Some generalizations of Pachpatte inequalities on time scales

Nasser, Bacem Ben; Boukerrioua, K.; Defoort, Michaël; Djemaï, Mohamed; Hammami, Mohamed

doi:10.1016/j.ifacol.2017.08.2531

Cited by 5 publications

(4 citation statements)

References 21 publications

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“…Moreover, Nasser et al [29] introduced some new generalizations and rectifications of many known results of Pachpatte kind, consolidating two nonlinear integral terms on time scales. These acquired consequences played a crucial role in reading a few lessons of integral and integro-differential equations.…”

Section: Introductionmentioning

confidence: 99%

Reconstruction of nonlinear integral inequalities associated with time scales calculus

Khan

2020

Adv Differ Equ

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In this paper, we build up some generalizations of nonlinear integral inequalities and recreate the results of some Pachpatte's inequalities on time scales. We not just settle new estimated bounds of a particular class of nonlinear retarded dynamic inequalities, but additionally determine and unify continuous analogs alongside a subjective time scale T. We demonstrate applications of the treated inequalities to reflect the benefits of our work. The key effects will be proven by using the analysis procedure and the standard timescale comparison theorem technique.

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Section: Introductionmentioning

confidence: 99%

Reconstruction of nonlinear integral inequalities associated with time scales calculus

Khan

2020

Adv Differ Equ

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“…Pachpatte's inequalities on time scale, both linear and nonlinear, have also been a subject of discussion for quite some time. These Pachpatte's inequalities have been developed by several authors [2,7,10,12], and [14]. Bohner formulated a number of dynamic inequalities that basically rely on Gronwall's inequality [2].…”

Section: Introductionmentioning

confidence: 99%

“…To illustrate the theoretical results, it has been shown that the obtained inequalities can be used as important tools in the study of certain properties of dynamic equations on time scales. Assuming arbitrary time scales, the authors of [14] established some new inequalities of Pachpatte type, incorporating two nonlinear integral terms, which are themselves the generalizations and refinements of many existing results. These obtained results played a vital role in studying some classes of integral and integro-differential equations.…”

Section: Introductionmentioning

confidence: 99%

Study of nonlinear Pachpatte’s inequalities on time scales

2019

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In this paper, we develop a fundamental dynamic inequality, a generalization of comparison theorem and reproduce the proofs of some nonlinear integral Pachpatte’s inequalities by using their continuous analogue. We also unify and extend these improved integral Pachpatte’s inequalities and their corresponding discrete analogues on arbitrary time scales. The results are used to make qualitative analysis of higher order dynamic equations.

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“…For example, the monographes [7,8] and the references cited therein. At the same time, in the papers [1], [4], [5], [9], [11], [14], [15], [16], [18], [20], [23], [24], [25] and references therein have studied the theory of integral inequalities on time scales. In [2] and [3] the author establishes some general nonlinear dynamic inequalities on general time scales involving functions in two independent variables and the author extends double sum and integral inequalities of Hilbert-Pachpatte type to general dynamic double integral inequalities on time scales.…”

Section: Introductionmentioning

confidence: 99%

Integral Equations on Time Scales

Georgiev¹

2016

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The "Atlantis Studies in Dynamical Systems" publishes monographs in the area of dynamical systems, written by leading experts in the field and useful for both students and researchers. Books with a theoretical nature will be published alongside books emphasizing applications. This book, or any parts thereof, may not be reproduced for commercial purposes in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system known or to be invented, without prior permission from the Publisher.Printed on acid-free paper PrefaceMany problems arising in applied mathematics or mathematical physics, can be formulated in two ways namely as differential equations and as integral equations. In the differential equation approach, the boundary conditions have to be imposed externally, whereas in the case of integral equations, the boundary conditions are incorporated within the formulation, and this confers a valuable advantage to the latter method. Moreover, the integral equation approach leads quite naturally to the solution of the problem as an infinite series, known as the Neumann expansion, the Adomian decomposition method, and the series solution method in which the successive terms arise from the application of an iterative procedure. The proof of the convergence of this series under appropriate conditions presents an interesting exercise in an elementary analysis.This book encompasses recent developments of integral equations on time scales. For many population models biological reasons suggest using their difference analogues. For instance, North American big game populations have discrete birth pulses, not continuous births as is assumed by differential equations. Mathematical reasons also suggest using difference equations-they are easier to construct and solve in a computer spreadsheet. North American large mammal populations do not have continuous population growth, but rather discrete birth pulses, so the differential equation form of the logistic equation will not be convenient. Age-structured models add complexity to a population model, but make the model more realistic, in that essential features of the population growth process are captured by the model. They are used difference equations to define the population model because discrete age classes require difference equations for simple solutions. The discrete models can be investigated using integral equations in the case when the time scale is the set of the natural numbers. A powerful method introduced by Poincaré for examining the motion of dynamical systems is that of a Poincaré section. This method can be investigated using integral equations on the set of the natural numbers. The total charge on the capacitor can be investigated with an integral equation on the set of the harmonic numbers.This book contains elegant analytical and numerical methods. This book is intended for the use in the field of integral equations and dynamic calculus on time v

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Some generalizations of Pachpatte inequalities on time scales

Cited by 5 publications

References 21 publications

Reconstruction of nonlinear integral inequalities associated with time scales calculus

Reconstruction of nonlinear integral inequalities associated with time scales calculus

Study of nonlinear Pachpatte’s inequalities on time scales

Integral Equations on Time Scales

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