Oscillatory Research Of Third Order Neutral Dynamic Equations On Time Scales

With the advancement of science and technology,the mathematical model described by the time-delay dynamic equations on the time scale has a wide range of applications in engineering fields such as control engineering,physics,oceanography,optics,biological environment,and medicine.It has developed rapidly,so it has received extensive attention from mathematics researchers at home and abroad.In this paper,the vibration and asymptotic properties of several types of third-order delay dynamic equations on the time scale are investigated,some new vibration criteria for the equations studied are obtained,and some results in the existing literature are generalized and improved.The first chapter introduces the research background,the current research situation at home and abroad,and the main research content of this article on the time scale of the third-order delay dynamic equations.The second chapter studies the oscillation and asymptotic behavior of a class of third-order neutral delay dynamic equations on the time scale.Considering the case where the coefficient of the neutral term is positive,several new oscillation and asymptotic behaviors of this type of equation are established.Judging criteria,popularizing,improving and unifying the related results of such differential equations and difference equations,and giving specific examples to illustrate the effect of the main conclusions in this chapter.The third chapter considers the case where the coefficient of the neutral term of the equation studied in Chapter 2 is negative.Using Riccati transformation and inequality techniques,inspired by the existing literature,several new decision criteria are obtained and specific examples are given to carry out the results.Argument.The fourth chapter studies the oscillation of the third-order nonlinear neutral distributed delay dynamic equations on the time scale.Using the generalized Riccati transformation and inequality techniques,sufficient conditions are established to ensure that each solution of the equation vibrates or converges to zero.The results of the existing literature are generalized.The fifth chapter summarizes the research content of the full text,analyzes some existing problems,and looks forward to future research directions.