A Variation-of-Constants Formula for Abstract Functional Differential Equations in the Phase Space

Abstract

For linear functional differential equations with infinite delay in a Banach space, a variation-of-constants formula is established in the phase space. As an application one applies it to study the admissibility of some spaces of functions whose spectra are contained in a closed subset of the real line. © 2002 Elsevier Science

On the Asymptotic Periodic Solutions of Abstract Functional Differential Equations

“…By the superposition principle, the combination of these two steps allows us to study the inhomogeneous equation (1). The obtained results Theorems 3.4,3.7,3.15,3.16,3.18 extend the known ones in [15,17,22,29,30] and complement the ones in [4,7,10,21,27,28,31,40,41].…”

“…Let yðtÞ be the S-valued function defined in [21, Since s i ðDÞ is bounded by Lemma 2.7, the rest part is proved by repeating the argument in the proof of Theorem 3.4. r Remark 3.8. In the paper [21] the variation of constants formula is proved under the following condition: F is represented as…”

Decomposition and Variation-of-Constants Formula in the Phase Space for Integral Equations

“…This approach uses extensively functional analytic methods. For more information in this direction we refer the reader to the following references [3,7,8,10,12,13,14,15,18,19,20,21] and the references therein. To the best of our knowledge, it is not the case for integral equations, especially, for integral equations with infinite delay of the form (1).…”

“…Our VCF is di¤erent from the one in [2]. Indeed, to establish our VCF we take an approach similar to that in [5,13,14,15,20], without using the notion of ''sun-star''-like space. Our VCF can be applied to equations with any principal linear part, so it seems to be an e¤ective and simple tool in the study of several subjects in integral equations.…”

“…Our VCF can be applied to equations with any principal linear part, so it seems to be an e¤ective and simple tool in the study of several subjects in integral equations. For example, we can study Massera type results on the existence of almost periodic solutions for equations with almost periodic forcing term, several invariant manifolds for nonlinear equations, and so on, in the way that is discussed in [5,14,15,18,19,20,21]. In this paper as an application of the VCF we will restrict ourself to studying the admissibility of a translation-invariant closed space of functions with respect to homogeneous integral equations (Theorem 9).…”