Kato's theorem on the integration of non-autonomous linear evolution equations

TL;DR

This paper compares Kato's and Yosida's approaches to integrating non-autonomous linear evolution equations in Banach spaces, clarifying assumptions and establishing their equivalence.

Contribution

It clarifies the assumptions in Yosida's classic work and proves their equivalence to Kato's differentiability condition for the evolution equations.

Findings

Yosida's assumptions are equivalent to Kato's differentiability condition.

The paper clarifies the regularity conditions needed for integration.

It provides a detailed comparison of foundational works in the field.

Abstract

This paper is devoted to a comparison of early works of Kato and Yosida on the integration of non-autonomous linear evolution equations $\dot{x} = A(t)x$ in Banach space, where the domain $D$ of $A(t)$ is independent of $t$. Our focus is on the regularity assumed of $t\mapsto A(t)$ and our main objective is to clarify the meaning of the rather involved set of assumptions given in Yosida's classic and highly influential \emph{Functional Analysis}. We prove Yosida's assumptions to be equivalent to Kato's condition that $t\mapsto A(t)x$ is continuously differentiable for each $x\in D$.