Several Inequalities of Gronwall and Their Proofs
Abstract
It is well known that integral inequalities play a very important role in studying the properties of solutions to ordinary differential equations and integral equations. In 1919, Gronwall established a class of basic integral inequalities when he studied the dependence of differential equations on parameters, which is called Gronwall's inequalities. Gronwall's inequality play a very important role in ordinary differential equations, and it is also an important tool to study the properties of differential equations and integral equation solutions. There are several proofs of Gronwall's inequality, in particular, Agarwal, Deng and Zhang studied the Gronwall-Bellman inequality with multiple nonlinear terms, which made the adaptability of the Gronwall-Bellman inequality widely. Gronwall's inequality has various generalization forms and different proving methods, which is also a good tool for solving many mathematical problems. Different kinds of Gronwall's inequalities and their proofs are discussed in this paper. By researching the induction of Gronwall's inequality forms and their proofs, this paper aims to solve the problems of inequality as much as possible.
Copyright (c) 2022 Xueqing Wang
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
Copyright on any open access article in a journal published by PiscoMed Publishing is retained by the author(s).
Authors grant PiscoMed Publishing a license to publish, copy, distribute, and convey the article.
The current adopted license, the Creative Commons Attribution 4.0 International License (CC BY 4.0), formalizes these and other terms and conditions of publishing articles. The license (CC BY 4.0) means:
Share: Everyone can copy and redistribute the open-access content in the journal.
Adapt: Materials in the articles can be remixed, reused, and reanalyzed for any purpose.
Attribution: You must cite the source with the correct license if some changes to the materials are made, but that does not mean that the licensor endorses you or your use.
Authors should ensure that the content of the article is not involved in a copyright dispute before submitting it. For previously published articles, authors should obtain permission from the copyright holder if the material is under a more restrictive license.
References
Peng, LX., The Proof of Gronwall's Inequality and Related Applications[J]. Journal of Anqing Normal University (Natural Science Edition), 2015, 4: 117-119.
Sun, L., The note on Gronwall's Inequality Proof[J]. Advanced Mathematics Research, 2007, 10(1): 69-72.
Wu, X., A Proof and Extension of Gronwall's Inequality[J]. Journal of Yunnan Normal University, 1999, 6: 24-27.
Li, JM., A Note on Stochastic Gronwall's Inequality[J]. Journal of Zhanjiang Normal University, 2013, 34(6): 19-21.
Li, HC., Liu, ZM., Liang, S., Generalization and Application of Gronwall's Inequality[J]. Journal of Baoshan University, 2010, 2: 55-56.
Zhao, Y., A Note on Gronwall's Inequality[J]. Advanced Mathematics Research, 2011, 14(4): 17-18.
