Study On The Oscillation Asymptotic Behavior And The Existence Of Positive Solutions For Difference Equations

In recent years, with the development of management science, delayed network system, biological science, computer science, control theory, modern physics, ect, People have put forward a lot of mathematical models that described by difference equations in scientific research and social practice. The approximate solution of the difference equation can be very convenient asked out by computer, so as to meet the needs of numerical analogue, differential equations are often transformed into difference equations to ask for the numerical solution.qualitative analysis of the difference equations with delayed has attracted the attention of large number of scholars. The oscillation theory, asymptotic theory and existence of positive solution being the key content of qualitative study of the difference equations, it is of great theoretical significance and practial value to research the oscillatory properties, asymptotic behavior and existence of positive solution.This paper is devoted to the problems of qualitative behavior of a second order and third order of difference equations with maxima, first order and third order difference equations with multiply delay and a kind of forced difference equation. Some results in the previous papers are expanded and improved.Firstly, we study the oscillation of second order difference equation with maxima.on this basic, we study the asymptotic and oscillation behavior of third order equation with maxima.Then ,we study the existences of positive solutions of first order difference equation with multiply delay, Meanwhile, by using of Banach contracting mapping principle, we study the existence of positive solutions of third order difference equation with multiply delay.Finally, In the last paper of this thesis, by using an important inequality, we establish some criterias for the oscillation of forced difference equation, By using formation of functions and reduction to absurdity, one sufficient conditions are obtained, we also generalize it to higher order.