Periodicity in the αα-norm for partial functional differential equations in fading memory spaces

The aim of this work is to study the existence of periodic solutions in the αα-norm for some partial functional differential equations of neutral type in fading memory spaces. We assume that a linear part is densely defined and is the generator of an analytic semigroup. The delayed part is assumed to be periodic with respect to the first argument. In the nonhomogeneous linear case, we show that the existence of a bounded solution in R+R+ implies the existence of the periodic solution. In the nonlinear case, we use two approaches, the first one is based on the ultimate boundedness of solutions and the second one is based on the multivalued fixed point theorem.