$\ell_{p}$ -maximal regularity for a class of fractional difference equations on UMD spaces: The case $1\lt \alpha\leq2$

Abstract

Abstract. By using Blunck's operator-valued Fourier multiplier theorem, we completely characterize the existence and uniqueness of solutions in Lebesgue spaces of sequences for a discrete version of the Cauchy problem with fractional order 1 < α ≤ 2. This characterization is given solely in spectral terms on the data of the problem, whenever the underlying Banach space belongs to the U M D-class.

Well posedness for semidiscrete fractional Cauchy problems with finite delay

“…See for instance [4], [10], [11], [12] and [31]. More recently, methods from operator-valued Fourier multipliers were used in [23], [24], and [25] to successfully characterize the existence and uniqueness of p -solutions for discrete time fractional models.…”

“…In [24], the authors considered the discrete time fractional operator corresponding to the discretization, via the Poisson distribution, of the continuous Riemann-Liouville fractional derivative of order α defined on R + ; see [22,Theorem 3.5] and Definition 2.3 below. The same discrete time fractional operator has been considered in the paper [23], where it appears in connection with the study of well posedness for semidiscrete fractional problems without delay.…”

Maximal ℓp-regularity for discrete time Volterra equations with delay

“…We now show that (iii) =⇒ (ii) We claim that, by hypothesis, the set {(e it − 1)(e it + 1)M (t)} t∈T is R-bounded. Indeed, given t ∈ T and observing that Now, considering ϕ(t) = η(t) δ τ (−t) in (19) we obtain from (19) and (21) the identity w,η = u, (η • δ τ − )ˇ . Observe that ϕ ∈ C ∞ per (T) because τ ∈ Z.…”

“…The second author is supported by MEC, grant MTM2016-75963-P and GVA, Grant 18I264.01/1. [13,21,20,22]. This study is strongly connected with the necessity of optimal p − q time-space estimates for the corresponding linearized problem [2,11,13,16,14,18,17,24,25].…”

Maximal regularity in l spaces for discrete time fractional shifted equations

“…This characterization was given in terms of boundedness of the associated resolvent operator, but only in case that A is a bounded operator and 0 < α ≤ 1. See also the recent paper [26] for the case 1 < α ≤ 2. However, the study of maximal regularity for A unbounded was left open.…”