Existence and multiplicity of solutions for a class of damped vibration problems with impulsive effects

Some sufficient conditions on the existence and multiplicity of solutions for the damped vibration problems with impulsive effects {u ''(t) + g(t)u'(t) + f(t,u(t)) =0(,) a.e. t is an element of [0,T], u(0) = t(T) = 0, Delta u'(t(j)) = u'(t(j)(+)) - u'(t(j)(-)) = I-j(u(t(j))), j= 1, ... ,p, are established, where t(0) = 0 < t(1) < ... < tp < t(p+1) = T,g is an element of L-1 (0,T;R), f: [0, T] x R -> R is continuous, and I-j : R -> R, j = 1, ... ,p, are continuous. The solutions are sought by means of the Lax-Milgram theorem and some critical point theorems. Finally, two examples are presented to illustrate the effectiveness of our results.