Bisectorial operator pencils and the problem of bounded solutions
type: Item
instance of: scholarly article
title: Bisectorial operator pencils and the problem of bounded solutions
zbMATH Open document ID: 1272.47051
DOI: 10.3103/S1066369X1203005X
author: A. V. Pechkurov
published in: Russian Mathematics
publication date: 2012-11-20T00:00:00Z
review text: The starting point of the paper is the abstract equation \(u'- Au= f\). There are well-known results about unique solvability for any right-hand side and properties of the spectrum of \(A\), for example, some sectorial condition for \(A\). The paper is devoted to bi-sectorial pencils (bi-semigroups) of the form \(\lambda\mapsto\lambda F- G\), where the operators \(F\) and \(G\) mapping a Banach space \(X\) to a Banach space \(Y\) are bounded. Some results for \(Y^1\) bi-sectorial pencils are explained in Section 1. Here, \(Y^1\) is a complete subspace of \(Y\). The Green function for \((F\cdot)'-G\) is introduced in Section 2. Additionally, some auxiliary results for the Green function are proved there. In Section 3, a result about unique existence of abstract equations of the form \((Fu)'(t)- Gu(t)= f(t)\) is proved. Moreover, a representation of the solution by using the Green function is given. Some applications complete the paper.
reviewed by: Michael Reissig
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 47A05
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 47A56
zbMATH DE Number: 6108792
zbMATH Keywords: abstract operator equations
zbMATH Keywords: spectrum of bisectorial operator pencils
zbMATH Keywords: Green function
zbMATH Keywords: representation of solutions
zbMATH Keywords: resolvent
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6108792