Generalization And Application Of Gronwall - Bellman Type Integral Inequality

Differential equation is a branch of mathematics and it has closely connection with application. Using the theory of differential equation, we can describe and explain the various problems of many fields of natural and social science. In 1943, the famous Gronwall-Bellman integral inequality was proven. Then the research on this field appears one after another. In recent years, a number of scholars has established many forms of Gronwall-Bellman type integral inequalities, which provides a favorable tool to study the existence, uniqueness, boundedness and other qualitative properties of the solution of the differential and integral equations.At the beginning, scholars concerned more about the continuous functions rather than the discontinuous functions with regard to Gronwall-Bellman type integral inequalities. Not too long ago, we began to pay attention to the case of discontinuous functions. Based on the existing study, this thesis generalizes and improves some new integral inequalities of both continuous and discontinuous functions and it also obtains some new results.The thesis is divided into four chapters according to the contents.Chapter 1 We establish a new type of retarted integral inequality for discontinuous functions,and extend the above inequality.Chapter 2 On the basis of Chapter 1 and motivited by [17, 19], we change the limits of integration of the inequalities as follows,Chapter 3 In this chapter, we mainly study the estimates on the solution of the following two integral inequalities,Chapter 4 In this chapter, we focus on a class of new nonlinear retarted VolterraFredholm type integral inequality,...