Oscillation And Nonoscillation Of Seconed-Order Delay Differential Equations On Time Scales

The theory of time scales, which was established on the basis of the theory of measure chains, has received a lot of attention since Stefan Hilger introduced it in his Ph.D. in 1988. But few studies consider the oscillation of the second differential equatioans on time scales. In this article, we mainly discuss the oscillation, existence and classification of nonoscillatory solutions of seconed-order differential equations on time scales. The article is composed of four chapters. Chapter one, we give a short introduction to the time scales. Chapter two, based on the results of oscillation theory for differential equations and difference equations, we consider the oscillation and nonoscillation of a second-order differential equation on time scales. Furthermore, we apply comparison theorem, some of the oscillatory and nonoscillatory results for the second-order delay (or advanced) dynamic equation on time scales. Chapter three, we consider the second-order half-linear differential equation on time scales, and obtain some sufficient conditions for every solution of the equations which is oscillatory. Chapter four, we mainly study asymptotic of a class of second-order neutral differential equation on time scales. All solutions are classed into four types by means of their asymptotic behavior. Some necessary and/or sufficient conditions are obtained for such equations to possess a solution of each of the four types.