Study On The Qualitative Behavior Of Several Kinds Neutral Delay Difference Equations

The theory of qualitative behavior of difference equations,which is the important consititute part of the theory of difference equations. With the development of science and technology, the research of the theory, nowadays, has already become a more active research field. Moreover, the differevce equation has extensively been applied to many important problems being studied and handled by the natural science and the social science field, such as control theory, modern physics, medical science, economics, bio-science, and so on. A lot of mathematical models, described by difference equations, have been put forward in the field of scientific research and social practice. The key content of qualitative difference equations is oscillation theory, asymptotic theory and existence of positive solution. Therefore, doing deeply research on it is not only of great theoretical significance but also practial value as well.This paper is devoted to the problem of qualitative behavior of even order and high order difference equations with positive and negative coefficientsn and second order of delayed difference equations with maxima. The relative topics extend the problems studied in the literatures and give separately some sufficient conditions for oscillation, asymptotic behavior of solutions and existence of positive solutions.Firstly, the background and applications of difference equations were introduced, and the achievements of this field were summarized.Secondly, by using of Banach contracting mapping principle, the difference equations with positive and negative coefficients were studied, and the existence of positive solutions of even order and high order difference equation with positive and negative coefficientsn was given.Finally, as to the difference equations with maxima, the asymptotic and oscillation behavior of second order equations and several sufficient conditions were given.