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.