| The integral inequality is a useful tool when we study the stab theory ofdifferential equation, especially when we study the stability, estimate of solutionand the boundedness of differential equation. In past few years, many scholarstake on the research of this field, they have achieved many good results(see [1]-[12]).The oscillation theory of difference equation is one of important branch of difference equations. In the field of modern applied mathematics, it has made considerable headway in recent years, because all the structures of its emergence have deep physical background and realistic mathematical models. With the increasingdevelopment of science and technology, there are many problems relating to difference equation derived from lots of real applications and practice, such as whether difference equation has a oscillating solution or not, and whether all of its solutions are oscillatory or not. In very resent years, great changes of this field have taken place. Especially, the second order nonlinear difference equation has been paid more attentions and investigated in various classes by using different methods(see [13]-[36]).The thesis is divided into four sections according to contents.In Chapter 1, Preface, we introduce the main contents of this paper.In Chapter 2, we study the retarded integral inequalityandwhere m＞n＞0. Meanwhile, we extends the main results of Zhao [3] and Xu In Chapter 3, we investigate the oscillation criteria of second-order nonlinear neutral delay difference equation,we mainly employed a generalized Riccati transformation shall further the investigationand improve the main results of Sun [18], we obtained several new oscillation criteria at the end of this section.In Chapter 4, we study the exsitence of nonoscillatory solutions of second order nonlinear neutral delay difference equations,we obtain some sufficient conditions for the existence of nonoscillatory solutions of Eq (4.1.1).Meanwhile, we extends the main results of Cheng [30].
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