Researches About The Oscillations Of Several Classes Of Second-Order Delay Differential Equations

The content of this paper is composed of four chapters.In the first part of this paper, we mainly introduce the background, the status of recent researches and the tendency of development of oscillation theory for delay differential equations. We also introduce some basic notions on oscillation. In addition, we briefly introduce the research content, the structure of this paper and the research conclusion.In the second part of this paper, we firstly consider the second-order nonlinear neutral delay differential equation with invariable sign oscillation factorWhere m is nonnegative integer, r(t)∈C([t₀,∞), (0,∞)), p(t), q(t) e C([t₀,∞), [0,+∞)), p(t) < 1, f∈C(R, R), uf(u) > 0 (u≠0).σandÏ„are nonnegative constant. And we study the oscillation of this equation and obtain three sufficient conditions for this equation.Next we give another class of second-order nonlinear neutral delay differential equation with invariable sign oscillation factorWhere m and n are nonnegative integer andis a reduced fraction,andÏ„are nonnegative constant.In the same time, we study the oscillation of this equation and obtain three sufficient conditions for the oscillation of this equation. The equation in this chapter including the equation in Liukaien' s paper and the results obtained in this chapter generalize the theories in Liukaien' s paper.In the third part of this paper, we study a class of second-order nonlinear delay differential equation with variable sign oscillation factor Where p(t)∈ C([t₀,∞),R), f ∈ C(R,R), uf(u)>0(u≠0),Ï„ is nonnegative constant.Then we obtain five sufficient conditions for the oscillation of this equation. The results obtained in this chapter generalize the theories in Fuxilin and Yuyuanhong' s paper, Liukaien' s paper and Wangqiru' s paper.In the forth part of this paper, we deeply study the more general second-order nonlinear variable delay differential equation with variable sign oscillation factorWhere p(t),g(t)∈ C([t₀,∞),R), f ∈ C(R,R), uf(u) > 0 (u ≠ 0). And we acquire five sufficient conditions for the oscillation of this equation.