Several New Inequalities And Their Applications

As is well known, Gronwall-Bellman type inequalities, Henry-Gronwall type inequalities and Volterra-Fredholm type inequality play a very important role in the study of differential equations, integral equations and differential equations of qualitative and quantitative properties. Therefore, many mathematics experts have a great interest in various forms of promotion. In recent years, with the progress of the differential equation theory, more and more people concerned the discrete form of inequality,which provides a useful tool to study some properties of difference equations. This paper establishes three new classes of nonlinear discrete inequalities and Henry-Gronwall type inequalities, and applies the research conclusion to study the boundedness, uniqueness and certain properties of solutions of fractional differential equations.In this paper, we establish some new nonlinear discrete inequalities and integral inequalities, and obtain some new results. The purpose of generalizing inequalities is to study all kinds of the difference equations and integral equations, so as to prove the importance of inequalities in curse of studying equations.This paper is divided into five chapters according to the contents.Chapter 1 We introduce the development about inequality and the main content of this paper.Chapter 2 Some integral inequalities in reference [15], we extend to the discrete inequalities of Gronwall-Bellman type:(?)and applies the research conclusion to study the boundedness of the solutions of some differential equations.Chapter 3 Based on the literature [8], we extend them into the following nonlinear(?)discrete inequalities:and applies the research conclusion to study the boundedness of the solutions of nonlinear finite difference equations.Chapter 4 On the basis of literature [23], we establish some new Volterra-Fredholm type discrete inequalities:(?)then we apply the above conclusions to study the boundedness and uniqueness of the solutions of certain difference equations.Chapter 5 According to some inequalities in [33], we study some new HenryGronwall type integral inequalities:(?)then we apply the above conclusion to study some properties of solutions of fractional differential equations.