Studies On Oscillation For One Class Superlinear Second-order Differential Equations

The oscillation theory of differential equation is one of important branchof differential equations. In the field of modern applied mathematics, it hasmade considerable headway in recent years, because all the structures of itsemergence have deep physical background and realistic mathematical models.Many scholars take on the research of this field, they have achieved manygood results. With the increasing development of science and technology,there are many problems relating to differential equation derived from lotsof real applications and practice, such as whether differential equation has aoscillating 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, thesecond order superlinear differential equation has been paid more attentionsand investigated in various classes by using different methods(see [1]-[36]).The present paper employs a generalized Riccati transformation, Integralaverage technique and the monotone of functions to investigate the oscillationcriteria for some class of superlinear differential equations, the results of whichgeneralized and improved some known oscillation criteria.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 study the oscillation of the second-order nonlinear dampeddifferential equation, (r(t)k₁(x(t),x′(t)))′+p(t)k₂(x(t),x′(t))x′(t)+q(t)f(x(t))=0,t≥t₀,(2.1.1)we mainly employed a generalized Riccati transformation and Integral averagetechnique shallfurther the investigation and improve the main results of Zhao[8], we obtained several new oscillation criteria at the end of this section. In Chapter 3, the chapter is divided into two sections to investigate theoscillation criteria for some class the forced second-order nonlinear dampeddifferential equation,(r(t)k₁(x(t),x′(t)))′+p(t)k₂(x(t),x′(t))x′(t)+q(t)f(x(t))=e(t),t≥t₀,(3.1.1)First, we mainly employed average function H(t,s)∈C(D,R), in thestudy of oscillatory properties differential equation (3.1.1).Second, we employ a integral operator A_a^b of second-order nonlinear dif-ferential equation (3.1.1) in this section. We obtained several new oscillationcriteria at the end of this paper.In Chapter 4, we study the interval oscillation of second order nonlineardifferential equations with delayed argument,(4.1.1)In this chapter, our results aslo including that the order of the function f(x)don't influence the oscillation of equation (4.1.1). We shall further the inves-tigation and improve the main results of D. Cakmak and A. Tiryaki [22] andobtained several new oscillation criteria in the subinterval of [t₀,∞) at the endof this chapter.