Regularity of Difference Equations on Banach Spaces

Maximal regularity in l spaces for discrete time fractional shifted equations

“…We immediately obtain a characterization of maximal p -regularity for Hilbert spaces, since each Hilbert space is U M D and, in such case, R-boundedness coincides with boundedness [2].…”

“…Now, we recall the following Fourier multiplier theorem for operator-valued symbols given by S. Blunck [7,2]. This theorem corresponds to the discrete version of a notable result independently proven by Weis [33] and Amann [3] which provides sufficient conditions to ensure when an operator-valued symbol is a multiplier.…”

“…Now, we recall the following Fourier multiplier theorem for operator valued symbols given by S. Blunck [7,2]. Since we are dealing with unbounded operators we need to state a slight modification of Blunck's theorem concerning two Banach spaces instead of one.…”

$\ell_{p}$ -maximal regularity for a class of fractional difference equations on UMD spaces: The case $1\lt \alpha\leq2$

“…For a recent review of this topic in the context of discrete models on Banach spaces, see the monograph [3] and references therein. Although research in this area has been done, there are many interesting questions related to the study of fractional difference equations that remain unanswered.…”

“…We point out that characterizations of maximal regularity for evolution equations using methods of operator valued Fourier multiplier theorems has been already studied (see for example [3]). For instance, S. Bu in [11] and [12] used Fourier multipliers to characterize the Lebesgue maximal regularity of fractional evolution equations in compact intervals.…”