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

Abstract

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 ) + A u ( t ) = f ( t , u ( t ) ) , 0 < t < ω , u ( 0 ) = u ( ω ) , where A : D ( A ) ⊂ H → H is a positive definite self-adjoint operator, ω > 0 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.