Existence and multiplicity of periodic solutions for second-order systems at resonance

The purpose of this paper is to prove the existence and multiplicity of periodic solutions for the following second-order system at resonance: {ü+m2w2u=∇F(t,u),a.e.t∈[0,T],u̇(0)−u̇(T)=u(0)−u(T)=0, where m>0m>0. Some new existence theorems are obtained by the variational methods.