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 [GRAPHICS] for [GRAPHICS] where [GRAPHICS] is the generator of a [GRAPHICS] -semigroup defined on a Banach space [GRAPHICS] , [GRAPHICS] denote fractional difference in Weyl-like sense and [GRAPHICS] satisfies Lipchitz conditions of global and local type. We introduce the notion of [GRAPHICS] -resolvent sequence [GRAPHICS] and we prove that a mild solution of [GRAPHICS] corresponds to a fixed point of [GRAPHICS] We 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.