑥 , 𝑡 ) 𝑥 ′ + 𝜔 𝑝 Φ 𝑝 ( 𝑥 ) + 𝛼 | 𝑥 | 𝑙 𝑥 + 𝑒 ( 𝑥 , 𝑡 ) = 0 , where Φ 𝑝 ( 𝑠 ) = | 𝑠 | ( 𝑝 − 2 ) 𝑠   ( 𝑝 > 1 ), 𝛼 > 0 and 𝜔 > 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.