The Study Of Two Kinds Of Integral Inequalities And Their Applications

For a long time,integral inequalities as a useful tool,play an important role in the properties analysis of the solutions to differential and integral equations.Therefore,to study of various inequalities has been widely concerned by the researchers,and many scholars have spent a lot of effort to study more general type of integral inequalities.Based on the existing research,this dissertation will study several kinds of fractional integral inequalities and dynamic integral inequalities on time scales.This work not only useful and significant for further investigation of the theory of differential and integral equations,but also for the study of various version inequalities,and for the study of the solutions to various equations.This dissertation has the following chapters:In the first chapter,we will introduce the research background of fractional differential equations,integral equations,dynamic equations on time scales,and discusses the research significance of integral inequalities under these background.We also describe the research status,and the main research content.In the second chapter,we given the preliminary knowledge with needed in this dissertation,including some definitions,theorems,lemmas,basic properties and conclusions of fractional differential equations,integral equations,dynamic equations on time scales.In chapter 3,we show several new types of Gronwall-Bellman integral inequalities,which mixture of integer-order and fractional-order integrals.This results can be used to study the boundednesss of solutions of several special kinds of differential and integral equations.In chapter 4,we generalize some nonlinear Volterra type integral inequalities with two independent variables,and study the boundednesss of the solutions of dynamic differential and integral equations.Finally,we will summarizes this work,and discuss some possible research directions.