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

http://dx.doi.org/10.4153/CJM-2010-005-2
Canad. J. Math. 62(2010), 74-93

  Published:2009-12-04
 Printed: Feb 2010

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.