Convergence theorems for integrated semigroups
type: Item
instance of: scholarly article
title: Convergence theorems for integrated semigroups
zbMATH Open document ID: 0786.47036
author: Binghui Wu
author: Stavros N. Busenberg
published in: Differential and Integral Equations
publication date: 1992-10-04T00:00:00Z
review text: The paper provides approximation results for the abstract homogeneous \[ {du\over dt}=Au,\quad u(0)= u_ 0,\tag{1} \] and nonhomogeneous \[ {du\over dt}= Au(t)+ f(t),\quad u(0)=x\tag{2} \] Cauchy problems. The novelty consists in the search for integral solutions to (1) and (2). Then \(u(t)\) satisfies the integrated equation \[ u(t)= u_ 0+ A \int^ t_ 0 u(s)ds,\tag{3} \] in the case (1), and \[ u(t)= x+ A\int^ t_ 0 u(s)ds+ \int^ t_ 0 f(s)ds,\tag{4} \] in the second case. This idea was introduced by \textit{G. Da Prato} and \textit{E. Sinestrari} [Ann. Sc. Norm. Sup. Pisa, Cl. Sci., IV. Ser. 14, No. 2, 285-344 (1987; Zbl 0652.34069)]. It is useful when \(u_ 0\) and \(x\) are not sufficiently regular. The main theorem formulated and proved in the present paper concerns the generalization of the famous Trotter-Kato theorem of semigroup theory. In the case under study it states that resolvent consistency, under appropriate conditions, implies convergence of integral solutions. One simple example illustrates the possibility of application to the population model. The paper is clearly written and of interest to specialists in differential equations in Banach spaces.
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 35A35
Mathematics Subject Classification ID: 35C15
Mathematics Subject Classification ID: 35G10
Mathematics Subject Classification ID: 35K25
zbMATH DE Number: 68010
zbMATH Keywords: integrated semigroups
zbMATH Keywords: generator
zbMATH Keywords: nonhomogeneous Cauchy problem
zbMATH Keywords: integrated equation
zbMATH Keywords: Trotter-Kato theorem
zbMATH Keywords: semigroup theory
zbMATH Keywords: integral solutions
zbMATH Keywords: population model
reviewed by: J. J. Telega
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 68010