الفهرس 
ContentsOnly 14 pages are availabe for public view
 |  | from | 95 |  |  |
| | | from | 95 | | |
Abstract
Since the dynamic inequalities on time scales unifies the study of continuous and discrete inequalities so it was natural and desirable to introduce reverse dynamic inequalities on time scales and use them to prove some higher integrability theorems for nonincreasing functions on time scales and then obtain the continuous and the discrete cases as special cases. The general idea is to prove a result for an inequality where the domain of the unknown function is a so-called time scale, which is an arbitrary nonempty closed subset of the real numbers R. This idea goes back to its founder Stefan Hilger which started the study of dynamic equations on time scales. The study of dynamic inequalities on time scales helps avoid proving results twice, once for differential inequality and once again for difference inequality. The main objective of this thesis is to study some inequalities of weighted Cesàro and Copson operators and deduce the generalization of Hardy’s type inequality, which we use to prove some reverse dynamic inequalities and apply them to prove some new higher integrability theorems. The thesis is divided into three chapters and is organized as follows : Chapter 1. This chapter contains some preliminaries, definitions, basic concepts concerning time scale calculus and basic dynamic inequalities. Next, in the rest of the chapter, we present some related results of discrete and integral inequalities of Hardy’s type that serve and motivate the contents of the next chapters. Chapter 2. In this chapter, we establish some factorization theorems for weighted Cesàro and Copson spaces, obtain two sided norm dynamic inequalities, and give conditions for the boundedness of the Hardy and Copson dynamic operators on the weighted space. We obtain, as special cases, the classical integral inequalities and the discrete inequalities. The results in this chapter help in proving generalization of Hardy’s type inequality, which can be used in the proof of some results of reverse Hardy’s inequality. The results in this chapter are published in The Bulletin of the Brazilian Mathematical Society. Chapter 3. In this chapter, we prove some reverse dynamic inequalities and apply them to prove some new higher integrability theorems for nonincreasing functions on time scales. As a special case, we will derive the classical integral inequalities due to Alzer and formulate the discrete versions of these inequalities which are essentially new.