Maximal Regularity in Lp Spaces for Discrete Time Fractional Shifted Equations

In this paper, we are presenting a new method based on operator-valued Fourier multipliers to characterize the existence and uniqueness of ℓp-solutions for discrete time fractional models in the form Δαu(n,x)=Au(n,x)+∑j=1kβju(n−τj,x)+f(n,u(n,x)),n∈Z,x∈Ω⊂RN,βj∈Randτj∈Z where A is a closed linear operator defined on a Banach space X and Δα denotes the Grünwald–Letnikov fractional derivative of order α>0. If X is a UMD space, we provide this characterization only in terms of the R-boundedness of the operator-valued symbol associated to the abstract model. To illustrate our results, we derive new qualitative properties of nonlinear difference equations with shiftings, including fractional versions of the logistic and Nagumo equations. © 2017 Elsevier Inc.