Maximal regularity of discrete second order Cauchy problems in Banach spaces

Abstract

Abstract. We characterize the discrete maximal regularity for second order difference equations by means of spectral and R-boundedness properties of the resolvent set.

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

“…In Ref. [28], the authors used a Fourier multiplier theorem, due to Blunck (see Ref. [13], Theorem 1.3), to study maximal regularity of equation (1.2) in the border case r ¼ 1.…”

Semilinear evolution equations on discrete time and maximal regularity

“…Furthermore, the authors investigated maximal regularity in discrete Hölder spaces for the Crank-Nicolson scheme. In [18] maximal regularity for linear parabolic difference equations is treated, whereas in [13] a characterization in terms of R-boundedness properties of the resolvent operator for linear second order difference equations was given. See also the recent paper by Kalton and Portal [21], where they discussed maximal regularity of power-bounded operators and relate the discrete to the continuous time problem for analytic semigroups.…”

Well-posedness of second order evolution equation on discrete time

“…A motivation for our studies in this paper stems in the recent article by Arendt and Bu [4] and Cuevas and Lizama [19]. In the first one the authors consider the operator-valued Marcinkiewich multiplier theorem and maximal regularity, and the second one the authors shown a characterization for the maximal regularity for a second order difference equation by R-boundedness properties of the resolvent operator which defines the equation.…”

“…The sequences of linear and bounded operators C(n) and S(n) were introduced in [19] to represent the solution of (2.2) in the border case r = 1. …”

“…In recent years it has become apparent that one need not only the classical theorems but also vector-valued extensions with operator-valued multiplier functions or symbols. In particular, in [19] the authors used a vector-valued Fourier multiplier theorem due to Blunck (see [12], Theorem 1.3), to study maximal regularity of second order difference equations. These approach will be also our method in this paper.…”