APPROXIMATION OF INTEGRATED SEMIGROUPS

Abstract

Abstract. The purpose of this paper is to show an integratedsemigroup on a Banach space can be approximated by a sequenceof integrated semigroups acting on di erent Banach spaces. 1. IntroductionThe initial value problem in a Banach space Xu 0 (t) = Au(t); u(0) = xhas been extensively studied if Ais the generator of a C 0 semigroup.Hille-Yosida theorem gives the necessary and sucient conditions in or-der that Ais the generator of a C 0 semigroup [4]. One of these conditionsis the density of the domain of A. But there are many examples that isformulated in the above problem without the density of the domain ofA(see [3]). In this case the concept of integrated semigroup introducedby Arendt [1] is very useful to treat the above problem.In this paper we study the approximation of an integrated semigroupon a Banach space X by a sequence of the integrated semigroups onBanach spaces X n . In order to prove our result, we use Theorem 2.2 in[5] that the convergence of the sequence of functions ff