Some Generalizations Of Integral Inequalities And Difference Inequalities

Differential Equation and Difference Equation are important tools in studying of Natural Science, Engineering Technology, and the laws of social economic development, through the research of various attributes of solutions of Differential Equation and Dif-ference Equation, we can explain some phenomena, and we may make predictions for the future development trends. In the process of studying solutions of qualitative properties of difference equation and differential equation, the famous Gronwall-Bellman type in-equality and their various generalizations have become important tools in the study of existence, uniqueness, boundedness, stability and other qualitative properties of differen-tial equation and difference equation. In recent years, some generalizations of Gronwall-Bellman type inequality were made by many experts and scholars at home and abroad, and they establish some integral inequalities, differential inequalities and integral inequal-ities with non-continuous functions, and they make applications of Gronwall-Bellman type inequalities more widely.The aim of this article is to further generalize Gronwall-Bellman type integral in-equality, discrete difference inequality and the integral inequality with non-continuous function. The article is composed of five chapters.In chapter one, we introduce the background and current research status of Gronwall-Bellman type inequality, and we summarize the main work of this paper.In chapter two, in section one, we consider retarded integral inequality including some nonlinear terms. The unknown function of the inequality is a two-variable function, the first term of the right hand of the inequality is a nondecreasing and positive function, and the integrand of the second term contains nonlinear factor of unknown function, and it also contains a nonconstant factor outside integral sign. Thus, We further generalize the results of Agarwal et al.[4] and Chen et al.[17].In section two, we consider a class of more general form of nonlinear retarded inte-gral inequality. The inequality contains a nonconstant factor outside integral sign integral sign. The composite function which contains unknown function is not required mono-tonicity, we employ a technique of monotonicity to give the upper bound estimation of the unknown function. Thus, We further generalize the results of Cheung [16], Kim[28].In chapter three, we study a class of nonlinear sum-difference inequality with two variables. Based on the upper bound of the unknown function, we give the estimation of solutions of the corresponding difference equation.In chapter four, we discuss a class of nonlinear integral inequality with discontinuous function, namely the integral inequality with impulse item. The purpose of studying integral inequality with discontinuous function is to study of the qualitative properties of solutions of differential equation, integral equation and functional differential equation system with impulsive perturbation. For example, boundedness, Lyapunov stability, attractability, etc. We establish a new class of integral inequality with discontinuous function, and we generalize the previous results. Our results can be used as important tools to study some impulsive differential systems and some impulsive integral systems.In chapter five, we sum up the whole paper, and we prospect the research work in the future.