Existence theorems of periodic solutions for a class of damped vibration problems

In the present paper, we research the following damped vibration problem(1.1)u¨(t)+g(t)u˙(t)=∇F(t,u(t))a.e.t∈[0,T]u(0)-u(T)=u˙(0)-u˙(T)=0,where T>0, g∈L∞(0,T;R), G(t)=∫0tg(s)ds, G(T)=0 and F:[0,T]×RN→R. The variational principle is given, and three existence theorems for periodic solutions are obtained.