%0 Journal Article 
%A Lizama, Carlos
%A Murillo Arcila, Marina
%T Well posedness for semidiscrete fractional Cauchy problems with finite delay
%D 2018
%@ 0377-0427
%U http://hdl.handle.net/10498/35420
%X 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.
%K Fractional di erences
%K Delay equations
%K Well-posedness
%K Maximal regularity
%K Operator-valued Fourier multiplier
%~ Universidad de Cádiz