Qualitative Study On Several Classes Of Differential And Difference Equations

The thesis of Master, which is composed of six chapters, mainly studies the oscillation of several classes of Differential equation and Difference equation, and obtains some sufficient conditions of the existence of oscillatory solutions by introducing analysis techniques, Riccati transformation, parameter and generalized integral averaging technique. Some of our results presented in this paper are original while others only extend and improve some known results in the literature.In chapter 1, we introduce the background of the problem and the recent development of the research in this field. In chapter 2, we discuss the forced oscillation of the second order nonlinear delay differential equations and obtain some general sufficient conditions for the oscillation of solutions by making use of analysis techniques and integral operator. Some of results generalize and improve some known results in the literature.Chapter 3 deals with the oscillation of the first order neutral differential equation of Euler type. The known literature provides sufficient condition for the oscillation of all solutions of equation in the case of 0 < c < 1 by means of characteristics equation. This thesis gives several conditions for the oscillation of equation according to the different range value of c in new ways and some examples of application.Chapter 4 discusses the oscillation of the second order nonlinear neutral delay differential equations. Some known results in the literature are generalized.Chapter 5 deals with global asymptotic stability for a kind of rational difference equations with higher order, solves the open problem of M.R.S.Kulenovic and GLadas, and moreover proves that there do not exist periodic solutions for this kind of equation; then it extends these results to a general case.In the sixth chapter, the oscillation of the second order nonlinear neutral difference equations, we investigate the oscillation of equation by using analysis techniques and Riccati transformation, and obtain some new results.