Nonoscillatory Solutions Of Certain Neutral Difference Equation And Oscillation Of Certain Delay Differential Equation On Time Scales

This thesis is composed of two parts.The first part is introduction, in which the backgrounds of this subject and development of studies are summarized simply. To begin with, we introduce the development of oscillation theory of ordinary differential equations and functional differential equations briefly. Secondly, we also introduce the development of oscillatory theory of difference equations. Finally, we introduce the development of theory of time scales which unify continuous and discrete analysis. On the other hand, the main study of this thesis is introduced simply.The second part is the main body of the article. It is divided into two chapters, which study the nonoscillatory solutions for certain difference equation and the oscillation of certain delay differentional equation on time scales respectively.The Chapter one debates nonoscillatory solutions of certain first order neutral difference equation. To begin with, some basic concepts from the theory of difference equations are introduced. Secondly, we will give the classification of nonoscillatory solutions of the equation according to their asymptotic behaviors as t-. In the end, sufficient conditions are obtained for the existence of nonoscillatory solutions of the equation appearing in the classification by the fixed point theorem.In Chapter two,oscillation of the first order delay differential eqution on time scales is considered. To begin with, some basic concepts from the theory of the time scales are introduced. Secondly, we will give the proofs of some lemmas. Finally, we obtain some new oscillation criteria.