Inequalities for differential and integral equations
type: Item
instance of: scholarly article
title: Inequalities for differential and integral equations
zbMATH Open document ID: 1032.26008
author: B. G. Pachpatte
published in: Mathematics in Science and Engineering
publication date: 1999-03-02T00:00:00Z
review text: The present monogragh consists of five chapters. Chapter 1 contains the well-known Gronwall-Bellman linear integral inequality and many one-variable linear generalizations of this inequality. The remarkable results among others given in this chapter are various linear inequalities involving iterated integrals which are initiated by the author in 1973 with a special inequality involving a two-fold iterated integral as follows. If \(u,f,g\in C(\mathbb{R}_+, \mathbb{R}_+)\), \(\mathbb{R}_+= [0,\infty)\), and \(u_0\geq 0\) is a constant, then \[ u(t)\leq u_0+ \int^t_0 f(s)\Biggl(u(s)+ \int^s_0 g(k) u(k) dk\Biggr) ds,\quad t\in\mathbb{R}_+, \] implies \[ u(t)\leq u_0\Biggl[1+ \int^t_0 f(s)\exp\Biggl(\int^s_0 [f(k)+ g(k)]dk\Biggr) ds\Biggr],\quad t\in\mathbb{R}_+. \] Pachpatte and many other authors including the present reviewer have contributed in this direction in recent years and the general result was proved in the reference [308] of the book by this reviewer [\textit{En-Hao Yang}, J. Math. Anal. Appl. 103, 184-197 (1984; Zbl 0573.26008)]. These linear integral inequalities cited herein find widespread applications in the study of various differential and integral equations. Chapter 2 and Chapter 3 contain a number of one-variable nonlinear integral inequalities which can be used to achieve great diversity in both the theory and applications of various types of nonlinear differential and integral equations. Chapter 3 deals with a series of remarkable results due to the author and his Indian co-workers which belong to two different types. Namely, generalizations of the Ou-Iang and Dafermos integral inequality and extensions of the Haraux-Engler integral inequality. The features of these inequalities are: inequalities of the first kind possess a nonlinear power of the unkown function on their left-hand side, and an extra logarithmic factor of the unknown function appears in the integrands of all integral inequalities of the second kind. Chapters 4 and 5 deal with the l…
Mathematics Subject Classification ID: 26D10
Mathematics Subject Classification ID: 26-02
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 34A40
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35R45
Mathematics Subject Classification ID: 45A05
Mathematics Subject Classification ID: 45G10
Mathematics Subject Classification ID: 45J05
zbMATH DE Number: 1257653
zbMATH Keywords: Qu-Iang integral inequality
zbMATH Keywords: bounds on solutions
zbMATH Keywords: Gronwall-Bellman linear inequality
zbMATH Keywords: iterated integrals
zbMATH Keywords: differential and integral equations
zbMATH Keywords: nonlinear integral inequalities
zbMATH Keywords: Dafermos integral inequality
zbMATH Keywords: Haraux-Engler integral inequality
zbMATH Keywords: iterated integral inequalities
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1257653