Existence and multiplicity of periodic solution for non-autonomous second-order systems with linear nonlinearity

The purpose of this paper is to study the existence and multiplicity of periodic solution for the following non-autonomous second-order systems:u¨(t)=∇F(t,u(t))a.e.t∈[0,T],u(0)-u(T)=u˙(0)-u˙(T)=0.Some new existence and multiplicity theorems are obtained by using the least action principle and the minimax method.