Well-posedness of second order evolution equation on discrete time

Abstract

Abstract. We characterize the well-posedness for second order discrete evolution equations in U M D spaces by means of Fourier multipliers and R-boundedness properties of the resolvent operator which defines the equation. Applications to semilinear problems are given.

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

“…Theorem 3.2 ( [18]). Let X be a UMD space and let T [ BðXÞ be an analytic operator; assume that (2.7) is fulfilled.…”

“…In particular to facilitate access to the individual topics, we review some of the standard fact on R-boundedness, UMD spaces and l p -multipliers. In the third section, we focus our attention on Castro's et al (see [18], Theorem 3.4) characterization of well-posedness of (1.2) in terms of R-boundedness and l p -multipliers property (see Theorem 3.2). In the fourth section, we prove the first main result of this work which ensures the existence and uniqueness of solutions whose second discrete r-derivative is in l p ð1 , p , þ1Þ of the semilinear problem (1.3).…”

“…On the other hand, we note that the relation between operatorvalued Fourier multiplier and R-boundedness of their symbols, in the discrete time setting, is too incipient and limited a very few articles (see, e.g. [13,14,18,40,49,50]). …”

“…In Ref. [18], the authors have characterized the well-posedness in weighted spaces l r p ðZ þ ; XÞ U {ðx n Þ : ðr 2n x n Þ [ l p ðZ þ ; XÞ} ðr . 0Þ for the following discrete second order evolution equation: Banach spaces with UMD (see Theorem 3.2).…”

Regularity of Difference Equations on Banach Spaces

“…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.…”