Research On Volterra-Fredholm Type Dynamic Integral Inequalities On Time Scales

In the natural and social sciences,differential equation theory,difference equation theory and dynamic equation theory on time scales have very wide applications.In fact,not all differential equations,difference equations and dynamic equations on time scales can be accurately solved,which causes trouble for us to study the qualitative properties of solutions,so the emergence of inequalities lays a foundation for estimating the solutions of such equations.Under the assumption that the equation has a solution,we can directly use inequalities to study the boundedness,uniqueness and asymptotic properties of the solution.In recent years,the research on Volterra-Fredhlom type integral equations and integral inequalities on time scales has received much attention.Many scholars have extended their research field to study the properties of solutions of dynamic equations on time scales by using Volterra-Fredhlom integral inequalities.At present,linear Volterra-Fredhlom type integral inequalities,various continuous and discrete Volterra-Fredholm type integral inequalities,continuous and discrete linear Volterra-Fredholm type integral inequalities,some continuous and discrete nonlinear Volterra-Fredholm type integral inequalities,and some new generalized Volterra-Fredholm discrete fractional inequalities,etc.In this paper,we mainly study the Volterra-Fredhlom dynamic integral inequalities on several time scales,extend the existing results and give some applications.The full text is divided into the following four chapters:The first chapter is the introduction,which introduces the background and current research progress of the problem studied in this paper.In Chapter 2,we give some basic symbols and preliminary results about the theory of time scales.In Chapter 3,a class of linear Volterra-Fredhlom type dynamic integral inequalities on time scales is studied.In Chapter 4,we consider a class of nonlinear Volterra-Fredhlom type dynamic integral inequalities on time scales.