Nonlinear perturbations of positive semigroups
type: Item
instance of: scholarly article
title: Nonlinear perturbations of positive semigroups
zbMATH Open document ID: 0835.47024
DOI: 10.1007/BF02573678
author: H. Šikić
published in: Semigroup Forum
publication date: 1996-04-16T00:00:00Z
full work available at URL: https://eudml.org/doc/135319
review text: The main problem of this paper is to study the perturbation of a positive \(M\)-Pettis integrable linear semigroup. Let \(B\) be a Banach space and a linear lattice with a positive cone. Let \(B^+= \{x\in B;\) \(x\geq 0\}\). Let \((T_i)\) be a linear, positive, \(M\)-Pettis integrable semigroup. Let \(L: B^+\to B\) be an \(M\)-measurable (not necessarily linear) operator such that \(L[x(t)]\) is bounded on bounded subintervals whenever \(x(t)\in B^+\) is bounded on bounded subintervals. The author establishes the existence and uniqueness of the following integral equation. \[ V_t x= T_tx+ (MP) \int_{(0, t)} T_s [L(V_{t- s} x)] ds. \] With the help of this and other preliminaries developed in the earlier sections of the paper, the author explains how the supersemigroups and superprocesses due to \textit{F. Fitzsimmons} [Isr. J. Math. 64, No. 3, 337-361 (1988; Zbl 0673.60089)] and \textit{S. Watanabe} [J. Math. Kyoto Univ. 8, 141-167 (1968; Zbl 0159.462)] are particular cases of his results.
Mathematics Subject Classification ID: 47D03
zbMATH DE Number: 563282
zbMATH Keywords: perturbation of a positive \(M\)-Pettis integrable linear semigroup
zbMATH Keywords: linear lattice with a positive cone
zbMATH Keywords: integral equation
zbMATH Keywords: supersemigroups
zbMATH Keywords: superprocesses
reviewed by: D. Somasundaram
MaRDI profile type: Publication
cites work: Q3755956
cites work: Q5556844
cites work: Q3876158
cites work: An introduction to the theory of point processes
cites work: Superprocesses and their Linear Additive Functionals
cites work: Construction and regularity of measure-valued Markov branching processes
cites work: Q3244654
cites work: Q5723419
cites work: Semigroups of linear operators and applications to partial differential equations
cites work: Q4114083
cites work: Q4049667
cites work: Q4040434
cites work: A limit theorem of branching processes and continuous state branching processes
description: scientific article; zbMATH DE number 563282