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Projectors on the Generalized Eigenspaces for Neutral Functional Differential Equations in Lp Spaces

Part of: Parabolic equations and systems Functional-differential and differential-difference equations Nonlinear operators and their properties General theory of linear operators

Published online by Cambridge University Press:  20 November 2018

Arnaud Ducrot ,
Zhihua Liu  and
Pierre Magal
Arnaud Ducrot
Affiliation:
UMR CNRS 5251 IMB and INRIA Bordeaux sud-ouest Anubis, Université de Bordeaux, 33000 Bordeaux, France, e-mail: arnaud.ducrot@u-bordeaux2.fr, pierre.magal@u-bordeaux2.fr
Zhihua Liu
Affiliation:
(Liu) Department of Mathematics, Beijing Normal University, Beijing 100875, PR China e-mail: zhihualiu@bnu.edu.cn
Pierre Magal
Affiliation:
(Liu) Department of Mathematics, Beijing Normal University, Beijing 100875, PR China e-mail: zhihualiu@bnu.edu.cn
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Abstract

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We present the explicit formulas for the projectors on the generalized eigenspaces associated with some eigenvalues for linear neutral functional differential equations $\left( \text{NFDE} \right)$ in ${{L}^{p}}$ spaces by using integrated semigroup theory. The analysis is based on the main result established elsewhere by the authors and results by Magal and Ruan on non-densely defined Cauchy problem. We formulate the $\text{NFDE}$ as a non-densely defined Cauchy problem and obtain some spectral properties from which we then derive explicit formulas for the projectors on the generalized eigenspaces associated with some eigenvalues. Such explicit formulas are important in studying bifurcations in some semi-linear problems.

Keywords

neutral functional differential equationssemi-linear problemintegrated semigroupspectrumprojectors

MSC classification

Secondary: 34K05: General theory 35K57: Reaction-diffusion equations 47A56: Functions whose values are linear operators (operator and matrix valued functions, etc., including analytic and meromorphic ones) 47H20: Semigroups of nonlinear operators
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 62 , Issue 1 , 01 February 2010 , pp. 74 - 93
Copyright
Copyright © Canadian Mathematical Society 2010

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