RT journal article
T1 Well posedness for semidiscrete fractional Cauchy problems with finite delay
A1 Lizama, Carlos
A1 Murillo Arcila, Marina
A2 Matemáticas
K1 Fractional di erences
K1 Delay equations
K1 Well-posedness
K1 Maximal regularity
K1 Operator-valued Fourier multiplier
AB 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.
PB Elsevier
SN 0377-0427
YR 2018
FD 2018
LK http://hdl.handle.net/10498/35420
UL http://hdl.handle.net/10498/35420
LA eng
DS Repositorio Institucional de la Universidad de Cádiz
RD 10-may-2026