Oscillatory Behavior For Several Classes Of Second Order Delay 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 theorem is an important investigation area of dynamic equations. In this paper, oscillation of solutions of several classes of second-order delay dynamic equations with dimping on time scales is considered. The results given in this paper extend and improve the corresponding ones in the literatures.In Chapter one, we introduce the fundamental concepts and theories of dynamic equations on time scales.In Chapter two, we study the oscillatory behavior of all solutions of second order nonliear delay dynamic equation under some conditions, we establish five new oscillatory theories.In Chapter three, we deal with the oscillatory behavior of all solutions of second order nonliear delay dynamic equations with damping under some conditions, and consider both the cases and we present two new oscillatory theories and two conditions that each solution of the equation is either oscillatory or converges to zero.In Chapter four, we consider the oscillatory behavior of all solutions of second order half-liear delay dynamic equations with damping under some conditions, by using the generalized Riccati transformation and the inequality technique, we establish four new oscillatory criteria for second-order half-linear delay dynamic equations with damping on time scales. The results extend and improve some known theorems, but also unify the oscillation of second-order nonlinear differential equations and second-order nonlinear difference equations with damping on time scales.