On Some Dynamic inequalities on time scales

Abstract

Fatma Mostafa Khamis Ahmed” On Some Dynamic Inequalities on Time Scales (Pure Mathematics). Arts, Science, and Education, Ain Shams University. The main purpose of thesis is to study ; The dynamic inequalities on time scales which contains most recent basic dynamic inequalities such as Young’s inequality, Jensen’s inequality, Holder’s inequality, Minkowski’s inequality, Stef-fensen’s inequality, Hilbert’s inequality, Hardy’s inequality, and Lyapunov’s in¬equality. This thesis is devoted to prove some new inequalities of type Opial’s on time scales. This thesis consists of five chapters and is organized as follows: Chapter 1: Preliminaries. We present some basic concepts, definitions, the basic inequalities and preliminary results of the calculus on a time scale T which are absolutely essential for completing the results and techniques used in subsequent chapters. Chapter 2: Opial Type Inequalities. This chapter focus on the previous work and existing research effort in the field of Opial’s inequalities. Some for¬mer famous opial’s inequalities in some text books that have been established by different researchers and some known results in some recent periodicals are given as well. We collect some of the applications of Opial’s inequality and its many generalized versions. Chapter 3: On Some Generalizations of Dynamic Opial-Type In-equalities. In this chapter, we will prove some new generalizations of dynamic