Generalization of an inequality of Gronwall-Reid
type: Item
instance of: scholarly article
title: Generalization of an inequality of Gronwall-Reid
zbMATH Open document ID: 0684.34011
DOI: 10.1016/0021-9045(89)90003-8
author: Dean S. Clark
published in: Journal of Approximation Theory
publication date: 1989-01-01T00:00:00Z
review text: The paper provides an extension of Gronwall-Reid inequality in the form: If \(\mu_{\lambda_ j}\) \((j=1,2,...,n)\) is a set of finite measure on \([t_ 0,+\infty)\) and f is non-negative integrable on finite intervals with respect to the measures \(\mu_{\lambda_ j}\) and \[ f(t)\leq \alpha +\beta \sum^{m}_{j=1}\int^{t}_{t_ 0}f^{q_ j}(s)d\mu_{\lambda_ j}+\epsilon \sum^{n}_{j=m+1}\int^{t}_{t_ 0}f(s)d\mu_{\lambda_ j}, \] where \(\alpha\), \(\beta\) are non-negative constants and \(0\leq q_ j\leq\) and \(\mu_{\lambda_ j}[t_ 0,+\infty)\leq M<\infty\) for all \(\lambda_ j\) and \(0<nM\epsilon <1\), then \[ f(t)\leq \frac{\alpha +\beta cmM}{1-nM(\epsilon +\beta c^{-1})},\quad t\geq t_ 0 \] and \(c\geq \max [1,(\beta nm)/(1-nm\epsilon]\). The result finds effective use in the stability analysis of the system of difference equations \(X_{n+1}=X_ n+E_ nF_ n(X_ n,\Phi_ n)\) where \(\Phi_ n\) are perturbations and the \(E_ n\) are diagonal matrices with positive elements on the diagonal.
reviewed by: Gandikota L. N. Rao
Mathematics Subject Classification ID: 34A40
Mathematics Subject Classification ID: 28A25
zbMATH DE Number: 4120551
zbMATH Keywords: Gronwall-Reid inequality
zbMATH Keywords: stability analysis
zbMATH Keywords: system of difference equations
MaRDI profile type: Publication
full work available at URL: 0021-9045(89)90003-8
OpenAlex ID: W2000598697
cites work: Stability in a system of difference equations
description: scientific article; zbMATH DE number 4120551