On regularity of mild solutions for autonomous linear retarded functional differential equations

TL;DR

This paper investigates the regularity of mild solutions for autonomous linear retarded functional differential equations, establishing their local Lipschitz continuity and connecting them with existing solution concepts.

Contribution

It proves the regularity of mild solutions for autonomous linear RFDEs, clarifying their relation to classical solution concepts and extending previous results.

Findings

Mild solutions are locally Lipschitz continuous on [0, ∞)

Established a connection between mild solutions and classical solutions

Extended the understanding of solution regularity for RFDEs

Abstract

The notion of mild solutions for autonomous linear retarded functional differential equations (RFDEs) has been introduced in [J. Nishiguchi, Electron.\ J. Qual.\ Theory Differ.\ Equ.\ \textbf{2023}, No.~32, 1--77] for the purpose of defining fundamental matrix solutions and obtaining a variation of constants formula for the RFDEs. This notion gives a straightforward definition of solutions to the RFDEs under discontinuous history functions compared with previous studies in the literature. For a given autonomous linear RFDE, it holds that the fundamental matrix solutions are locally Lipschitz continuous on the interval $[0, \infty)$. However, it is not apparent whether a similar property is true for the mild solutions. Here we obtain a result which shows the regularity of mild solutions on $[0, \infty)$ for autonomous linear RFDEs. The result makes clear a connection between the mild solutions and solution concepts in previous studies.