Oscillation And Asymptotic Behavior Of Several Kinds Of Delay Differential Equations

In the study of differential equations theory,the qualitative properties is one of the most important problems.As part of qualitative research,oscillation and asymptotic behavior have been paid much attention.This paper studied the oscillation and asymptotic conditions of the second order,third order and high order neutral differential equation with delay under the canonical case.By using the Riccati-type functions,comparison principle,the integral mean value theorem,differential operator,the methods of the chain rule,we given some sufficient conditions of equations which is oscillation.Based on previous research and current research,we give some corollaries and estimates which is helpful for judgment or calculation.This paper is organized as follows.In the first chapter,the introduction part mainly introduces the development back-ground and related theoretical sources of delay differential equation.In the second chapter,the oscillation criterion of the second order delay differential equation with multiple time delays is studied under different restricted conditions.Four commonly used inequalities are given,which will pave the way for the subsequent proof of the oscillation conditions of the delay differential equations.We consider the more general constraints respectively,in the case of ??1 and in the case of ? bounded,and obtain the new oscillation criteria.In the third chapter,the oscillatory and asymptotic properties of the third order differ-ential equations with damped form are considered.By constructing the proper exponential function,the conclusions of the existing researches are extended,and the relevant inferences are given in special cases.In the fourth chapter,considered the oscillation of even order distributed delay differ-ential equation in the following form.A large number of oscillation results with parameter estimation are generalized by using the research process and related definitions in the third Chapter.