A Functional Analytic Perspective to Delay Differential Equations

TL;DR

This paper extends the solution theory for delay differential equations from Hilbert spaces to Banach spaces by treating differentiation as an operator over the entire real line, emphasizing causality and functional analysis techniques.

Contribution

It generalizes existing delay differential equation solutions to Banach spaces using a novel operator approach and causality considerations.

Findings

Successful extension of solution theory to Banach spaces

Demonstration of causality in delay differential equations

Application of contraction mapping theorem in this context

Abstract

We generalize the solution theory for a class of delay type differential equations developed in a previous paper, dealing with the Hilbert space case, to a Banach space setting. The key idea is to consider differentiation as an operator with the whole real line as the underlying domain as a means to incorporate pre-history data. We focus our attention on the issue of causality of the differential equations as a characterizing feature of evolutionary problems and discuss various examples. The arguments mainly rely on a variant of the contraction mapping theorem and a few well-known facts from functional analysis.