Well Posedness for Semidiscrete Fractional Cauchy Problems with Finite Delay - Lizama-Yanez, C (Elsevier B.V., 2018)
We address the study of well posedness on Lebesgue spaces of sequences for the following fractional semidiscrete model with finite delay 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<p<∞) and Δα 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 symbol ((z−1)αz1−αI−βz−τ−T)−1,|z|=1,z≠1,whenever 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. © 2017 Elsevier B.V.