Semi-Fredholm operators and periodic solutions for linear functional-differential equations
type: Item
instance of: scholarly article
title: Semi-Fredholm operators and periodic solutions for linear functional-differential equations
zbMATH Open document ID: 0922.34058
author: Jong Son Shin
author: Toshiki Naito
published in: Journal of Differential Equations
publication date: 1999-10-03T00:00:00Z
review text: The authors deal with the linear functional-differential equation with infinite delay of the form \[ \frac{dx(t)} {dt}=Ax(t)+B(t,x_t)+F(t). \] Denote by \(E\) a Banach space with the norm \(|\cdot|\). Let \(\mathcal B\) be a Banach space, consisting of functions \(\psi : (-\infty, 0]\to E\), which satisfies some axioms demonstrated in the paper. Assume that \(A: \mathcal D(A)\subset E\to E\) is the infinitesimal generator of a \(C_0\)-semigroup \(T(t)\) on \(E\); \(B:\mathbb{R}\times\mathcal B\to E\) is continuous and \(B(t,\cdot):\mathcal B\to E\) is linear; \(F:\mathbb{R}\to E\) is continuous. The authors obtain several criteria on existence and uniqueness of periodic solutions to the equation in the general phase space \(\mathcal B\) and in the concrete phase space \(\mathcal B=UC_g\). The technique used is established on perturbation theory of semi-Fredholm operators.
Mathematics Subject Classification ID: 34K13
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 1286238
zbMATH Keywords: qualitative theory
zbMATH Keywords: semi-Fredholm operators
zbMATH Keywords: linear functional-differential equations
zbMATH Keywords: infinite delay
zbMATH Keywords: infinitesimal generator
zbMATH Keywords: \(C_0\)-semigroup
reviewed by: Jan Ohriska
MaRDI profile type: Publication
full work available at URL: jdeq.1998.3547
OpenAlex ID: W2066157314
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DOI: 10.1006/JDEQ.1998.3547
description: scientific article; zbMATH DE number 1286238