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 (∗) ∆u(n) = Au(n+ 1) + f(n, u(n)), n ∈ Z, for 0 < α ≤ 1 where A is the generator of a C0-semigroup defined on a Banach space X, ∆ denote fractional difference in Weyl-like sense and f satisfies Lipchitz conditions of global and local type. We introduce the notion of α-resolvent sequence {Sα(n)}n∈N0 ⊂ B(X) and we prove that a mild solution of (∗) corresponds to a fixed point of u(n+ 1) = n ∑ j=−∞ Sα(n− j)f(j, u(j)), n ∈ Z. 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.