Research Of Oscillation Properties Of Functional Dynamic Equations On Time Scales

In the field of science and control engineering, many problems are studied by using functional dynamic equations. Hence functional dynamic equations are important to be investigated systematically. This paper is concerned with some oscillation properties of some kinds of functional dynamic equations on time scales.Chapter 1 briefly summarizes the development of functional dynamic equations. Some calculus theory on time scales and the main results in this thesis are also included.Chapter 2 concerns the oscillatory behavior of second-order neutral functional dynamic equations on time scales. Firstly, by using inequalities technique and Riccati substitution technique, the oscillation of certain second-order nonlinear neutral delay dynamic equation on time scales is studied. Some oscillation criteria are presented. These results extend and improve those results given in the literature. Secondly, the oscillatory behavior of a class of second-order neutral functional dynamic equation with mixed arguments on time scales is examined. Some comparison theorems for the oscillation of the studied equation are obtained. Thirdly, by applying the Riccati transformation and some inequalities, the oscillation nature of a kind of second-order neutral functional dynamic equation of mixed type on time scales is considered. Some sufficient conditions which ensure that all solutions of studied equation are either oscillatory or tend to zero are given.Chapter 3 is concerned with the oscillation of higher-order functional dynamic equations on time scales. Firstly, by using inequalities and Riccati transformation technique, the oscillation of a class of third-order linear delay dynamic equation on time scales is studied. A comparison theorem which guarantees that all solutions of studied equation are either oscillatory or converge to zero is obtained. The main result improves those results in the literature. Secondly, the oscillatory property for some third-order nonlinear retarded dynamic equation on time scales is investigated. Several sufficient conditions which ensure that all solutions of studied equation are either oscillatory or tend to zero are given. These criteria improve those results given by the literature. Thirdly, by using inequalities and Riccati transformation technique, the oscillation of a kind of third-order Emden-Fowler delay dynamic equation of neutral type is considered. Some sufficient conditions which ensure that all solutions of studied equation are either oscillatory or tend to zero are given. Fourthly, by means of inequalities and Riccati transformation, the oscillatory behavior of a class of third-order neutral functional dynamic equation of mixed type on time scale is examined. Some new oscillation results are established. Fifthly, by employing calculus theory on time scales and some inequalities, the oscillation problem of a class of fourth-order Emden-Fowler delay dynamic equation on time scales is considered. Some new oscillation criteria are obtained. These criteria improve those results in the literature. Sixthly, by using inequalities, the oscillatory behavior of a kind of higher-order nonlinear functional dynamic equation on time scales is investigated. Some new criteria which ensure that all bounded solutions of studied equation are either oscillatory or tend to zero are established.Chapter 4 is concerned with the oscillation of neutral functional differential equations. Firstly, by using inequalities, the oscillatory behavior of a class of second-order Emden-Fowler neutral delay differential equation is considered. Some new oscillation criteria are presented. These results extend those results obtained in the literature. Secondly, by employing some inequalities, the oscillation of a kind of second-order neutral differential equation of mixed type is studied. Some new oscillation results are obtained. These criteria extend some results given in the literature. Thirdly, by employing a new Riccati transformation and some inequalities, the oscillatory nature of a class of higher-order neutral delay differential equation is examined. A new theorem which guarantees that all solutions of studied equation are either oscillatory or converge to zero is established.In chapter 5, the main results obtained in this note are summarized, some unsolved problems are given.