Skip to main content

Advertisement

SpringerLink
Log in

Further Nonlinear Retarded Integral Inequalities for Gronwall–Bellman Type and Their Applications

  • Research Paper
  • Published:
Iranian Journal of Science and Technology, Transactions A: Science Aims and scope Submit manuscript

Abstract

In this article, we will derive several new nonlinear retarded integral inequalities for Gronwall–Bellman type and then apply them as handy tools to examine explicit bounds for solutions of initial value problems, and will also use to evaluate the qualitative as well as the quantitative behavior of solutions of nonlinear retarded differential and integral equations. These inequalities extend some present inequalities to the current literature. An application is also given to demonstrate the strength of our results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  • Abdeldaim A (2016) Nonlinear retarded integral inequalities of Gronwall–Bellman type and applications. J Math Inequal 10:285–299

    Article  MathSciNet  MATH  Google Scholar 

  • Abdeldaim A, Yakout M (2011) On some new integral inequalities of Gronwall–Bellman—Pachpatte type. Appl Math Comput 217:7887–7899

    MathSciNet  MATH  Google Scholar 

  • Abdeldaim A, El-deeb AA (2014) Some new retarded nonlinear integral inequalities with iterated integrals and their applications in retarded differential equations and integral equations. J Fract Calc Appl 5(9):1–9

    MathSciNet  Google Scholar 

  • Abdeldaim A, El-deeb AA (2015) On some new nonlinear retarded integral inequalities with iterated integrals and their applications in integro-differential equations. Br J Math Comput Sci 5:479–491

    Article  MATH  Google Scholar 

  • Agarwal RP, Deng S, Zheng W (2005) Generalization of a retarded Gronwall-like inequality and its applications. Appl Math Comput 165:599–612

    MathSciNet  MATH  Google Scholar 

  • Agarwal RP, Kim YH, Sen SK (2008) New retarded integral inequalities with applications. J Inequal Appl Article ID 908784

  • Bainov DD, Simeonov PS (1992) Integral inequalities and applications. Kluwer, New York

    Book  MATH  Google Scholar 

  • Bellman R (1943) The stability of solutions of linear differential equations. Duke Math J 10:643–647

    Article  MathSciNet  MATH  Google Scholar 

  • Bihari IA (1956) A generalization of a lemma of Bellman and its application to uniqueness problem of differential equation. Acta Math Acad Sci Hung 7:81–94

    Article  MathSciNet  MATH  Google Scholar 

  • Cheung WS (2006) Some new nonlinear inequalities and applications to boundary value problems. Nonlinear Anal 64:2112–2128

    Article  MathSciNet  MATH  Google Scholar 

  • El-owaidy H, Ragab AA, Abdeldaim A (1999) On some integral inequalities of Gronwall–Bellman type. Appl Math Comut 106:289–303

    MATH  Google Scholar 

  • El-owaidy H, Abdeldaim A, El-deeb AA (2014a) On some new nonlinear retarded integral inequalities with iterated integrals and their applications in differential–integral equations. Math Sci Lett J 3(3):157–164

    Article  Google Scholar 

  • El-owaidy H, Ragab AA, Abuelela W, El-deeb AA (2014b) On some new nonlinear integral inequalities of Gronwall–Bellman type. Kyungpook Math. J. 54:555–575

    Article  MathSciNet  MATH  Google Scholar 

  • Gronwall TH (1919) Note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Ann Math 20:292–296

    Article  MathSciNet  MATH  Google Scholar 

  • Kim YH (2005) On some new integral inequalities for functions in one and two variables. Acta Math Sin 21:423–434

    Article  MathSciNet  MATH  Google Scholar 

  • Lipovan O (2000) A retarded Gronwall-like inequality and its applications. J Math Anal Appl 252:389–401

    Article  MathSciNet  MATH  Google Scholar 

  • Pachpatte BG (1998) Inequalities for differential and integral equations. Academic Press, London

    MATH  Google Scholar 

  • Pachpatte BG (2002) Explicit bound on certain integral inequalities. J Math Anal Appl 267:48–61

    Article  MathSciNet  MATH  Google Scholar 

  • Rashid MHM (2012) Explicit bound on retarded Gronwall–Bellman inequality. Tamkang J Math 43:99–108

    Article  MathSciNet  MATH  Google Scholar 

  • Wang WS (2007) A generalized retarded Gronwall-like inequality in two variables and applications to BVP. Appl Math Comput 191:144–154

    Article  MathSciNet  MATH  Google Scholar 

  • Wang WS, Shen C (2008) On a generalized retarded integral inequality with two variables. J Inequal Appl. Article ID 518646

  • Wang WS, Li Z, Li Y, Huang Y (2010a) Nonlinear retarded integral inequalities with two variables and applications. J Inequal Appl. Article ID 240790

  • Wang WS, Luo RC, Li Z (2010b) A new nonlinear retarded integral inequality and its application. J Inequal Appl. Article ID 462163

  • Wang WS (2012a) Some generalized nonlinear retarded integral inequalities with applications. J Inequal Appl 2012:31

    Article  MathSciNet  MATH  Google Scholar 

  • Wang WS (2012b) Some new generalized retarded nonlinear integral inequalities with iterated Integrals and their applications. J Inequal Appl 2012:236

    Article  MathSciNet  MATH  Google Scholar 

  • Zareen AK (2015) Integral inequalities of Gronwall–Bellman Type. Appl Math 5:3484–3488

    Google Scholar 

Download references

Acknowledgements

The authors would like to thank the Editor and the advisors/reviewers for their very detailed comments and valuable suggestions which greatly improved our paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, Khwaja Fareed University of Engineering and Information Technology, Rahim Yar Khan, 64200, Pakistan

    Abdul Shakoor

  2. Basic Sciences and Humanities Department, University of Engineering and Technology Lahore, Faisalabad Campus, Faisalabad, 38000, Pakistan

    Ilyas Ali & Abdur Rehman

  3. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, China

    Muhammad Azam

  4. National College of Business Administration and Economics, Rahim Yar Khan, 64200, Pakistan

    Muhammad Zahid Iqbal

Authors
  1. Abdul Shakoor
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ilyas Ali
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Muhammad Azam
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Abdur Rehman
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Muhammad Zahid Iqbal
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

The final manuscript was read and approved by all the authors of this article.

Corresponding author

Correspondence to Abdul Shakoor.

Ethics declarations

Conflict of interest

The authors announce that they do not have any competing interests.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shakoor, A., Ali, I., Azam, M. et al. Further Nonlinear Retarded Integral Inequalities for Gronwall–Bellman Type and Their Applications. Iran J Sci Technol Trans Sci 43, 2559–2568 (2019). https://doi.org/10.1007/s40995-019-00745-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40995-019-00745-x

Keywords

Mathematics Subject Classification

Search

Navigation