Series expansions for functional differential equations
type: Item
instance of: scholarly article
title: Series expansions for functional differential equations
zbMATH Open document ID: 0821.34064
DOI: 10.1007/BF01195491
author: Sjoerd M. Verduyn Lunel
published in: Integral Equations and Operator Theory
publication date: 1995-10-03T00:00:00Z
review text: The question is studied whether the solution of a linear functional equation \({d \over dt} Dx_ t = Lx_ t\), \(t \geq 0\), \(x_ 0 = \varphi\), \(\varphi \in {\mathcal C}\) \(({\mathcal C}\) is the Banach space of continuous functions endowed with the supremum norm, \(x_ t\) is the state at time \(t)\) can be represented by a series of elementary solutions. Expansion of the state into a linear combination of eigenvectors and generalized eigenvectors is discussed. Estimates for the inverse of the characteristic matrix \(\Delta (z) = z[I - \int^ h_ 0 e^{- zt} d \mu (t)] - \int^ h_ 0 e^{- zt} d \zeta (t)\) are presented. Series expansions for retarded and neutral equations are discussed. Using Cesàro summation a more general Laplace inversion formula is obtained. The theory of hyperbolic semigroups is used to extend the results to general neutral equations.
Mathematics Subject Classification ID: 34K05
Mathematics Subject Classification ID: 35P10
Mathematics Subject Classification ID: 47A70
zbMATH DE Number: 765296
zbMATH Keywords: expansion
zbMATH Keywords: linear functional equation
zbMATH Keywords: eigenvectors
zbMATH Keywords: retarded and neutral equations
zbMATH Keywords: Laplace inversion formula
zbMATH Keywords: hyperbolic semigroups
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 765296