Well posedness for semidiscrete fractional Cauchy problems with finite delay

Abstract

We address the study of well posedness on Lebesgue spaces of sequences for the following fractional semidiscrete model with finite delaywhere T is a bounded linear operator defined on a Banach space X (typically a space of functions like L p (Ω), 1 < p < ∞) and ∆ α corresponds to the time discretization of the continuous Riemann-Liouville fractional derivative by means of the Poisson distribution. We characterize the existence and uniqueness of solutions in vector-valued Lebesgue spaces of sequences of the model (0.1) in terms of boundedness of the operator-valued symbolwhenever 0 < α ≤ 1 and X satisfies a geometrical condition. For this purpose, we use methods from operator-valued Fourier multipliers and resolvent operator families associated to the homogeneous problem. We apply this result to show a practical and computational criterion in the context of Hilbert spaces.

LebesguE Regularity for Nonlocal Time-Discrete Equations with Delays

“…, are equivalent. This characterization coincides perfectly as the counterpart of the result achieved in the paper [29] where also an R-boundedness condition on two sets is needed. We note that in practice, tools to check this condition are generally not easy to find.…”

“…In [28] p -maximal regularity for the equation (1.1) with infinite delay was studied in Z for all α > 0 when T is an unbounded operator. Recently, in [29] the authors characterized the p -maximal regularity for the finite delayed equation ∆ α u(n) = T u(n) + βu(n − τ ) + f (n), n ∈ N 0 , n ≥ 1, β ∈ R, u(j) = 0, j = −τ, ..., 0, τ ∈ N 0 , (1.1) whenever 0 < α ≤ 1. However, the validity of such characterization for the case of 1 < α ≤ 2 was left as an open problem.…”

Convexity, monotonicity, and positivity results for sequential fractional nabla difference operators with discrete exponential kernels

“…In the late 2000s, the work of Atici and Eloe [10][11][12][13][14] provided many new insights into discrete fractional calculus and open the door to numerous further investigations. Many other papers have subsequently appeared, such as those by Abdeljawad, 15 Agarwal et al, 16 Ahrendt et al, 17 Anastassiou, 18 Atici and Abdeljawad, 19 Eloe and Ouyang, 20 Ferreira, 21,22 Ghorbanian and Rezapour, 23 Hein et al, 24 Holm, 25 Huang et al, 26 Leal et al, 27 Lizama, 28 Lizama and Murillo-Arcila, 29,30 Ma et al, 31 Reunsumrit and Sitthiwirattham, 32 Rezapour and Salehi, 33 and Xu and Zhang, 34 each of which has further explored various fundamental properties of discrete fractional differences and sums-for example, Gronwall-type inequalities, boundary and initial value problems, and various modifications of the fractional sum and difference used in above mentioned papers of Atici and Eloe. We also note that there have also been some interesting attempts to use fractional difference calculus in modeling-for example, see the papers by Atici et al, 35 Wang et al, 36 and Wu et al 37 In particular, the recent paper by Atici et al 38 discussed the use of discrete fractional calculus in cancer and tumor modeling. An earlier discrete fractional model for tumor growth can be found in the paper by Atici andŞengül.…”

Maximal ℓp‐regularity of multiterm fractional equations with delay

“…The first reference in the context of discrete maximal regularity for fractional equations was given by Lizama 30 where he handles this study for fractional difference differential equations using methods of functional analysis and operator theory. Following this research line, other studies 26,27,31,32 correspond to studies of maximal ℓ p ‐regularity in the context of fractional equations with time variable both in and (see also Kovács et al 33 and Lizama and Murillo‐Arcila 34 where a connection between maximal ℓ p ‐regularity and nonlocal time steppings is established).…”