Almost automorphic mild solutions to fractional partial difference-differential equations

We study existence and uniqueness of almost automorphic solutions for nonlinear partial difference-differential equations modeled in abstract form as(*) for where is the generator of a -semigroup defined on a Banach space , denote fractional difference in Weyl-like sense and satisfies Lipchitz conditions of global and local type. We introduce the notion of -resolvent sequence and we prove that a mild solution of corresponds to a fixed point ofWe show that such mild solution is strong in case of the forcing term belongs to an appropriate weighted Lebesgue space of sequences. Application to a model of population of cells is given.