Well posedness for semidiscrete fractional Cauchy problems with finite delay

We address the study of well posedness on Lebesgue spaces of sequences for the following fractional semidiscrete model with finite delay (0.1)Δαu(n)=Tu(n)+βu(n−τ)+f(n),n∈N,0<α≤1,β∈R,τ∈N0,where T is a bounded linear operator defined on a Banach space X (typically a space of functions like Lp(Ω),1