Advances in Difference Equations
Volume 2010 (2010), Article ID 727486, 27 pages
doi:10.1155/2010/727486
Research Article

Existence of Solutions for a Class of Damped Vibration Problems on Time Scales

Yongkun Li and Jianwen Zhou

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

Received 3 June 2010; Revised 20 November 2010; Accepted 24 November 2010

Academic Editor: Kanishka Perera

Copyright © 2010 Yongkun Li and Jianwen Zhou. 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 present a recent approach via variational methods and critical point theory to obtain the existence of solutions for a class of damped vibration problems on time scale 𝕋 , 𝑢 Δ 2 ( 𝑡 ) + 𝑤 ( 𝑡 ) 𝑢 Δ ( 𝜎 ( 𝑡 ) ) = ∇ 𝐹 0 𝑥 0 0 6 0 𝑐 ( 𝜎 ( 𝑡 ) , 𝑢 ( 𝜎 ( 𝑡 ) ) ) , Δ -a.e. 𝑡 ∈ [ 0 , 𝑇 ] 𝜅 𝕋 , 𝑢 ( 0 ) − 𝑢 ( 𝑇 ) = 0 , 𝑢 Δ ( 0 ) − 𝑢 Δ ( 𝑇 ) = 0 , where 𝑢 Δ ( 𝑡 ) denotes the delta (or Hilger) derivative of 𝑢 at 𝑡 , 𝑢 Δ 2 ( 𝑡 ) = ( 𝑢 Δ ) Δ ( 𝑡 ) , 𝜎 is the forward jump operator, 𝑇 is a positive constant, 𝑤 ∈ ℛ + ( [ 0 , 𝑇 ] 𝕋 , ℝ ) , 𝑒 𝑤 ( 𝑇 , 0 ) = 1 , and 𝐹 ∶ [ 0 , 𝑇 ] 𝕋 × ℝ 𝑁 → ℝ . By establishing a proper variational setting, three existence results are obtained. Finally, three examples are presented to illustrate the feasibility and effectiveness of our results.