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, u(Delta 2) (t) + w(t)u(Delta)(sigma(t)) = del F(sigma(t), u(sigma(t), u(sigma(t))), Delta-a.e. t. is an element of [0, T](T)(kappa), u(0) - u(T) = 0, u(Delta)(0) - u(Delta)(T) = 0, where u(Delta)(t) denotes the delta (or Hilger) derivative of u at t, u(Delta 2)(t) = (u(Delta))(Delta)(t), sigma is the forward jump operator, T is a positive constant, w is an element of R+ ([0, T](T), R), e(w)(T, 0) = 1, and F : [0, T](T) x 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.