Periodic solutions of abstract functional differential equations with infinite delay

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)=Ax(t)+L(xt)+N(xt)+f(t),t∈R, where A:D(A)⊂X→XA:D(A)⊂X→X is a closed operator defined on a Banach space XX, LL is a bounded linear map, N:B→XN:B→X is a continuous function defined on an appropriate phase space BB and f∈Lp(T,X)f∈Lp(T,X).