Gronwall–Bellman Type Inequalities


Published: 2021-11-13

Page: 122-129

Kemi Iyabo Apanpa

Department of Mathematics, University of Jos, Jos, Plateau State, Nigeria.

Kamilu Rauf

Department of Mathematics, University of Ilorin, Ilorin, Kwara State, Nigeria.

Oladipo Ebenezer Olaide

Department of Mathematics, University of Ilorin, Ilorin, Kwara State, Nigeria.

*Author to whom correspondence should be addressed.


In mathematics, an integro-differential equation is an equation that covers various aspect of integrals and derivatives of an unknown function. Integral inequality act as an important role in the study of differential, integral and partial differential equations and have been of great use to Gronwall inequality.

In succession to our earlier work, we further provide a new generalized Gronwall inequality by giving two important lemmas to establish the important of our results. We further apply the inequality to a nonlinear integro-differential equation where we established that, the important of Gronwall-Bellman inequality to differential equations cannot be overlook.

Keywords: Gronwall-Bellman type inequality, non-negative continuous function, nonlinear integro-differential equations, inequalities, best possible constant

How to Cite

Apanpa, K. I., Rauf, K., & Olaide, O. E. (2021). Gronwall–Bellman Type Inequalities. Asian Journal of Pure and Applied Mathematics, 3(1), 122–129. Retrieved from


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Bainvov D, Simeonov D. Integral inequalities and applications. Kluwer Academic Publishers, Dordrech. 1992.

Oregan D, Samet B. Lyapunov-type inequalities for a class of fractional differential equations. J. Inequal. Appl. 2015(2015);1-10. Doi:10.1186/s13660-015-0769-2.

Baleanu D, Diethelm K, Scalas E. Fractional Calculus: Models and Numerical Methods. World Scientific, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ; 2012. Doi: 10.1142/9789814355216.

Rahman M. Integral equations and their application. Ashurst Lodge, Ashurst, Southampton, S040, 7AA, UK. 2007.

Rashid S, Hammouch Z, Kalsoom K, Ashraf R, Chu YM. New Investigation on the generalized K-fractional integral operators. Front. Phys. 2020;25. Doi:10.3389/fphy.2020.00025.

Wazwas A. Linear and nonlinear integral equations, methods and applications. Higer Education Press, Bengng and Springer-verlag Berlin Heidelberg; 2011. DOI:

Pachpatte BG. A note on Gronwall-Bellman inequality. Journal of Mathematical Analysis and Application. 1973; 44;758-762.

Agarwal P, Jleli M, Tomar M. Certain Hermite-Hadamard type inequalities via generalized K-fractional integrals. J. Inequal. Appl. 2017(2017);Paper No. 55:10. Doi:10.1186/s13660-017-1318-y.

Gronwall TH. Note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Annan of Mathematics. 1919;20(2):292-296.

Hazewinkel M. ed. Continuous function (Unpublished) Encyclopedia of Mathematics; 2001.

Oguntuase JA. On an Inequality of Gronwall. Journal of Inequalities in Pure and Applied Mathematics. 2001;2(1):1-6.

Apanpa KI, Rauf K, Aminu L. A note on Gronwall-Bellman type integral inequalities. Nigeria Journal of Mathematics and Applications. 2018;27:116-131.

Yu-Ming Chu, Saima Rashid, Fahd Jarad, Muhammad Aslam Noor, Humaira Kalsoom. More new results on integral inequalies for generalized K-Fractional conformable integral operators. Discrete and Continuous Dynamical Systems Series S. 2007;14. DOI: 10.3934/dcdss.2021063

Bainov D, Simeonov D. Integral inequalities and applications. Kluwer Academic Publisher, Dordrech; 1992.

Brauer F. A nonlinear variation of constants formula for Volterra equations. Mat. Systems Th. 1972;6:226-234.