Oscillation And Asymptotic Behavior Research Of Some Classes Of Delay Dynamic Equations On Time Scales

Recently, with the development of science and technology, many models that are described by delay dynamic equations of the actual research. Therefore, delay dynamic equations have important significance to study.Oscillation theory of dynamic equations has developed quickly, including the oscillation of differential equations and the oscillation of the difference equations. The theory of analysis on measure chains, which has recently received lots of attention, was established by Hilger in 1988 in order to unify continuous and discrete analysis. The time scale is a special case of measure chains, the theory of dynamic equations on time scales establishes a unified theory for the differential equations and the difference equations.In this paper, the works focus on some oscillatory and asymptotic behavior for the dynamic equations on time scales, the results obtained in this paper extend and improve some results in the literature. The main results are described as follows:In chapter 1, we briefly summarize background and development of the dynamic equations. Some calculus theory on time scales is introduced. The main results in this thesis are also introduced.In chapter 2, we study the oscillation, nonoscillation and asymptotic behavior of second-order dynamic equations on time scales. In section 2.1, by means of Riccati transformation technique, we consider the oscillation of second-order nonlinear delay dynamic equations on time scales. The main results are published in Advances in Difference Equations which is reviewed by Science Citation Index, the results obtained extend and improve the results established by Hassan in [34]. In section 2.2, by defining a class of functions, we examine the oscillation of the second-order dynamic equations with forced term on time scales. The main results are published in Electronic Journal of Qualitative Theory of Differential Equations which is reviewed by Science Citation Index, the results improve the results given in Saker [84]. In section 2.3, we employ Kranoselskii's fixed point theorem to study the existence of nonoscillatory solutions for the second-order neutral dynamic equations with forced term. The main results are published in Advances in Difference Equations which is reviewed by Science Citation Index, the results obtained extend and improve the results established by Kulenovi? and Had?iomerspahi? in [92].In chapter 3, we study the oscillation and asymptotic behavior of third-order delay dynamic equations on time scales. In section 3.1, by means of Riccati transformation technique and inequalities, we investigate the oscillation and asymptotic behavior of third-order delay dynamic equations on time scales. The main results are accepted by Annales Polonici Mathematici which is reviewed by Science Citation Index, the results established improve the results given in Hassan [69]. In section 3.2, we consider the oscillation and asymptotic behavior of the third-order Emden-Fowler neutral dynamic equations on time scales. The main results are published in Advances in Difference Equations which is reviewed by Science Citation Index, the results solve a problem in [74]. In section 3.3, we study the oscillation and asymptotic behavior of third-order nonlinear delay dynamic equations on time scales. The main results are accepted by Bulletin of the Malaysian Mathematical Sciences Society which is reviewed by Science Citation Index, the results obtained improve the results given in Hassan [69].In chapter 4, we study the oscillation of second-order neutral differential equations. In section 4.1, by using the new transformation technique, we consider the oscillatory behavior of second-order linear neutral delay differential equations. The results are new, and the main results are published in Advances in Difference Equations which is reviewed by Science Citation Index. In section 4.2, by employing the new method, we consider the oscillatory behavior of second-order nonlinear neutral delay differential equations. The main results are published in Applied Mathematics and Computation which is reviewed by Science Citation Index, the results obtained enrich and correct the results given in [96].In chapter 5, the main research contents and main results are introduced and summarized, and the future research work is prospected.