A Functional Analytic Perspective to Delay Differential Equations

2012/11/16 by Rainer Picard, Picard, Rainer, Sascha Trostorff +3 · 1 citation
Engineering · Mathematics · Computer Science · #Stability and Controllability of Differential Equations #Numerical methods for differential equations #Advanced Mathematical Modeling in Engineering

paper · pdf · doi:10.48550/arxiv.1211.3894

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.

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• Evolutionary equations with state-dependent delay

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