Lebesgue Regularity for Differential Difference Equations with Fractional Damping

We provide necessary and sufficient conditions for the existence and uniqueness of solutions belonging to the vector-valued space of sequences lp(ℤ, X) for equations that can be modeled in the form (Formula presented.) where X is a Banach space, f ∈ lp(ℤ, X), A is a closed linear operator with domain D(A) defined on X, and G is a nonlinear function. The operator Δγ denotes the fractional difference operator of order γ>0 in the sense of Grünwald-Letnikov. Our class of models includes the discrete time Klein-Gordon, telegraph, and Basset equations, among other differential difference equations of interest. We prove a simple criterion that shows the existence of solutions assuming that f is small and that G is a nonlinear term. Copyright © 2018 John Wiley & Sons, Ltd.