Using some fixed point theorems, the authors investigate the existence of periodic solutions in the α-norm for the partial functional differential equation with infinite delay
dudt(t)=−Au(t)+f(t,ut),t∈ℝ.
Here f:ℝ×α→X is a continuous function, σ-periodic in its first argument, X is a Banach space, α is the phase space, A:D(A)⊆X→X is a closed linear operator, for which −A generates an analytic semigroup, and ut, t∈ℝ, is the historic function defined on (−∞,0]. An application of the achieved results for a reaction-diffusion system with infinite delay is also presented.

analytic semigroup; partial functional differential equations; α-norm; multivalued maps; condensing maps; periodic solutions; fractional power of operators