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Projectors on the Generalized Eigenspaces for Neutral Functional Differential Equations in $L^{p}$ Spaces

  Published:2009-12-04
 Printed: Feb 2010
  • Arnaud Ducrot,
    Université de Bordeaux, France
  • Zhihua Liu,
    Beijing Normal University, China
  • Pierre Magal,
    Université de Bordeaux, France
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Abstract

We present the explicit formulas for the projectors on the generalized eigenspaces associated with some eigenvalues for linear neutral functional differential equations (NFDE) 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 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 equations, semi-linear problem, integrated semigroup, spectrum, projectors neutral functional differential equations, semi-linear problem, integrated semigroup, spectrum, projectors
MSC Classifications: 34K05, 35K57, 47A56, 47H20 show english descriptions General theory
Reaction-diffusion equations
Functions whose values are linear operators (operator and matrix valued functions, etc., including analytic and meromorphic ones)
Semigroups of nonlinear operators [See also 37L05, 47J35, 54H15, 58D07]
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 [See also 37L05, 47J35, 54H15, 58D07]
 

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