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 delay \begin{equation}\label{abstractlabel} \Delta^{\alpha}u(n) = Tu(n) + \beta u(n-\tau) +f(n), \quad n\in \mathbb{N},\,\,\ 0<\alpha\leq1,\,\,\,\beta\in\mathbb{R},\,\,\,\tau \in \mathbb{N}_0, \end{equation} where $T$ is a bounded linear operator defined on a Banach space $X$ (typically a space of functions like $L^p(\Omega), 1<p<\infty$) and $\Delta^{\alpha}$ 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 \eqref{abstractlabel} in terms of boundedness of the operator-valued symbol $$ ((z-1)^{\alpha}z^{1-\alpha}I -\beta z^{-\tau} -T)^{-1}, \quad |z|=1, \,\, z \neq 1, $$ whenever $0<\alpha \leq 1$ 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.