Oscillation Of Higher Order Dynamic Equations With Deviating Arguments On Time Scale

With the development of science and technology,the study of dynamic equation on time scale has developed rapidly and become an important research field.The study of oscillation of dynamic equation on time scale has extensive theoretical significance and great research value.It has been widely studied and concerned by scholars at home and abroad.This is not only the requirement of its own theoretical development,but also the practical problem,especially in the fields of physics,mechanics,chemical engineering,communication and control engineering etc.In this paper,we mainly study the oscillation of high-order dynamic equation with deviating arguments on time scale.We mainly study the oscillation of delay dynamic equation,advanced dynamic equation and mixed dynamic equation,and obtain some oscillation criteria of the equation.In Chapter 1,we briefly introduce the research background and development status of oscillatory theory for high-order dynamic equations with deviating arguments on time scale.In Chapter 2,the oscillation of second-order nonlinear neutral delay dynamic equation is studied.In Section 2.1,we consider the oscillation of second-order dynamic equation with non-linear neutral term on time scale.In Section 2.2,we study the oscillation of Emden-Fowler type non-linear neutral delay dynamic equation on time scale.Using Riccati transformation and inequality techniques,we obtain some new oscillation and asymptotic properties of the equation.In Chapter 3,we consider the oscillation of third-order delay dynamic equation with damping term on time scale.According to the oscillation of second-order dynamic equation without damping on time scale,a new characterization of the oscillation of the third order dynamic equation is given.We also study the oscillation of dynamic equation by Riccati and integral mean method.In Chapter 4,we consider the oscillation of the advanced dynamic equation and give the oscillation criteria of the second-order neutral dynamic equation with advanced arguments on time scale.Based on the new comparison theorem,some new oscillation results of the equation are given,so that we can reduce the oscillation problem of the second-order equationto that of the first-order one.In Chapter 5,the oscillation of the mixed dynamic equation is studied.In Section 5.1,we study the oscillation of the third-order nonlinear damped dynamic equation with mixed arguments on time scale.Some new criteria for the oscillation of the equation are given by using Riccati transformation,integral averaging technique and comparison theorem.In Section 5.2,we study the oscillation of the second-order neutral dynamic equations with deviating arguments on time scale.By using Riccati transformation and inequality technique,we give a new criterion of the equation.Many known results of oscillation of second order dynamic equation are generalized and improved.In Chapter 6,we summary the research contents of the full text,analyse the existing problems,and look forward to the direction worthy of further study.