Maximal regularity for fractional difference equations with finite delay on UMD space

In this paper, we study the ᵖ-maximal regularity for the fractional difference equation with finite delay: {equation*} \ \ \ \ \ \ \ \ \{{array}{cc} Δαu(n)=Au(n)+γ u(n-λ)+f(n), \ n∈ N_0, λ ∈ N, γ ∈ R; u(i)=0,\ \ i=-λ, -λ+1,⋯, 1, 2, {array} . {equation*} where A is a bounded linear operator defined on a Banach space X, f: N₀→ X is an X-valued sequence and 2<α<3. We introduce an operator theoretical method based on the notion of α-resolvent sequence of bounded linear operators, which gives an explicit representation of solution. Further, using Blunck's operator-valued Fourier multipliers theorems on ᵖ(Z; X), we completely characterize the ᵖ-maximal regularity of solution when 1 < p < ∞ and X is a UMD space.