New properties of the D-operator and its applications on the problem of periodic solutions to neutral functional differential system
type: Item
instance of: scholarly article
title: New properties of the D-operator and its applications on the problem of periodic solutions to neutral functional differential system
zbMATH Open document ID: 1226.34078
author: Yong-Cai Geng
author: Sumit K. Garg
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2011-04-19T00:00:00Z
review text: The object of study is a neutral functional differential equation of the form \[ \frac{dDx_t}{dt}=f(t,x_t),\tag{1} \] where \(x_t(\theta)=x(t+\theta),\;\theta\in [-\tau,0],\;\tau>0\), and \(f\in C\left( \mathbb{R}\times (C[-\tau,0],\mathbb{R}^n),\mathbb{R}^n \right)\) is \(\omega\)-periodic in \(t\) and maps bounded sets on bounded sets. Here, the \(D\)-operator is defined as \[ D(x_t)=x(t)-Bx(t-\tau) \] for some given \(n\times n\) real matrix \(B\). It is proved that, if \(|\lambda|\neq 1\) for any eigenvalue \(\lambda\) of \(B\), then any \(\omega\)-periodic solution of (1) is continuously differentiable on \(\mathbb{R}\). Finally, it is given a continuation-type result to assure the existence of some \(\omega\)-periodic solution of (1). The poof is based on coincidence degree theory.
reviewed by: Adriana Buică
Mathematics Subject Classification ID: 34K40
Mathematics Subject Classification ID: 34K13
Mathematics Subject Classification ID: 47N20
zbMATH DE Number: 5880257
zbMATH Keywords: neutral functional differential equation
zbMATH Keywords: \(D\)-operator
zbMATH Keywords: periodic solution
zbMATH Keywords: Mawhin's continuation theorem
MaRDI profile type: Publication
full work available at URL: j.na.2011.01.023
OpenAlex ID: W2094736741
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DOI: 10.1016/J.NA.2011.01.023
description: scientific article; zbMATH DE number 5880257