Abstract and Applied Analysis
Volume 2011 (2011), Article ID 489148, 8 pages
http://dx.doi.org/10.1155/2011/489148
Research Article

The Lagrangian Stability for a Class of Second-Order Quasi-Periodic Reversible Systems

Yanling Shi and Jia Li

Department of Mathematics, Southeast University, Nanjing 211189, China

Received 3 July 2011; Accepted 22 September 2011

Academic Editor: Martin D. Schechter

Copyright © 2011 Yanling Shi and Jia Li. 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 two-order differential equation, ( Φ 𝑝 ( 𝑥 ′ ) ) ′ + 𝑓 ( 𝑥 , 𝑡 ) Φ 𝑝 ( 𝑥 ′ ) + 𝑔 ( 𝑥 , 𝑡 ) = 0 , where Φ 𝑝 ( 𝑠 ) = | 𝑠 | ( 𝑝 − 2 ) 𝑠 , 𝑝 > 0 . 𝑓 ( 𝑥 , 𝑡 ) and 𝑔 ( 𝑥 , 𝑡 ) are real analytic functions in 𝑥 and 𝑡 , 2 𝑎 𝜋 𝑝 − periodic in 𝑥 , and quasi-periodic in 𝑡 with frequencies ( 𝜔 1 , … , 𝜔 𝑚 ) . Under some odd-even property of 𝑓 ( 𝑥 , 𝑡 ) and 𝑔 ( 𝑥 , 𝑡 ) , we obtain the existence of invariant curves for the above equations by a variant of small twist theorem. Then all solutions for the above equations are bounded in the sense of s u p 𝑡 ∈ 𝑅 | 𝑥 ( 𝑡 ) | < + ∞ .