Study On The Oscillation Problem Of Solutions Of Second Order Quasilinear Delayed Dynamic Equations On Two Spaces

Quasi-linear second-order differential equation is a second order differential equation for the development of the past, and is based on second-order equations rich evolved in recent decades, the research results of second order differential equations of the theory in a number of areas development has been widely applied, especially with regard to the second order vibrations quasilinear ordinary differential equation and its solution of Of theories guiding role in the practical application of speci?c applied to physics, chemistry,polymer rheology, power system control, use more of the factors taken into account the actual situation of the proposed second-order linear equations to establish mathematical model, and thus better value.This is the method to study the Riccati transformation equations of second order quasilinear ordinary vibration problems in the theory of time-scale space of solutions in use on the basis of many of the older studies, studied two types of solutions of quasi-linear ordinary differential equations of vibration su?cient condition.This paper is divided into three chapters:In the ?rst chapter of this paper is to introduce the required time scale space of some basic concepts and some sophisticated basic theorems, and the literature [2] [3] [4] Lemma.In the second chapter, under the literature [6] [7] inspired to literature [6] of the second-order terms in the equation(r(t)x?(t))?into a positive odd’s power of α, using the generalized Riccati transformation study su?cient conditions for its second order quasi Oscillation of Solutions of linear equations.(r(t)|x?(t)|α-1x?(t))?+ q(t)f(x(τ(t))) = 0,(2.1.1)Where α is a positive odd business, τ is Delay time scale space conversion, r(t) and q(t)is a positive function, f : R → R, and for x ?= 0, f(x) ≥ Kxαsgnx, K is a positive real number.In the third chapter, on the basis of the second chapter, try adding damping the third chapter of the equation by damping term study added Riccati transformation(3.1.1)vibration resistance of the solution, the resulting equation(3.1.1) solution Oscillation su?cient condition. accordingly, get Kamenev type oscillation criteria for the inference of this chapter among the main conclusions will be applied to R, Z speci?c space h Z.(r(t)|x?(t)|α-1x?(t))?+ p(t)|x?(t)|α-1x?(t) + q(t)f(x(τ(t))) = 0,(3.1.1)Where α, is a positive odd providers,τ is a time lag transformation,and f ∈ C([R, R], R),and x ?= 0, f(x) ≥ Kxαsgnx, K is a positive real number.