Improved Gronwall-Bellman Inequalities For Discontinuous Function With Impulsive Term

The inequality theory of integral equation is one of important branch of differential equations. Because it has deep physical background and realistic mathematical models. In recent years, this theory has made quickly development and widely considerable in the field of modern applied mathematics. Many scholars take on the research of this field, they have achieved many good results. In very resent years, this field developed very fast, Especially, the research of the Gronwall-Bellman inequality. So it researched more widely and more deeply than other inequalities in type and method.(some results you can see [1]-[35]).The article is divided into three sections according to contents.In chapter 1, we introduce the main contents of this paper.In chapter 2, this chapter is divided into two sections to investigate some kinds of improved Volterra type integral equation with impulsive terms.We state the main results as follows: First, we are concerned with the damped Volterra type integral equation inequalityand the nonlinear Volterra type integral equation with impulsive termsIn the first section, we improved the result of Olivia Lipovan in the article [17] the equation is and we get the proof of the new inequality.In the second section, we mainly discuss the equation:which improved the integral inequality (2.1.1) to nonlinear situation and get some new inequality.In chapter 3, we study the Bihari type inequality of integral equation with retarded term, we consider the inequality of integral equation as follows:In this section we will get some new result about the inequality (3.1.1) on [t₀,∞). we give an inverse example to correct the result of reference [5],and get some new results.