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Kato’s Theorem on the Integration of Non-Autonomous Linear Evolution Equations

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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↦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 Functional Analysis. We prove Yosida’s assumptions to be equivalent to Kato’s condition that t↦A(t)x is continuously differentiable for each x∈D.

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  1. Fachbereich Mathematik, Universität Stuttgart, D-70569, Stuttgart, Germany

    Jochen Schmid & Marcel Griesemer

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  1. Jochen Schmid
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  2. Marcel Griesemer
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Correspondence to Marcel Griesemer.

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Schmid, J., Griesemer, M. Kato’s Theorem on the Integration of Non-Autonomous Linear Evolution Equations. Math Phys Anal Geom 17, 265–271 (2014). https://doi.org/10.1007/s11040-014-9154-5

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  • DOI: https://doi.org/10.1007/s11040-014-9154-5

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