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Hindawi Publishing Corporation Advances in Difference Equations Volume 2010, Article ID 727486, 27

Hindawi Publishing Corporation Advances in Difference Equations Volume 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 Correspondence should be addressed to Yongkun Li, yklie@ynu.edu.cn Received 3 June 2010; Revised 20 November 2010; Accepted 24 November 2010 Academic Editor: Kanishka Perera Copyright q 2010 Y. Li and J. 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. | Hindawi Publishing Corporation Advances in Difference Equations Volume 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 Correspondence should be addressed to Yongkun Li yklie@ynu.edu.cn Received 3 June 2010 Revised 20 November 2010 Accepted 24 November 2010 Academic Editor Kanishka Perera Copyright 2010 Y. Li and J. 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. 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 uA2 t w t uA ơ fỴ VF ơ t u ơ t A-a.e. f e 0 T Ị u 0 - u T 0 uA 0 - uA T 0 where uA t denotes the delta or Hilger derivative of u at t uA2 t uA A t Ơ is the forward jump operator T is a positive constant w e R 0 T t R ew T 0 1 and F 0 T t X Rn 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. 1. Introduction Consider the damped vibration problem on time-scale T uA2 t w tfuA ơ t VF ơ t u ơ tyy A-a.e. t e 0 T Ị _ A._. 1.1 u 0 - u T 0 uA 0 - uA T 0 where uA t denotes the delta or Hilger derivative of u at t uA2 t uA A t Ơ is the forward jump operator T is a positive constant w e R 0 T T R ew T 0 1 and F 0 T T X RN R satisfies the following assumption. A F t x is A-measurable in t for every x eV and continuously differentiable in x for t e 0 T T and there exist a e C R R b e LA 0 T t R such that F t x a x b t VF t x a x b t 1.2 for all x MĨN and t e 0 T T where VF t x denotes the gradient of F t x in x. 2 Advances in Difference Equations .