Boundary Value Problems
Volume 2011 (2011), Article ID 736093, 11 pages
doi:10.1155/2011/736093
Research Article

Three Solutions for Forced Duffing-Type Equations with Damping Term

Yongkun Li and Tianwei Zhang

Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China

Received 16 December 2010; Revised 6 February 2011; Accepted 11 February 2011

Academic Editor: Dumitru Motreanu

Copyright © 2011 Yongkun Li and Tianwei 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

Using the variational principle of Ricceri and a local mountain pass lemma, we study the existence of three distinct solutions for the following resonant Duffing-type equations with damping and perturbed term 𝑢 ″ ( 𝑡 ) + 𝜎 𝑢 ′ ( 𝑡 ) + 𝑓 ( 𝑡 , 𝑢 ( 𝑡 ) ) + 𝜆 𝑔 ( 𝑡 , 𝑢 ( 𝑡 ) ) = 𝑝 ( 𝑡 ) , a.e. 𝑡 ∈ [ 0 , 𝜔 ] , 𝑢 ( 0 ) = 0 = 𝑢 ( 𝜔 ) and without perturbed term 𝑢 ″ ( 𝑡 ) + 𝜎 𝑢 ′ ( 𝑡 ) + 𝑓 ( 𝑡 , 𝑢 ( 𝑡 ) ) = 𝑝 ( 𝑡 ) , a.e. 𝑡 ∈ [ 0 , 𝜔 ] , 𝑢 ( 0 ) = 0 = 𝑢 ( 𝜔 ) .