Ordinary Differential Equation Boundedness And Oscillation

The theory of boundedness 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 boundedness of solutions of differential equations first arise in the fields of biology, ecology, physiology, physics, neural network and so on. It is also one of important areas in the study.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 paper is divided into four chapters according to contents.The first chapter is a preface.In the second chapter, we consider the n-dismensional nonautonomous system of differential equationdx/dt = f(t,x), (2.1.1)where f(t, x) ∈ C[J × Rⁿ, Rⁿ) and f(t, x) is a real continous funtion. This chapter make great improvement and generalization toward the paper [1] and [2] by means of improving the limitations of dV/dt .In the third chapter, we consider the oscillatory behavior of second-order neutral delay differential equation[a(t)|(x(t) +p(t)x(t -Υ))'|^α-1(x(t)+P(t)x(t - Υ))']' + q(t)f(x(t - σ))g(x'(t)) = 0,this chapter is divided into two sections. In the first section we employ interval oscillation criteria of this second-order differential equations. In the second sec- tion we consider the behavior of solution of this equation. The obtained results generalized and improved some known oscillation criteria.In the fourth chapter, we consider the oscillatory behavior of higher order non-linear differential equationWe obtained some new results.