On the existence and uniqueness of periodic solutions of linear differential equation with shifted argument in Banach space
type: Item
instance of: scholarly article
zbMATH Open document ID: 0986.47009
author: A. Chajkovs'kyj
publication date: 2002-01-30T00:00:00Z
Mathematics Subject Classification ID: 47A50
zbMATH DE Number: 1698086
zbMATH Keywords: existence and uniqueness
zbMATH Keywords: periodic solutions
zbMATH Keywords: linear differential equation
zbMATH Keywords: shifted argument
zbMATH Keywords: Banach space
MaRDI profile type: Publication
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title: On the existence and uniqueness of periodic solutions of linear differential equation with shifted argument in Banach space
published in: Visnyk. Matematyka. Mekhanika. Kyïvs'kyĭ Universytet Imeni Tarasa Shevchenka
review text: Let \((B,\|\cdot\|)\) be a complex Banach space. Let \(L(B)\) be a space of linear continuous operators acting in space \(B\), let \(T\) be a positive rational number, and let \(\{A_{i}, i=-N,\ldots,N\}\subset L(B)\). The author obtains necessary and sufficient conditions for the existence and uniqueness of a solution \(x\in C^1(R,B)\), \(\|x\|_{\infty}=\sup_{t\in R}\|x(t)\|<\infty\), \(\forall t\in R: x(t+T)=x(t)\) of the equation \(x'(t)=\sum_{k=-N}^{N}A_{k}x(t-k)+y(t)\), \(t\in R\) for any \(y(t)\in C(R,B)\), \(\|y\|_{\infty}<\infty\), \(\forall t\in R: y(t+T)=y(t)\). The explicit form of the solution is found.
reviewed by: Aleksandr D. Borisenko
description: scientific article; zbMATH DE number 1698086