Linearization of solutionoperators for state-dependent delay equations: A simple example

Odo  Diekmann; Karolína  Korvasová

doi:10.3934/dcds.2016.36.137

Article Contents

2016, Volume 36, Issue 1: 137-149. Doi: 10.3934/dcds.2016.36.137

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Linearization of solution operators for state-dependent delay equations: A simple example

Odo Diekmann^1, and
Karolína Korvasová^1,

1.
Department of Mathematics, Utrecht University , Budapestlaan 6, 3584 CD Utrecht

Received: August 31, 2014

Revised: March 31, 2015

Published: May 2017

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Abstract

For state-dependent delay equations, it may easily happen that the equation is not differentiable. This hampers the formulation and the proof of the Principle of Linearized Stability. The fact that an equation is not differentiable does not, by itself, imply that the solution operators are not differentiable. And indeed, the aim of this paper is to present a simple example with differentiable solution operators despite of lack of differentiability of the equation. The example takes the form of a renewal equation and is motivated by a population dynamical model.

Keywords:

State-dependent delay,
linearized stability,
size structure,
maturation delay,
differentiability of solution operators,
consumer-resource.

Mathematics Subject Classification: Primary: 37N25, 92D25, 34K20; Secondary: 45D05, 58D07.

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