Periodic solutions of abstract functional differential equations with infinite delay

Abstract

We characterize the existence of periodic solutions for a class of abstract retarded functional differential equations with infinite delay. We apply our results to the nonlinear equation x ′ ( t ) = A x ( t ) + L ( x t ) + N ( x t ) + f ( t ) , t ∈ R , where A : D ( A ) ⊂ X → X is a closed operator defined on a Banach space X , L is a bounded linear map, N : B → X is a continuous function defined on an appropriate phase space B and f ∈ L p ( T , X ) .