Opial type integral inequalities

Integral inequalities have been frequently employed in the theory of functional analysis, differential equations and applied sciences such as probability and statistics. There are a lot of types of inequalities such as Hermite-Hadamard type inequalities, Lyapunov type inequalities, Wirtinger type inequalities and Qi type inequalities. They have been the center of attention in many papers. Especially, Opial inequalities have been studied by many authors. In this thesis, we are concerned with Opial type integral inequalities. We present some theorems on generalizations of Opial inequality and Opial type inequalities on time scale. We survey the literature on those type inequalities. The organization of this thesis is as follows: In Chapter 1, we give a brief overview of what this thesis is about and also introduce the necessary inequalities and some definition. In Chapter 2, we investigate results on generalizations of Opial inequality. In Chapter 3, we consider discrete Opial inequalities. In Chapter 4, we present Opial dynamic inequalities on time scales. In Chapter 5, we give some general weighted Opial type inequalities on time scales. In Chapter 6, we establish some Opial type inequalities on time scales via the notion of the diamond-alpha derivative. In Chapter 7, we consider dynamic Opial type diamond-alpha inequalities in time scales.