Well-posedness of initial value problems for       functional differential and algebraic equations of mixed type

Hermen Jan Hupkes; Emmanuelle Augeraud-Véron

doi:10.3934/dcds.2011.30.737

Article Contents

2011, Volume 30, Issue 3: 737-765. Doi: 10.3934/dcds.2011.30.737

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Well-posedness of initial value problems for functional differential and algebraic equations of mixed type

Hermen Jan Hupkes^1, and
Emmanuelle Augeraud-Véron^2,

1.
Division of Applied Mathematics - Brown University, 182 George Street, Providence, RI 02912
2.
Laboratory of Mathematics, Images and Applications - University of La Rochelle, Avenue Michel Crépeau, 17042 La Rochelle

Received: December 31, 2009

Revised: December 31, 2010

Published: August 2011

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Abstract

We study the well-posedness of initial value problems for scalar functional algebraic and differential functional equations of mixed type. We provide a practical way to determine whether such problems admit unique solutions that grow at a specified rate. In particular, we exploit the fact that the answer to such questions is encoded in an integer n^#. We show how this number can be tracked as a problem is transformed to a reference problem for which a Wiener-Hopf splitting can be computed. Once such a splitting is available, results due to Mallet-Paret and Verduyn-Lunel can be used to compute n^#. We illustrate our techniques by analytically studying the well-posedness of two macro-economic overlapping generations models for which Wiener-Hopf splittings are not readily available.

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Mathematics Subject Classification: Primary: 34K06, 34A12; Secondary: 91B64.

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