Bisectorial operator pencils and the problem of bounded solutions (Q1759250)

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.