Semilinear evolution equations on discrete time and maximal regularity

Abstract

This paper deals with the existence and stability of solutions for semilinear equations on Banach spaces by using recent characterizations of discrete maximal regularity. As application we examine the asymptotic behavior of discrete control systems.

Regularity of Difference Equations on Banach Spaces

“…An application of Blunck's theorem to difference equations is in their infancy. Most contributions in this area are due to Portal [160][161][162] and the works of the authors [53][54][55] and collaborators.…”

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

“…We note that conditions of type (i) have previously been considered in the literature, see, e.g. [27,29,30].…”

“…[28]) to investigate maximal regularity of the harmonic oscillator Equation (see Examples 7.5 and 8.5). In Cuevas and Lizama [30], the authors considered an application of an abstract result similar to our Theorem 5.1 for digital communication channels (see Ref. [48]), they also considered applications to more general semilinear discrete control systems.…”

Well-posedness of second order evolution equation on discrete time

“…In [7], well-posedness of difference schemes for abstract elliptic problems in L p spaces was studied. In [26], maximal regularity for linear parabolic difference equations is treated while in [21,22] the authors has made a perturbation theory for semilinear evolution equations on discrete time for first and second order using discrete maximal regularity; see also the recent paper by Kalton and Portal [30], where they discussed maximal regularity of power-bounded operators and relate the discrete to the continuous time problem for analytic semigroups. Recently, discrete maximal regularity for functional difference equations with infinite delay was considered in [20].…”