Research For The Oscillation Of Differential Equations Of Euler Type With Unbounded Delays

In this thesis of Master, we investigate some delay and neutral differential equations of Euler type with unbounded delays. Some oscillation ciriteria are established. It consists of four chapters.As the introductions, in Chapter 1, we briefly address the main work of this paper and the background.In the chapter 2, we investigates the following delay differentail equation in Euler type with positive and negative coeffiecientswhere p_i＞, q_i≥0,α_i,β_i∈(0, 1), i=1, 2,..., n. Some sufficient conditions for the oscillation of all solutions of Eq. (0.1) are obtained by use of analytic technique. Our results improve the relavent ones in the literature.Chapter three deals with the first order neutral differential equation of Euler type with unbounded delayswhere 0≤c＜1, p_i＞0,α,β_i∈(0, 1), i=1.2,...,n. For n=1, we introduce its characteristic equation and establish the necessary and sufficient condition for the oscillation of all sulotions of Eq. (0.2), moreover, an explicit oscillation result is presented. And for n＞1, some sufficient conditions for the oscillation of all solutions are setablished. The relavent results in the publication are improved.In the fourth chapter, we investigate the following neutral differential equation with positive and negative coefficientswhere P, Q, R∈C([t₀,∞), R⁺),α,γ∈(0, 1),β∈(0, 1]. Some oscillation criteria, including "Hille" type oscillation criterion, are obtained by means of a new technique.