Oscillatory retarded functional systems.
type: Item
instance of: scholarly article
title: Oscillatory retarded functional systems.
zbMATH Open document ID: 1036.34078
DOI: 10.1016/S0022-247X(03)00421-9
author: J. M. Ferreira
author: S. Pinelas
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-10-14T00:00:00Z
review text: The authors consider a functional system \[ x(t)= \int_{-1}^0 x(t-r(\theta))\, d[\nu(\theta)], \tag{1} \] where \(x(t)\in \mathbb{R}^n\), \(r(\theta)\) is a real continuous function on \([-1,0]\) and \(\nu(\theta)\) is a real \(n\times n\)-matrix real-valued function of bounded variation on \([-1,0]\). System (1) includes the important class of delay functional systems of the form \[ x(t)= \sum_{j=1}^p A_jx(t-r_j). \tag{2} \] Also, as it is well known that the systems (1) and (2) can be considered as a particular case of differential systems of neutral type. The authors use the spectral properties and prove some oscillation theorems. One of them is: Every solution of (1) is oscillatory if and only if \(s(\lambda)< 1\) where \(s(\lambda)\in \sigma(\int_{-1}^0 \exp(-\lambda r(\theta))\, d[\nu(\theta))])\). The authors also use the matrix measure and establish some sufficient conditions for oscillations. Some examples are considered to illustrate the main results.
Mathematics Subject Classification ID: 34K11
zbMATH DE Number: 1992562
zbMATH Keywords: retarded functional systems
zbMATH Keywords: delay functional systems
zbMATH Keywords: oscillation
reviewed by: S. H. Saker
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1992562