Maximal Regularity of the Discrete Harmonic Oscillator Equation

Abstract

We give a representation of the solution for the best approximation of the harmonic oscillator equation formulated in a general Banach space setting, and a characterization of l p -maximal regularity-or well posedness-solely in terms of R-boundedness properties of the resolvent operator involved in the equation.

Regularity of Difference Equations on Banach Spaces

“…Notice that the results of this section also hold in a more general form without the assumption 1 2 .T / (see [38] for details).…”

“…Notice that the above result in a more general setting and without the assumption 1 2 .T / appeared in author's work [38].…”

“…Maximal regularity has also been studied in the finite difference setting by Portal [160][161][162], Ashyralyev et al [12,13], Geissert [86], Guidetti and Piskarev [91], Kalton and Portal [111], Castro et al [37][38][39], and Cuevas et al [52][53][54][55]59]. In [160,162], the author has discussed discrete analytic semigroup and maximal regularity on discrete-time scales, respectively.…”

“…5.3 is called the fundamental solution of (5.4.1) (see [38]) and satisfies the following equation: We emphasize that from a more theoretical perspective a similar analysis as in Sect. 5.4 can be carried out when we consider more general initial conditions, but the price to pay for this is that the proof would certainly require additional l p -summability condition on fB.n/g n 5 .…”

Perturbation theory, stability, boundedness and asymptotic behaviour for second order evolution equation in discrete time

“…In Ref. [19], the authors used the Cuevas and Lizama's approach about maximal regularity of abstract second order difference Equations (see Ref. [28]) to investigate maximal regularity of the harmonic oscillator Equation (see Examples 7.5 and 8.5).…”