A note on certain integral inequalities with delay
type: Item
instance of: scholarly article
title: A note on certain integral inequalities with delay
zbMATH Open document ID: 0854.26012
DOI: 10.1007/BF01882199
published in: Periodica Mathematica Hungarica
publication date: 1997-01-19T00:00:00Z
review text: The author establishes explicit bounds for solutions of three delay integral inequalities. The typical inequality considered in the paper is of the form \[ x^2(t) \leq c^2 + 2 \int^t_0 [f(s) x(\sigma(s)) W(x(\sigma(s))) + h(s) x(\sigma(s))]ds, \tag{A} \] for \(t \in [0,\infty)\), with the condition \(x(t) = \psi(t)\), \(t \in [a,0]\), \(\psi(\sigma(t)) \leq c\) for every \(t \geq 0\) and \(\sigma(t) \leq 0\), where \(c \geq 0\) is a constant, \(f, h \in C(R_+, R_+)\), \(W \in C(R_+,R_+)\) is nondecreasing, with \(W(0) = 0\) and \(W (x)>0\) for \(x > 0\), \(\sigma \in C(R_+, R_+)\) with \(\sigma(t) \leq t\) and \(-\infty < a:= \inf\{\sigma(t) : t\in R_+\} \leq 0\), \(x \in C([a,\infty), R_+)\). The key step of the argument used in all proofs of the paper is to show that, under the assumed conditions, all functions \(x(\sigma(s))\) involved in the right member of the delay inequality satisfy the relation \(x(\sigma(x)) \leq x(s)\) for all \(s \in R_+\). Hence, after replacing all \(x(\sigma(s))\) by \(x(s)\), the delay integral inequality under consideration yields a corresponding integral inequality without delay, and it can be treated easily by a known method. Some generalizations of Ou-Iang's inequality given by the same author [J. Math. Anal. Appl. 182, No. 1, 143-157 (1994; Zbl 0806.26009)] can be obtained from the Theorem of the paper as consequences for the case when \(\sigma (t) = t\) for \(t \in R_+\). The argument in the proofs of the paper had been used by \textit{P. Ch. Tsamatos} and \textit{S.K. Ntouyas} in Period. Math. Hung. 23, No. 1, 91-94 (1991; Zbl 0746.26011)].
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 26D20
zbMATH DE Number: 892563
zbMATH Keywords: upper bounds on the solutions
zbMATH Keywords: delay integral equation
zbMATH Keywords: delay integral inequalities
author: B. G. Pachpatte
MaRDI profile type: Publication
cites work: Nonlinear evolution equations - global behavior of solutions
cites work: On a nonlinear convolution equation occurring in the theory of water percolation
cites work: Q4287859
cites work: On a certain inequality arising in the theory of differential equations
cites work: Q3987665
cites work: On a Bellman-Bihari type integral inequality with delay
full work available at URL: https://doi.org/10.1007/bf01882199
OpenAlex ID: W2080726714
Recommended article: Explicit bounds on some new nonlinear integral inequalities with delay
Recommended article: Some new delay integral inequalities and their applications
Recommended article: Q4278918
Recommended article: Q5392254
Recommended article: Q4454451
Recommended article: Q3622924
Recommended article: On some delay nonlinear integral inequalities in two independent variables
Recommended article: Some delay integral inequalities on time scales and their applications in the theory of dynamic equations
Recommended article: Q5474511
Recommended article: On certain integral inequalities in two independent variables for retarded equations
description: scientific article; zbMATH DE number 892563