| Recently, the study of dynamic equations on time scales is an area of mathematics that has received a lot of attention. A time scale is an arbitrary nonempty closed subset of real numbers . Thus R, N, N+, hN := {hk\k Z,h > 0}, qZ := {qk|k 6 Z,q > 1} {0} are examples of time scales. The theory of time scales was introduced by Stefan Hilger in his PhD thesis ([1]) in 1988 (supervised by Bernd Aulbach), it has been created in order to unify continuous and discrete analysis. Hilger found many results concerning differential equations had carried over quite easily to corresponding results for difference equations. Then he defined the dynamic equation on time scales, i.e. the domain of the unknown function of the dynamic equations is a time scale. By choosing the tune scale to be the set of real numbers, it yields a differential equation, and by choosing the time scale to be the set of integers, it yields a difference equation. So one dynamic equation on time scale can probably be either differential equation or difference equation. However, since there are many other time scales that just the set of real numbers or the set of integers, we can study the dynamic equation on all kinds of time scales at the same time. Then our work can be greatly reduced, which is difficult to be done in the past.What especially interests us is the real sense of the dynamic equations on tune scales. For example, it can model a land of insect populations. These insects continuously grow at a certain rate during the months of April until September, while at the beginning of October, they suddenly die, but their eggs dormant and then hatch and start growing at a certain rate next April. So there are both continuous and discrete in the progress ( Id fact, we always meet such a progress with both continuous and discrete variation during studying the biological model). Then we can model the progress by a dynamic equation on tune scales to solve it easily.Since the last decade, there have been much advance on the dynamic equations on time scales. Many foreign mathematicians have respectively devoted to researching the Hamiltonian systems, Boundary value problem and the oscillation, the asymptotic behavior, the stability of the dynamic equation on time scales, such as Erbe L, Bonnet M, Peterson A, Agarwal R, D sl O, Kaymakcalan B, Lakshmikantham V etc. In general, the research methods they used are just as the following: Firstly, we compare the method of differential equation and with the corresponding difference equation, then unite them, finally, extend the results to other time scales.The paper is formed by three chapters:In chapter 1, we introduced the preliminaries of dynamic equation on tune scales.In chapter 2, the oscillation of second order nonlinear dynamic equationholds, then all solutions of equation (2.1) oscillate, WhereTheorem 2.2 If there exist functions L2[t1, ), C[T,R+], and the following conditions hold,then all solutions of equation (2.1) oscillate. Where the definition of A(t), B(t) is the same as the definition in Theorem 2.1, and +(t) = max{ (t),0}. Finally, we concerned the equation with disturbing and damping and some good results were obtained. We offered a new research method for second order dynamic equation on time scales in chapter 2, which united, improved and extended the corresponding results of differential equations and difference equations in essence (see paper[17-22j).In chapter 3, the asymptotic behavior of the nonoscillation solutions of some nonlinear dynamic equation(A2) f C[T R, R] is continuously increasing with the second variate and yf(t, y) > 0, y 0;(A3) g is a strictly increasing derivative function on R and yg(y) > 0, y 0. Four sufficient and necessary conditions were obtained under the conditions ofand a > 1, Then equation has a nondecreasing in absolute value solution y(t), such that y(t) - M.R(t, t0), t - for some |M | > 1, if and only ifholds.Theorem 3.3 If (A1 - A4), (3.4) hold, then y(t) is a solution of (3.1) nonincreasing in...
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