Existence and uniqueness of periodic solution for a class of semilinear evolution equations

This paper deals with the existence and uniqueness for the periodic boundary value problem of the semilinear evolution equation in a Hilbert space H{u′(t)+Au(t)=f(t,u(t)),00 and f:[0,ω]×H→H satisfy Caratheodory condition. We present some spectral conditions for the nonlinearity f(t,u) to guarantee the existence and uniqueness. These spectral conditions are the generalization for nonresonance condition of the self-adjoint elliptic boundary value problems.