Semi-Fredholm Operators and Periodic Solutions for Linear Functional Differential Equations

We deal with the inhomogeneous linear periodic equation with infinite delay of the formdx/dt=Ax(t)+B(t, xt)+F(t), whereAis the generator of aC0-semigroup on a Banach space. Assuming that it has a bounded solution, we obtain several criteria on the existence and the uniqueness of periodic solutions for the equation in the general phase space and in the concrete phase space =UCg. The key of our approach is the employment of the perturbation theory of semi-Fredholm operators to show that the period map satisfies the condition of the fixed point theorem by Chow and Hale (Funkcial. Ekvac.17(1974), 31–38).