Some Nonlinear Volterra - Fredholm Type Dynamic Integral Inequalities On Time Scales

The study of theory of dynamic equations on time scales, which goes back to its founder Hilger[1], is a new area of mathematics that has received a lot of attention.For example, we refer the reader to the monographs[2,3], and the references cited therein. During the last few years, some integral inequalities used in dynamic equations on time scales have been extended by many authors[4-12,18-22,24,25]. Recently in[13,14], Pachpatte has established the useful linear Volterra-Fredholm type integral inequalities and discrete inequalities. Ma[15-17] has developed the useful nonlinear Volterra-Fredholm type integral inequalities and discrete inequalities. Very recently in[23], Meng has extended the linear Volterra-Fredholm type inequalities into dynamic integral inequalities on time scales.However, it seems to us that is very little known results about some power nonlinear Volterra-Fredholm type dynamic integral inequalities on time scales.The aim of this paper is to give some explicit bounds to some new power nonlinear Volterra-Fredholm type dynamic integral inequalities on time scales, which can be used as handy and effective tools in the study of Volterra-Fredholm type dynamic equations on time scales.The thesis is divided into four sections according to contents.Chapter 1 Preference, we introduce the overall background and the calculus on time scales of this paper.Chapter 2 In this chapter, some new explicit bounds on solutions to a class of new nonlinear Volterra-Fredholm type dynamic integral inequalities on time scales are established, which can be used as effective tools in the study of certain dynamic equations. Some applications for dynamic equations are also indicated.Chapter 3 In this chapter, some new explicit bounds on solutions to a class of new nonlinear Volterra-Fredholm type dynamic integral inequalities with two variables on time scales are established, which can be used as effective tools in the study of certain dynamic equations. Some applications for dynamic equations are also indicated.Chapter 4 In this chapter, some explicit bounds on solutions to a class of new power nonlinear Volterra-Fredholm type dynamic integral inequalities on time scales are established, which can be used as effective tools in the study of certain dynamic equations. Application examples are also given.