On variational methods for a class of damped vibration problems

In the present paper, the following damped vibration problems: equation(1.1){ü(t)+q(t)u̇(t)=A(t)u(t)+∇F(t,u(t)),a.e. t∈[0,T]u(0)−u(T)=u̇(0)−eQ(T)u̇(T)=0, and equation(1.1λ){ü(t)+q(t)u̇(t)=A(t)u(t)+λ∇F(t,u(t)),a.e. t∈[0,T]u(0)−u(T)=u̇(0)−eQ(T)u̇(T)=0, are studied, where T>0T>0, λ>0λ>0, q∈L1(0,T;R)q∈L1(0,T;R), Q(t)=∫0tq(s)ds, A(t)=[aij(t)]A(t)=[aij(t)] is a symmetric N×NN×N matrix-valued function defined in [0,T][0,T] with aij∈L∞([0,T])aij∈L∞([0,T]) for all i,j=1,2,…,Ni,j=1,2,…,N and there exists a positive constant θθ such that A(t)ξ⋅ξ≥θ|ξ|2A(t)ξ⋅ξ≥θ|ξ|2 for all ξ∈RNξ∈RN and a.e. t∈[0,T]t∈[0,T]. The variational principles are given, and an existence theorem and three multiplicity theorems of periodic solutions are obtained.