Oscillation Of Second-order Half-linear Neutral Differential Equations

The oscillation of ordinary 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 mathematicalmodel. Many scholars take on the research of this field, they have achieved many good results . With the increasing development of science and technology , there are many problems relating to differential equation derived from lots of real applications and practice, such as whether differential equation has a oscillating solution or not, and whether all of its solutions are oscillatory or not. In very resent years, great changes of this field have taken place. Especially, the second order differential equation has been paid more attentions and investigated in various classes by using different methods(see [1]-[37]).The present paper employs a generalized Riccati technique, integral average ,the inequality of Yong and the monotone of functions to investigate the oscillation criteria for some class of half-linear netural differential equations, the results of which generalized and improved some known oscillation criteria.The thesis is divided into three sections according to contents.In chapter 1, Preface.we introduce the importance of the main contents of this paper.In chapter 2, This chapter is divided into four sections to investigate the oscillation criteria for some class of half-linear netural differential equations,We state the main results as follows:First,we are concerned with the second-order half-linear neutral differen-tial equation= 0, < > k, (2.1.1)where r,p,q € C([t0, oo), i?), /i € C1^ R), and a > 0 is a constant. Throughout this paper, We assume that (Hi) 00,r(t)>0;(H2) f.00 -±rdt = oo;(H3) h(t) < t, lim h(t) = oo.Definition 2.1.1 By a solution of (2.1.1), We mean a function y : [Ty, oo) >-R,Ty > t0 such that y(t) and r(t)\(y(t) +p(t)y(h(t)))'\a-1(y{t) + p(t)y(h(t)))' are continuously differentiable and satisfy equation (2.1.1) for t > Ty. We restrict our attention to the nontrivial solutions y(t) of (2.1.1) only, i.e. to solution y(t) such that sup{|y(t)| : t > T} > 0 for all T > Ty.Definition 2.1.2 A nontrivial solution of (2.1.1) is called oscillatory if it has arbitrarily large zeros;otherwise, it is said to be nonscillatory. Equation (2.1.1) is called oscillatory if all its solutions are oscillatory.Theorem 2.1.1 Assume that for any T > t0, there exist a, b with T < a < b, and if there exist H G D(a, 6)and a positive function p G C1([io,oo))such thatW_ p[a+1 'H + Pwhere tp(t) = g(t)(l - p(t))a, then Eq.(2.1.1)is oscillatory.Second, We are here concerned with the oscillations behavior of solutions of the second-order quasi-linear neutral delay differential equationsAba(...