Gronwall-Bellman-type inequalities for systems
type: Item
instance of: scholarly article
title: Gronwall-Bellman-type inequalities for systems
zbMATH Open document ID: 0611.26009
DOI: 10.1016/0022-247X(86)90184-8
author: B. George
published in: Journal of Mathematical Analysis and Applications
publication date: 1986-01-01T00:00:00Z
review text: In this paper the Gronwall-Bellman inequality is generalized as follows. If \[ u(x)\leq g(x)+\int^{x}_{0}K(x,tu(t))dt \] then \[ u(x)\leq g(x)+\int^{x}_{0}\exp \{\int^{x}_{t}K(\xi (s,x),s)ds\}K(\xi (t,x),t)g(t)dt \] (this is how the inequality (2.6) of the paper should have been). Here x, s, t are n vectors; u, g are continuous m vectors; \(K(x,t)\) is an \(m\times m\) matrix whose elements are continuous in x and t and monotone in x; and \(\int^{x}_{0}(\quad)dt=\int^{x_ 1}_{0}...\int^{x_ n}_{0}(\quad)dt_ 1...dt_ n.\) By an unfortunate oversight the condition that K be monotone got omitted in the statement of the hypotheses. Also note that the misprinted equation (2.6) is correct for \(K(x,t)\) non decreasing in x.
Mathematics Subject Classification ID: 26D10
zbMATH DE Number: 3987600
zbMATH Keywords: Gronwall-Bellman inequality
MaRDI profile type: Publication
cites work: Q3328767
cites work: Q5336417
cites work: Linear Generalizations of Gronwall's Inequality
cites work: On Gronwall's Inequality
cites work: Q5570370
cites work: Q5628697
cites work: A Linear Generalization of Gronwall's Inequality
cites work: The Gronwall-Bellman inequality in several variables
cites work: On some integral inequalities in n independent variables and their applications
cites work: Bellman-Bihari integral inequalities in several independent variables
cites work: Errata for: Bellman-Bihari integral inequalities in several independent variables
cites work: Higher dimensional gronwall-bellman type inequalities
description: scientific article; zbMATH DE number 3987600