Oscillation For Some Dynamic Equations On Time Scales

The theory of dynamic equations on time scales is a new branch of mathematics which appears as a natural description of evolution phenomena of several real world problems. Oscillatory properties is an important investigation area of dynamic equations. In this paper, oscillation of solutions of several classes of delay dynamic equations is considered. The results given in this paper extend and improve the corresponding ones in the literatures.In Chapter 1, we introduce the fundamental concepts and theory of time scales and dynamic equations.In Chapter 2, we study the first order linear delay dynamic equation It is proved that the existence of positive solution of the equation is equivalent to the existence of solution of the corresponding generalized characteristic equation. Furthermore, some sufficient conditions are obtained respectively for oscillation of all solutions of the equation or existence of non-oscillatory solution.In Chapter 3, firstly we consider the first order nonlinear neutral delay dynamic equation with variable sign coefficients By using appropriate transformation and auxiliary functions some sufficient conditions for oscillation of all solutions of the equation are obtained.Secondly we consider the second order nonlinear neutral delay dynamic equation Sufficient conditions for oscillation of all solutions of the equation or the existence of non-oscillatory solution of the equation are obtained respectively.Finally we consider the half-linear neutral delay dynamic equation A comparison result for the oscillation between the equation and a first order delay dynamical inequality is established. Furthermore some sufficient conditions for oscillation of all solutions of the equation are obtained.Chapter 4 is concerned with the second order nonlinear delay dynamic equations and By using general Riccati transformation and the results of dynamic inequalities, some sufficient conditions for oscillation of y~â–³(t) are obtained, where y(t) is the solution of the equations.Chapter 5 is concerned with the second order half-linear delay dynamic equation By using the different general Riccati transformation and the inequality some sufficient conditions for oscillation of all solutions of the equation are obtained.