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 (Formula presented.) for (Formula presented.) where (Formula presented.) is the generator of a (Formula presented.) -semigroup defined on a Banach space (Formula presented.) , (Formula presented.) denote fractional difference in Weyl-like sense and (Formula presented.) satisfies Lipchitz conditions of global and local type. We introduce the notion of (Formula presented.) -resolvent sequence (Formula presented.) and we prove that a mild solution of (Formula presented.) corresponds to a fixed point of (Formula presented.) 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. © 2015 Taylor & Francis.