The paper deals with the partial functional differential equation with finite delay in the α-norm
dudt(t)=−Au(t)+f(t,ut),t∈ℝ,
where f:ℝ×Cα→X is a continuous function, σ-periodic in the first argument, A:D(A)⊂X→X is a linear operator, −A is the infinitesimal generator of an analytic semigroup of linear operators (T(t))t≥0 on a Banach space X, and ut,t∈ℝ, is the historic function defined on [−r,0] by ut(θ)=u(t+θ). By using some fixed point theorems, the authors prove the existence of periodic solutions for the above equation in various cases (f linear or nonlinear). Some applications of the main results are also presented.

analytic semigroup; partial functional differential equations; α-norm; multivalued maps; periodic solution; Horn’s fixed point theorem