Study On Oscillation Of Several Kinds Of Delay Dynamic Equations On Time Scale

Oscillation is the characteristic of system motion.Oscillation theory is an important part of qualitative theory of dynamic equations.With the rapid development of scientific research and production practice,the calculus theory on time scales provides strong support for study on oscillation of equations,and oscillation of delay dynamic equations has been widely concerned.In the background of calculus theory on time scales,oscillation of two classes of second order and a class of third order delay dynamic equations on time scales is discussed by Riccati transformation,H function,the deformation of Young inequality and related lemmas,and the results of the existing literature are extended and improved.The main contents of the paper are summarized as follows:Firstly,on the basis of the existing literature,oscillation of a class of second order nonlinear delay dynamic equations is studied by expanding the range of the parameter,and four oscillatory theorems of such equations are obtained.Then four criteria are proved by the calculus theory,Riccati transformation,the deformation of Young inequality and the related lemmas on time scales,and some examples are given to verify the related results.It improves and generalizes the results of existing literatures,and enriches the oscillation research of second order nonlinear delay dynamic equations and corresponding differential and difference equations.Secondly,second order delay dynamic equations in the existing literature are popularized,the parameteris added,and a new class of second order nonlinear neutral delay dynamic equations with damping term is obtained.Oscillation of such equations is analyzed and studied,and Phlios type oscillation criteria for such equations and other oscillatory theorems and corollaries are obtained.Relevant theorems are proved by Riccati transformation,the deformation of Young inequality and H function.The concrete examples are given to verify the main results which enrich the study of oscillation of second order nonlinear delay dynamic equations with damping term.Finally,the parameteris added to some equations of the existing literature,the range of parameteris extended,and a new class of third order neutral nonlinear delay dynamic equations is obtained.Oscillation of these equations is studied,and three theorems and four corollaries of the oscillation of such equations are obtained.Moreover,three oscillatory theorems are proved by using related lemmas,the deformation of Young inequality and Riccati transformation.The main conclusions are verified by examples.When the parameters in the equation are taken for a specific value,those equations in other literatures can be obtained.The obtained results are applicable to these equations,and oscillation of third order neutral delay dynamic equations is enriched.