On the Oscillation and Non-oscillation of Solutions of Linear and Nonlinear Differential Equations.

Abstract

In recent years the study of the oscillatory behavior of solutions of second- order differential equations has received a lot of attention in the literature due to its importance in different applications in physics, mathematics, engineering and biology, etc. This thesis deals with the oscillatory behavior of different classes of differential equations. Also, it discusses the oscillation of impulsive differential equations due to its importance in dynamical models in the case of the discontinuous movements. This thesis is devoted to establish necessary and sufficient of oscillation and non-oscillation of linear and non-linear differential equations. This thesis consists of six chapters organized as follows: Chapter 1 In this chapter we outline some basic concepts of oscillation and non-oscillation of differential equations. Basic notations and definitions of second order linear and non-linear differential equations are given. Also, we present some preliminary results of a class of first order neutral differential equations. Chapter 2 In this chapter, we discuss the oscillatory behavior of solutions of general classes of second order forced nonlinear differential equations with and without damping effect. The results are proved by using Riccati transformation, integral average technique and Holder inequality. Also, examples are given to illustrate our results. Chapter 3 In this chapter, we establish the interval oscillatory behavior of solutions of more general classes of second order nonlinear forced differential equations.