| 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 oscillations. In addition, we briefly introduce researches and innovations of this paper.In the second part of this paper, we study the oscillation of the second-order linear neutral differential equation with variable sign oscillation factor. Its oscillation factor promotes from the original invariable sign to the variable sign, and this does not have in the former literature. This part mainly uses some theories and the method of the mathematical analysis,the ordinary differential equation,the real variable function,the functional analysis and non-linear functional analysis, and union the method of qualitatively and the stability theory of the differential equation and the oscillation research about the second-order neutral differential equation with variable sign oscillation factor. Then we obtain four sufficient conditions for the oscillation of this equation. The results obtained in this chapter generalize the theories in Fuxilin and Yuyuanhong's paper and Wangqiru'sIn the third part of this paper, we deeply study the oscillation of the second-order nonlinear neutral differential equation with variable sign oscillation factor. Its oscillation factor also promotes from the original invariable sign to the variable sign, and this does not have in the former literature. We also obtain four sufficient conditions for the oscillation of this equation. Opposite to the second chapter, the differential equation that this chapter studies is nonlinear. Of course it had the universality, and the obtained results of this chapter promote the related conclusion of the second chapter.The fourth part is the brief of this paper. It gives not only the brief of the paper, but also the aim of research in the future.
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