Oscillation, Classification And Existence Of Nonoscillatory Solutions Of Second-Order Nonlinear Dynamical Equations On Time Scales

Theory of time scales ,which was established on the basis of the theory of measure chains ,has received a lot of attention since Stafen Hilger[l] introduced it in his Ph.D.in 1988.In this area,Bohner,A.Peterson,Agarwal etc. have done some important work.But few studies consider the oscillation of the second order nonlinear dynamical equations on time scales .In this article ,we mainly discuss the oscillation and the existence of the nonoscillatory solutions of second-order nonlinear differential equation on time scales.The article composes of four chapters .In chapter two ,we give a introduction to the time scales and introduce the preliminaries of dynamic equations on time scales. Chapter three and Chapter four are the mainly results of this article.and we apply the behavior of strongly suplinear and strongly sublinear of a function.In chapter three ,we get some sufficicent and necessary conditions for all the oscillatory solutions of second-order nonlinear dynamical equation on time scales,and give proofs of them.We get four theorems in this chapter ,namely we get sufficient and necessary conditions for all the oscillatory solutions of equation (r(t)x~△(t))~△ + f(t, x(g1(t)), …, x(gm(t))) = 0 (E), ∫_α~∞ 1/(p(s))△s =∞ or ∫_α~∞ 1/(p(s)) < ∞ and f(t, x(g1(t)), …, x(gm(t))) are strongly suplinear and strongly sublinear separately herein. Furthermore,our results unify and improve some known results of neutral functional differential equations and delay difference equations in the references. In chapter four,with the references,we study the existence and asymptotic behavior of the nonoscillatory solutions of second-order nonlinear dynamical equation (E) on time scales.and give the classification according to their asymptotic behavior,namely,the classification of all the nonoscillatory solutions of (E) when ∫_α~∞ 1/(p(s))△s = ∞ and ∫_α~∞ 1/(p(s)) △s < ∞ separately. Some sufficient and necessary conditions are obtained for all the solutions of second-order nonlinear dynamical equation (E) to be nonoscillatory.