Twin semigroups and delay equations

O. Diekmann, S. M. Verduyn Lunel*

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

Abstract

In the standard theory of delay equations, the fundamental solution does not ‘live’ in the state space. To eliminate this age-old anomaly, we enlarge the state space. As a consequence, we lose the strong continuity of the solution operators and this, in turn, has as a consequence that the Riemann integral no longer suffices for giving meaning to the variation-of-constants formula. To compensate, we develop the Stieltjes-Pettis integral in the setting of a norming dual pair of spaces. Part I provides general theory, Part II deals with “retarded” equations, and in Part III we show how the Stieltjes integral enables incorporation of unbounded perturbations corresponding to neutral delay equations.

Original languageEnglish
Pages (from-to)332-410
Number of pages79
JournalJournal of Differential Equations
Volume286
DOIs
Publication statusPublished - 15 Jun 2021

Bibliographical note

Funding Information:
A referee provided detailed constructive feedback, leading to substantial improvement of the manuscript. We are most thankful to this anonymous referee.

Publisher Copyright:
© 2021 Elsevier Inc.

Funding

A referee provided detailed constructive feedback, leading to substantial improvement of the manuscript. We are most thankful to this anonymous referee.

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