Periodic solutions of abstract neutral functional differential equations
type: Item
instance of: scholarly article
title: Periodic solutions of abstract neutral functional differential equations
zbMATH Open document ID: 1255.34067
author: Michelle Pierri
author: Andréa Prokopczyk
author: Hernán R. Henríquez
published in: Journal of Mathematical Analysis and Applications
publication date: 2011-10-24T00:00:00Z
full work available at URL: http://hdl.handle.net/10533/130309
review text: This paper deals with the existence of periodic solutions for the class of linear abstract neutral functional differential equations the form \[ \frac{d}{dt}(x(t)-Bx(t-r))=Ax(t)+L(x_t)+f(t),\quad t\in {\mathbb R}, \] where \(x(t)\in X\) for any \(t\in {\mathbb R}\) with \(X\) a given Banach space. \(x_t\) is a function defined by \(x_t(\theta)=x(t+\theta)\) for \(\theta\) in an appropriate domain. \(A: D(A)\subset X\to X\) and \(B: D(B)\subset X\to X\) are closed linear operators, \(L\) is a bounded linear map defined on an appropriate space, and \(f\) is a locally \(p\)-integrable and \(2\pi\)-periodic function for \(1\leq p<+\infty\). When the underlying space is a UMD space, the authors characterize the existence of a periodic solution for the above mentioned equation with finite and infinite delay. In the case of finite delay, the existence and uniqueness of a periodic strong \(L^p\) solution, a periodic weak solution and periodic mild solution are discussed, respectively. In the case of infinite delay, existence and uniqueness of a periodic strong \(L^p\) solution are studied.
Mathematics Subject Classification ID: 34K13
Mathematics Subject Classification ID: 34K40
Mathematics Subject Classification ID: 34K30
Mathematics Subject Classification ID: 34K06
zbMATH DE Number: 5962638
zbMATH Keywords: abstract neutral functional differential equation
zbMATH Keywords: periodic solution
zbMATH Keywords: UMD space
MaRDI profile type: Publication
OpenAlex ID: W1967066071
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DOI: 10.1016/J.JMAA.2011.06.078
description: scientific article; zbMATH DE number 5962638