Study On Integral Inequalities Of Discontinuous Functions And The Hadamard Type Integral Inequalities

With the development of the theory of differential equation and integral equation,many specialists pay attention to relevant theories and applications.The Gronwall-Bellman,Hadamard type integral inequality are widely used in the qualitative and quantitative analysis of the solution of differential equations and integral equations,such as the uniqueness,the boundary,the continuous dependence of the initial value problem and the stability of the settlement,and some new research results have been obtained.In the paper,based on the works of[27-48],we consider the Gronwall-Bellman-Bihari type integral inequalities,Hadamard integral inequalities involving maxima of the unknown functions,and existence and uniqueness of solutions for Hadamard type fractional differential equations.This paper is divided into the following four chapters.Chapter 1 Preference,we introduce the main basic definitions and its back-ground of this paper.Chapter 2 Based on the works of[9-10],we consider the Gronwall-Bellman-Bihari type integral inequalities for discontinuous function:Then we apply the obtained results to study the qualitative properties of some differential equations and integral equations.Chapter 3 Motivated by the work of[4,19,43],we give some new Hadamard fractional integral inequalities with maxima as follows:Then,we give some examples to study the boundness of the solution of the Hadamard fractional integral equations with maxima.Chapter 4 Motivated by[6,20,23],we establish some Hadamard fractional differential equation with initial value problem as follows:Then we apply the results to research the existence and uniqueness of the solution of Hadamard fractional differential equation with initial value problem.