Quadratic Integrability And Boundedness Of The Solutions For Second Order Delay Differential Equation

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 acquired some new criteria about the boundedness and quadratic integrability for second-order nonlinear differential equation .The results obtained essentially improve and extend those of some other papers.In chapter 3, we study the nonlinear differential equations about its the boundedness and quadratic integrability .(r(t)x'(t))'+p(t)p₁(x'(t)) + q₁(t)x(t) + q₂(t)q₃(x(t -Î¥)) = f(t,x). (1.1) some sufficient conditions for the boundedness and quadratic integrability of(1) are obtained .At the last chapter, some sufficient conditions are are given about the oscillation of second order liner matrix differential equation system MY" + f(t)Y' + Q(t)Y = 0. The results can be regarded as extension of results of kamenev in the Sealar Case .