Studies On Oscillation Of Second-order Differential Equations

With the rapid development of science and technology, all sorts of nonlinear problems have derived from mathematics, physics, chemistry, biology and so on. On the one hand, many nonlinear problems come forth all kinds of applied subject which attracts many scholars to study them.On the other hand, great changes of nonlinear differential equations have taken place in several decade years. It's powerful and fruitful theoretical tools and advanced methods have become ripe step by step.The theory of normal differential equation is one of important branches of differential equations. In the field of modern applied mathematics, it has made considerable headway in recent years, because all the structure of its emergence has deep physical background and realistic mathematical model.The thesis is divided into four sections according to contents.In Chapter 1, Preface.we introduce the main contents of this paper.In Chapter 2, we will establish some new oscillation criteria for second-order quasi-linear differential equation with forced term(r(t)|y'(t)|^α-1y'(t))'+p(t)|y'(t)|^α-1y'(t)+q(t)|y(t)|^β-1y(t)=e(t).The results obtained essentially improve and extend those of Wan-Tong Li [Interval Oscillation criteria for second-order quasi-linear nonhomogeneous differential equations with damping, Appl. Math. Comput. 147(2004)753-763].In chapter 3, we study the oscillation of second-order nonlinear delay differential equations (r(t)Ψ(x)Φ(x'))'+p(t)Φ(x')+ q(t)f(t) = e(t)with Riccati techniques and interval averagence .The results obtained essentially improve and extend some theorems in reference [21-28].At the last chapter, we establish some new method to charge oscillation of neutral second-order nonlinear differential equation [r(t)(y(t)+p(t)y(σ(t)))']' + Σ_iⁿ=1 q_i(t)f_i(y(Τ_i(t))) =0,t≥t₀.Those methods are difference to others as we know before which are more simple.