Abstract comparison principles and multivariable Gronwall-Bellman inequalities
type: Item
instance of: scholarly article
title: Abstract comparison principles and multivariable Gronwall-Bellman inequalities
zbMATH Open document ID: 0613.47037
DOI: 10.1016/0022-247X(86)90251-9
published in: Journal of Mathematical Analysis and Applications
publication date: 1986-01-01T00:00:00Z
review text: The paper is a continuation of the previous papers by the author [Atti Accad. Naz. Lincei 8, 66, 189-193 (1979; Zbl 0452.47062), Bull. Acad. Polon. Sci. Ser. Sci. Math. 30, 161-166 (1982; Zbl 0497.47038), Demonst. Math. 15, 145-153 (1982; Zbl 0509.47045)]. The paper divided into four chapters contains seven theorems, four lemmas, many remarks, and examples. In the first chapter the author studies compactness of increasing sequences of elements in an ordered metrizable uniform space X with denumerable sufficient family of semi-metrics. In the second chapter the author studies comparison theorems between the solutions in Y of an operator inequality (O.I.) \(x\leq Tx\) and associated equation (O.E.) \(x=Tx\) where T:Y\(\to Y\) is increasing mapping on some ordered closed subset Y of X. The statement is e.g.: to any solution u of (O.I.) there corresponds solution v of (O.E.) such that \(u\leq v\) and if there exists other solution w of (O.I.) such that \(v\leq w\) then \(v=w.\) In the next two chapters applying the above described theorems the author studies multivariable (and scalars) Gronwall type inequalities \[ x(t)\leq p(t)+\int^{t}_{0}k(x)(s)ds,\quad t\in R^ n_+ \] and many of its particular cases. Here \(x,p\in X^ m_ n\), \(k:X^ m_ n\to X^ m_ n\) is an increasing mapping and \(X^ m_ n\) denotes the class of all continuous functions from \(R^ n_+\) to \(R^ m\).
Mathematics Subject Classification ID: 47B60
Mathematics Subject Classification ID: 26D10
zbMATH DE Number: 3992246
zbMATH Keywords: comparison principles
zbMATH Keywords: multivariable Gronwall type inequalities
zbMATH Keywords: compactness of increasing sequences of elements in an ordered metrizable uniform space
zbMATH Keywords: comparison theorems
author: Mihai Turinici
reviewed by: Jerzy Ryszard Popenda
MaRDI profile type: Publication
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Abstract comparison principles and multivariable Gronwall-Bellman inequalities (full work available at URL)
full work available at URL: 0022-247x(86)90251-9

Abstract comparison principles and multivariable Gronwall-Bellman inequalities (OpenAlex ID)
OpenAlex ID: W2041739046