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

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 , uΔ 2 t w t uΔ σ t ∇F σ t , u σ t , Δ-a.e. t ∈ 0, T κ , u 0 − u T 0, uΔ 0 − uΔ T 0, where uΔ t denotes the delta or Hilger derivative of u at t, uΔ 2 t uΔ Δ t , σ is the forward jump operator, T is a positive constant, w ∈ R 0, T , , ew T, 0 1, and F : 0, T × N → . 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.