Osclllation Of Second Order Nonlinear Differential Equations

Oscillation is an important branch of qualitative theory of differential equation.it be arrived from x"(t) + q(t)x(t) - 0 by Sturm.Therefore,linear and nonlinear equation have a rapid development. Fite,Hartman,Wintner and Philos get much rich results by classical technigue of Riccati,method of functional average and method of integral average.In recent years,classical oscillation of ODE has been extended to functional diffcricntial equation , differience equation and partial diffcrientialequation. The organization of the paper is as follows.In chapter 1, we introduce a history of oscillation of differential equation.Chapter 2 introduce research methodsof oscillation of differiential equation.Chapter 3 includes three section,First,in section 1, we introduce many research of semi-linear differential equation.Section 2 and section 3 study two types of nonlinear differential equation:andIt be all generation of semi-linear DE. Let us consider following nonlinear equation(E₁) where [t₀, +∞] ,a(t) > 0, (?)(x) > 0 is continuous,α> 0 is a constant,(?)(t, s)∈C¹[t_x, +∞]×[t_x, +∞], ,partial derivative of (?)(t, s) to indepenment variable 1 and indepenment variable 2 define (?)1 and (?)2 .Theorem 1 Suppose that there exists a continuous function H : D= {(t, s)|t≥s≥t₀}→R such that: (H₁) t≥t₀,H(t,t)=0 ;(t.s)∈D,H(t,s) > 0,whereh(t, s) is a nonnegative continuous function on D, (H₃) (H₄)whereβ=1/αk^α(α/(α+1))^α+1,Then equation (E₁) is oscillatory.Corollary Let condition (H₄) in Theorem 1 be replaced byThen the conclusion Theorem 1 holds.Theorem 2 Suppose the conditions (H₁)(H₂)(H₃) in Theorem 1 holds,and exists a differientiable functionρ: [t₀, +∞]→(0, +∞)such thatρ'(t)≥0 ,andwhereβis same in theorem 1,Then equation (E₁) is oscillatory.Theorem 3 Suppose the conditions (H₁)(H₂)(H₃)in Theorem 1 holds,and(H₆)There exists a continuousφ(x) : [t₀, +∞] such that T≥t₀,and(H₉) (H₁₀)whereφ₊(s)=max{φ(s),0). Then equation (E₁) is oscillatory.Consider following partial differiential equation with p-Lapelace (E₂), wherex = (x₁, x₂…x_n),‖.‖is the usual Euclidean norm in Rⁿ andâ–½is the usual nabla operator.Define the setsThe function q(x) is assumed to be intcrgrabl on every compact subsetΩ(1) .Thefunction f(u) is continuous and differentiable and satisfies f'(u)/(f(u))^q≥p-1>0,where1/p+1/q=1.we get following results:Lemma Letωbe the solution of(E₂) onΩ(a), The following statements are equivalent: (i)(ii)There exista a finite limit Q₀ such thatwhere(iii)Theorem 4 If eitheror Then equation (E₂) is oscillatory onΩ(a)(a > 1).Theorem 5 IfThen equation (E₂) is oscillatory onΩ(a)(a > 1).