Skip to main content
Top
Published in: Journal of Inequalities and Applications 1/2006

Open Access 01-12-2006 | Research Article

An impulsive nonlinear singular version of the Gronwall-Bihari inequality

Author: Nasser-Eddine Tatar

Published in: Journal of Inequalities and Applications | Issue 1/2006

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

We find bounds for a Gronwall-Bihari type inequality for piecewise continuous functions. Unlike works in the prior literature, here we consider inequalities involving singular kernels in addition to functions with delays.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Literature
1.
go back to reference Bainov DD, Hristova SG: Impulsive integral inequalities with a deviation of the argument. Mathematische Nachrichten 1995, 171: 19–27.MathSciNetCrossRefMATH Bainov DD, Hristova SG: Impulsive integral inequalities with a deviation of the argument. Mathematische Nachrichten 1995, 171: 19–27.MathSciNetCrossRefMATH
2.
go back to reference Bainov DD, Simeonov PS: Systems with Impulse Effect: Theory and Applications. Ellis Horwood, Chichister; 1989.MATH Bainov DD, Simeonov PS: Systems with Impulse Effect: Theory and Applications. Ellis Horwood, Chichister; 1989.MATH
3.
go back to reference Bainov DD, Simeonov PS: Integral Inequalities and Applications, Mathematics and Its Applications. Volume 57. Kluwer Academic, Dordrecht; 1992:xii+245.CrossRefMATH Bainov DD, Simeonov PS: Integral Inequalities and Applications, Mathematics and Its Applications. Volume 57. Kluwer Academic, Dordrecht; 1992:xii+245.CrossRefMATH
4.
go back to reference Butler G, Rogers T: A generalization of a lemma of Bihari and applications to pointwise estimates for integral equations. Journal of Mathematical Analysis and Applications 1971,33(1):77–81. 10.1016/0022-247X(71)90183-1MathSciNetCrossRefMATH Butler G, Rogers T: A generalization of a lemma of Bihari and applications to pointwise estimates for integral equations. Journal of Mathematical Analysis and Applications 1971,33(1):77–81. 10.1016/0022-247X(71)90183-1MathSciNetCrossRefMATH
5.
go back to reference Hristova SG: Nonlinear delay integral inequalities for piecewise continuous functions and applications. Journal of Inequalities in Pure and Applied Mathematics 2004,5(4):1–14. article 88 article 88MathSciNetMATH Hristova SG: Nonlinear delay integral inequalities for piecewise continuous functions and applications. Journal of Inequalities in Pure and Applied Mathematics 2004,5(4):1–14. article 88 article 88MathSciNetMATH
6.
go back to reference Kirane M, Tatar N-E: Global existence and stability of some semilinear problems. Archivum Mathematicum (Brno) 2000,36(1):33–44.MathSciNetMATH Kirane M, Tatar N-E: Global existence and stability of some semilinear problems. Archivum Mathematicum (Brno) 2000,36(1):33–44.MathSciNetMATH
7.
go back to reference Kirane M, Tatar N-E: Convergence rates for a reaction-diffusion system. Zeitschrift für Analysis und ihre Anwendungen. Journal for Analysis and Its Applications 2001,20(2):347–357.MathSciNetCrossRefMATH Kirane M, Tatar N-E: Convergence rates for a reaction-diffusion system. Zeitschrift für Analysis und ihre Anwendungen. Journal for Analysis and Its Applications 2001,20(2):347–357.MathSciNetCrossRefMATH
8.
go back to reference Krylov NN, Bogolyubov NN: Introduction to Nonlinear Mechanics. Izd. Acad. Sci. Ukr. SSR, Kiev; 1937. Krylov NN, Bogolyubov NN: Introduction to Nonlinear Mechanics. Izd. Acad. Sci. Ukr. SSR, Kiev; 1937.
9.
go back to reference Lakshmikantham V, Bainov DD, Simeonov PS: Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics. Volume 6. World Scientific, New Jersey; 1989:xii+273.CrossRef Lakshmikantham V, Bainov DD, Simeonov PS: Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics. Volume 6. World Scientific, New Jersey; 1989:xii+273.CrossRef
10.
go back to reference Mazouzi S, Tatar N-E: Global existence for some integro-differential equations with delay subject to non-local conditions. Zeitschrift für Analysis und ihre Anwendungen. Journal for Analysis and Its Applications 2002,21(1):249–256.MathSciNetCrossRefMATH Mazouzi S, Tatar N-E: Global existence for some integro-differential equations with delay subject to non-local conditions. Zeitschrift für Analysis und ihre Anwendungen. Journal for Analysis and Its Applications 2002,21(1):249–256.MathSciNetCrossRefMATH
11.
go back to reference Medved M: A new approach to an analysis of Henry type integral inequalities and their Bihari type versions. Journal of Mathematical Analysis and Applications 1997,214(2):349–366. 10.1006/jmaa.1997.5532MathSciNetCrossRefMATH Medved M: A new approach to an analysis of Henry type integral inequalities and their Bihari type versions. Journal of Mathematical Analysis and Applications 1997,214(2):349–366. 10.1006/jmaa.1997.5532MathSciNetCrossRefMATH
12.
go back to reference Medved M: Singular integral inequalities and stability of semilinear parabolic equations. Archivum Mathematicum (Brno) 1998,34(1):183–190.MathSciNetMATH Medved M: Singular integral inequalities and stability of semilinear parabolic equations. Archivum Mathematicum (Brno) 1998,34(1):183–190.MathSciNetMATH
13.
go back to reference Pachpatte BG: Inequalities for Differential and Integral Equations, Mathematics in Science and Engineering. Volume 197. Academic Press, California; 1998:x+611.MATH Pachpatte BG: Inequalities for Differential and Integral Equations, Mathematics in Science and Engineering. Volume 197. Academic Press, California; 1998:x+611.MATH
14.
go back to reference Samoilenko AM, Perestyuk NA: Stability of solutions of differential equations with impulse effect. Differential Equations 1977,13(11):1981–1992. Samoilenko AM, Perestyuk NA: Stability of solutions of differential equations with impulse effect. Differential Equations 1977,13(11):1981–1992.
15.
go back to reference Samoilenko AM, Perestyuk NA: Impulsive Differential Equations, World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises. Volume 14. World Scientific, New Jersey; 1995:x+462. Samoilenko AM, Perestyuk NA: Impulsive Differential Equations, World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises. Volume 14. World Scientific, New Jersey; 1995:x+462.
16.
go back to reference Tatar N-E: Exponential decay for a semilinear problem with memory. Arab Journal of Mathematical Sciences 2001,7(1):29–45.MathSciNetMATH Tatar N-E: Exponential decay for a semilinear problem with memory. Arab Journal of Mathematical Sciences 2001,7(1):29–45.MathSciNetMATH
17.
go back to reference Zhang W, Agarwal RP, Akin-Bohner E: On well-posedness of impulsive problems for nonlinear parabolic equations. Nonlinear Studies 2002,9(2):145–153.MathSciNetMATH Zhang W, Agarwal RP, Akin-Bohner E: On well-posedness of impulsive problems for nonlinear parabolic equations. Nonlinear Studies 2002,9(2):145–153.MathSciNetMATH
Metadata
Title
An impulsive nonlinear singular version of the Gronwall-Bihari inequality
Author
Nasser-Eddine Tatar
Publication date
01-12-2006
Publisher
Springer International Publishing
Published in
Journal of Inequalities and Applications / Issue 1/2006
Electronic ISSN: 1029-242X
DOI
https://doi.org/10.1155/JIA/2006/84561

Other articles of this Issue 1/2006

Journal of Inequalities and Applications 1/2006 Go to the issue

Premium Partner